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New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings

  • Humaira Kalsoom , Muhammad Amer Latif , Saima Rashid , Dumitru Baleanu and Yu-Ming Chu EMAIL logo
Published/Copyright: December 31, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In the article, we present a new ( p , q ) -integral identity for the first-order ( p , q ) -differentiable functions and establish several new ( p , q ) -quantum error estimations for various integral inequalities via ( α , m ) -convexity. We also compare our results with the previously known results and provide two examples to show the superiority of our obtained results.

Keywords: (p,q)-quantum calculus; Hermite-Hadamard inequality; Simpson’s type inequality; (α,m)-convex functions
MSC 2010: 26D15; 26D10; 26A51

1 Introduction

Integral inequalities are considered a fabulous tool for constructing the qualitative and quantitative properties in the field of pure and applied mathematics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. A continuous growth of interest has been occurring to meet the requirements for the wide applications of these inequalities. These applications are closely related to the convex functions and have been studied by many researchers using various techniques [21,22,23,24,25,26,27,28,29,30,31,32].

Now, we recall the definition of convex function as follows.

Let K ⊆ ℝ be an interval. Then a real-valued function g : K → ℝ is said to be convex if the inequality

g ( λ ϕ + ( 1 − λ ) ψ ) ≤ λ g ( ϕ ) + ( 1 − λ ) g ( ψ )

holds for all ϕ , ψ ∈ K and λ ∈ [ 0 , 1 ] .

For convex functions, many inequalities have been established by many authors, for example, Jensen inequality [33], Ostrowski inequality [34], hypergeometric function inequality [35], elliptic integral inequalities [36,37,38,39,40,41] and so on. But the most celebrated and significant inequality is the Hermite-Hadamard inequality [42,43], which is stated as follows.

Let ϕ < ψ and g : [ ϕ , ψ ] → ℝ be a convex function. Then the double inequality

(1.1) g ϕ + ψ 2 ≤ 1 ψ − ϕ ∫ ϕ ψ g ( λ ) d λ ≤ g ( ϕ ) + g ( ψ ) 2

The Hermite-Hadamard inequality (1.1) has been extensively discussed because it is essential in developing a connection between the theory of convex functions and integral inequalities. A number of researchers have dedicated their efforts to extend, generalize and refine the Hermite-Hadamard inequality (1.1) for different classes of convex functions and mappings. Some recent results on inequality (1.1) can be found in the literature [44,45,46].

Let g : [ ϕ , ψ ] → ℝ be a four times continuous and differentiable mapping on the interval [ ϕ , ψ ] such that | | g (4) | | ∞ = sup z ∈ ϕ , ψ | g ( 4 ) ( z ) | < ∞ . Then Simpson’s inequality [47]

1 3 g ( ϕ ) + g ( ψ ) 2 + 2 g ϕ + ψ 2 − 1 ϕ − ψ ∫ ϕ ψ g ( z ) d z ≤ ψ − ϕ 4 2,880 ∥ g ( 4 ) ∥ ∞

holds.

Quantum calculus is the study of calculus without limits and is also known as q-calculus [48]. In q-calculus, we obtain the initial mathematical formulas as q approaches 1. The commencement of the analysis of q-calculus can be dated back to the era of Euler (1707–1783), who first initiated the q-calculus in the tracks of Newton’s work on infinite series. Subsequently, Jackson [49] launched the concept of q-integrals and studied it in a systematic way. The aforementioned results lead to an intensive investigation on q-calculus in the twentieth century. The idea of q-calculus is used in numerous areas in mathematics and physics, especially in orthogonal polynomials, number theory, hypergeometric functions, mechanics and relativity theory. The concept of q-derivatives over the definite interval [ ϕ , ψ ] of ℝ is introduced by Tariboon et al. [50,51], and they addressed several problems on quantum analogs such as Hölder inequality, Ostrowski inequality, Cauchy-Schwarz inequality, Grüss-Chebyshev inequality, Grüss inequality and other integral inequalities by classical convexity.

From the last few years, q-calculus has become an interesting topic for many researchers and several new results have been established in the literature [52,53,54,55,56,57,58]. Furthermore, Tunç and Göv [59,60] derived the notion of ( p , q ) -calculus on the intervals [ ϕ , ψ ] of ℝ , found the formulae for ( p , q ) -derivative and ( p , q ) -integral and established their several fundamental properties. The results that depend on ( p , q ) -calculus are the Minkowski inequality, Hölder inequality, Grüss and Grüss-Chebyshev inequality and many others. Kunt et al. [61] gave the generalized ( p , q ) -Hermite-Hadamard inequalities on the finite interval and some important results which are connected with ( p , q ) -midpoint-type inequality. Recently, ( p , q ) -calculus has been the subject of intensive research, and its refinements and generalizations can be found in the literature [61,62].

Now, we recall the definitions and theorems for ( p , q ) -derivative and ( p , q ) -integral.

Definition 1.1

[59] Let 0 < q < p ≤ 1 and g : [ ϕ , ψ ] → ℝ be a continuous function. Then the ( p , q ) -derivative of g at λ ∈ [ ϕ , ψ ] is defined by

D p , q ϕ g ( λ ) = g ( p λ + ( 1 − p ) ϕ ) − g ( q λ + ( 1 − q ) ϕ ) ( p − q ) ( λ − ϕ ) ( λ ≠ ϕ )

and

D p , q ϕ g ( ϕ ) = lim λ → ϕ D p , q g ( λ ) .

Example 1.2

Define the function g : [ ϕ , ψ ] → ℝ by g ( λ ) = 2 λ 2 + 1 with 0 < q < p ≤ 1 . Then for λ ≠ ϕ we have

D p , q ϕ ( 2 λ 2 + 1 ) = ( 2 ( p λ + ( 1 − p ) ϕ ) 2 + 1 ) − ( 2 ( q λ + ( 1 − q ) ϕ ) 2 + 1 ) ( p − q ) ( λ − ϕ ) = 2 [ 2 ] p , q λ 2 + 4 ϕ λ [ 1 − [ 2 ] p , q ] + 2 ϕ 2 [ [ 2 ] p , q − 2 ] ( λ − ϕ ) = 2 λ [ 2 ] p , q ( λ − ϕ ) − 2 ϕ [ 2 ] p , q ( λ − ϕ ) + 4 ϕ ( λ − ϕ ) ( λ − ϕ ) = 2 [ 2 ] p , q ( λ − ϕ ) + 4 ϕ .

Definition 1.3

[59] Let 0 < q < p ≤ 1 and g : [ ϕ , ψ ] → ℝ be a continuous function. Then the ( p , q ) -integral on [ ϕ , ψ ] is defined by

∫ ϕ λ g ( x ) ϕ d p , q x = ( p − q ) ( λ − ϕ ) ∑ n = 0 ∞ q n p n + 1 g q n p n + 1 λ + 1 − q n p n + 1 ϕ

for λ ∈ [ ϕ , ψ ] .

If c ∈ ( ϕ , λ ) , then the ( p , q ) -definite integral on [ c , λ ] can be expressed as

∫ c λ g ( x ) ϕ d p , q x = ∫ ϕ λ g ( x ) ϕ d p , q x − ∫ ϕ c g ( x ) ϕ d p , q x .

Example 1.4

Define the function g : [ ϕ , ψ ] → ℝ by g ( x ) = 4 x + 1 with 0 < q < p ≤ 1 . Then one has

∫ ϕ λ ( 4 x + 1 ) ϕ d p , q x = ( p − q ) ( λ − ϕ ) 4 ∑ n =0 ∞ q n p n + 1 q n p n + 1 λ + 1 − q n p n + 1 ϕ + ∑ n =0 ∞ q n p n + 1 = ( λ − ϕ ) [ 4 ( λ − ϕ ( 1 − p − q ) ) + [2] p , q ] [2] p , q .

Theorem 1.5

[61] Let 0 < q < p ≤ 1 and g : [ ϕ , ψ ] → ℝ be a convex differentiable function on [ ϕ , ψ ] . Then

g q ϕ + p ψ [ 2 ] p , q ≤ 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ q g ( ϕ ) + p g ( ψ ) [ 2 ] p , q .

The definition of ( α , m ) -convex function was presented by Miheşan in [63] and is stated as follows.

Definition 1.6

Let α , m ∈ ( 0 , 1 ] and ψ ∗ > 0 . Then the function g : [ 0 , ψ ∗ ] → ℝ is said to be ( α , m ) -convex if the inequality

g ( x λ + m ( 1 − λ ) y ) ≤ λ α g ( x ) + m ( 1 − λ α ) g ( y )

holds for all x , y ∈ [ 0 , ψ ∗ ] and λ ∈ [ 0 , 1 ] .

Zhang et al. [64] investigated some inequalities about q-differentiable convex and quasi-convex functions which are linked with the different types of inequalities in q-calculus.

Lemma 1.7

[64] Let 0 < q < 1 and g : [ ϕ , ψ ] → ℝ be a q-differentiable function on ( ϕ , ψ ) such that D q ϕ g is continuous and integrable on [ ϕ , ψ ] . Then

γ [ μ g ( ψ ) + ( 1 − μ ) g ( ϕ ) ] + ( 1 − γ ) g ( μ ψ + ( 1 − μ ) ϕ ) − 1 ψ − ϕ ∫ ϕ ψ g ( x ) ϕ d q x = ( ψ − ϕ ) ∫ 0 μ ( q λ + γ μ − γ ) ϕ D q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d q λ + ∫ μ 1 ( q λ + γ μ − 1 ) ϕ D q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d q λ

for all γ , μ ∈ [ 0 , 1 ] .

The main purpose of the article is to provide an identity, which is the generalization of an identity presented in Lemma 1.7, and establish the ( p , q ) -analogues of different types of integral inequalities via the ( p , q ) -differentiable ( α , m ) -convex functions. By using the new identity with distinct parameters we obtain some new ( p , q ) -quantum error estimations for different types of inequalities such as the midpoint-type, the Simpson-type, the average of midpoint-trapezoid-type and the trapezoid-type inequalities via ( α , m ) -convexity.

2 Auxiliary results

In order to obtain different types of integral inequalities through ( p , q ) -differentiable ( α , m ) -convex functions, we need several lemmas which we present in this section.

Lemma 2.1

Let 0 < q < p ≤ 1 and g : [ ϕ , ψ ] → ℝ be a ( p , q ) -differentiable function on ( ϕ , ψ ) such that D p , q ϕ g is continuous and integrable on [ ϕ , ψ ] . Then

γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + (1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p μ ( q λ + γ p μ − γ ) ϕ D p , q g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ + ∫ p μ 1 ( q λ + γ p μ − 1) ϕ D p , q g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ

for all γ , μ ∈ [ 0 , 1 ] .

Proof

By an identical transformation, we get

(2.1) ( ψ − ϕ ) ∫ 0 p μ ( q λ + γ p μ − γ ) ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ + ∫ p μ 1 ( q λ + γ p μ − 1 ) ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ = ( ψ − ϕ ) ∫ 0 1 ( q λ + γ p μ − 1) ϕ D p , q g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ + (1 − γ ) ∫ 0 p μ D p , q ϕ g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ .

Applying Definitions 1.1 and 1.3, we have

(2.2) ∫ 0 1 λ ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ = ∫ 0 1 g ( p λ ψ + ( 1 − p λ ) ϕ ) − g ( q λ ψ + ( 1 − q λ ) ϕ ) ( p − q ) ( ψ − ϕ ) d p , q 0 λ = 1 ψ − ϕ ∑ n = 0 ∞ q n p n + 1 g q n p n ψ + 1 − q n p n ϕ − ∑ n = 0 ∞ q n p n + 1 g q n + 1 p n + 1 ψ + 1 − q n + 1 p n + 1 ϕ = 1 ψ − ϕ 1 p ∑ n = 0 ∞ q n p n g q n p n ψ + 1 − q n p n ϕ − 1 q ∑ n = 1 ∞ q n p n g q n p n ψ + 1 − q n p n ϕ = 1 ψ − ϕ 1 q g ( ψ ) − 1 q − 1 p × ∑ n = 0 ∞ q n p n g q n p n ψ + 1 − q n p n ϕ = 1 q ψ − ϕ g ( ψ ) − 1 p q ( ψ − ϕ ) 2 ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ,

(2.3) ∫ 0 1 D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ = ∫ 0 1 g ( p λ ψ + ( 1 − p λ ) ϕ ) − g ( q λ ψ + ( 1 − q λ ) ϕ ) λ ( p − q ) ( ψ − ϕ ) d p , q 0 λ = 1 ψ − ϕ ∑ n = 0 ∞ g q n p n ψ + 1 − q n p n ϕ − ∑ n = 0 ∞ g q n + 1 p n + 1 ψ + 1 − q n + 1 p n + 1 ϕ = g ψ − g ϕ ψ − ϕ ,

(2.4) ∫ 0 p μ D p , q ϕ g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ = ∫ 0 p μ g ( p λ ψ + (1 − p λ ) ϕ ) − g ( q λ ψ + (1 − q λ ) ϕ ) λ ( p − q ) ( ψ − ϕ ) d p , q 0 λ = 1 ψ − ϕ ∑ n =0 ∞ g q n p n p μ ψ + 1 − q n p n p μ ϕ − ∑ n =0 ∞ g q n + 1 p n + 1 p μ ψ + 1 − q n + 1 p n + 1 p μ ϕ = g p μ ψ + (1 − p μ ) ϕ − g ( ϕ ) ψ − ϕ .

Substituting (2.2), (2.3) and (2.4) into (2.1), we obtain the desired result.□

Remark 2.1

The following statements are true under the conditions of Lemma 2.1.

  1. If μ = 0 , then we get

    g ( ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 1 ( q λ − 1) ϕ D p , q g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ .

  2. If p = μ = 1 , then we have

    g ( ψ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 1 q λ ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ .

  3. If μ = 1 / [ 2 ] p , q , then one has

γ q g ( ϕ ) + p g ( ψ ) [ 2 ] p , q + ( 1 − γ ) g q ϕ + p ψ [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p [ 2 ] p , q q λ − γ q [ 2 ] p , q D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ + ∫ p [ 2 ] p , q 1 q λ + p γ − 1 − q [ 2 ] p , q D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ .

Remark 2.2

If all the conditions of Lemma 2.1 are satisfied, then the following four statements are true:

  1. If γ = 0 , then we get

(2.5) g ( p μ ψ + (1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p μ q λ ϕ D p , q g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ + ∫ p μ 1 ( q λ − 1) ϕ D p , q g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ .

Let μ = 1 / [ 2 ] p , q in (2.5). Then we acquire the midpoint-type integral identity

(2.6) g q ϕ + p ψ [2] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p [ 2 ] p , q q λ ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ + ∫ [ 2 ] p , q 1 ( q λ − 1 ) ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ ,

which was proposed by Kunt et al. in [61], and equation (2.6) leads to Lemma 11 of [65] if p = 1 .

  1. If γ = 1 / 3 , then we get

    (2.7) 1 3 [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) + 2 g ( p μ ψ + ( 1 − p μ ) ϕ ) ] − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p μ q λ + p μ − 1 3 D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ + ∫ p μ 1 q λ + p μ − 3 3 D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ .

    In particular, if μ = 1 / [ 2 ] p , q , then equation (2.7) leads to the Simpson-type integral identity

    1 3 q g ( ϕ ) + p g ( ψ ) [ 2 ] p , q + 2 g q ϕ + p ψ [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p [ 2 ] p , q q λ − q 3 [ 2 ] p , q D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ + ∫ p [ 2 ] p , q 1 q λ − 2 p + 3 q 3 [ 2 ] p , q D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ .

  2. If γ = 1 / 2 , then one has

    (2.8) 1 2 [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) + g ( p μ ψ + (1 − p μ ) ϕ ) ] − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p μ q λ + p μ − 1 2 D p , q ϕ g ( λ ψ + (1 − λ ) ϕ ) 0 d p , q λ + ∫ p μ 1 q λ + p μ − 2 2 D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ .

    In particular, if μ = 1 / [ 2 ] p , q , then equation (2.8) gives the average of midpoint-trapezoid-type integral identity

    1 2 q g ( ϕ ) + p g ( ψ ) [2] p , q + g q ϕ + p ψ [2] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 p [ 2 ] p , q q λ − q 2 [ 2 ] p , q D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ + ∫ p [ 2 ] p , q 1 q λ − p + 2 q 2 [ 2 ] p , q D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ .

  3. Let γ = 1 . Then we get

(2.9) p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 1 ( q λ + p μ − 1 ) ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ .

Let μ = 1 / [ 2 ] p , q . Then equation (2.9) leads to the trapezoid-type integral identity

(2.10) q g ( ϕ ) + p g ( ψ ) [2] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x = ( ψ − ϕ ) ∫ 0 1 q λ − q [ 2 ] p , q ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ ,

which was proposed by Latif et al. in [66].

In particular, Lemma 3.1 of [55] can be derived from equation (2.10) if we take p = 1 .

The following Lemma 2.2 can be obtained immediately from Definition 1.3.

Lemma 2.2

Let 0 < q < p ≤ 1 , 0 ≤ μ ≤ 1 and ξ ∈ [ 0 , ∞ ) . Then we have

∫ 0 p μ λ ξ d p , q 0 λ = ( p − q ) μ ξ + 1 ∑ n = 0 ∞ q p ( ξ + 1 ) n = μ ξ + 1 ( p − q ) p ξ + 1 p ξ + 1 − q ξ + 1

and

∫ 0 p μ (1 − λ ) ξ d p , q 0 λ = ( p − q ) μ ∑ n =0 ∞ q n p n 1 − q n p n μ ξ .

Lemma 2.3

Let 0 < q < p ≤ 1 , γ , μ ∈ [ 0 , 1 ] and ξ ∈ [ 0 , ∞ ) . Then we get

ν 1 ( γ , p μ , ξ ) = ∫ 0 p μ λ ξ | q λ − ( γ − γ p μ ) | 0 d p , q λ = μ ξ + 1 ( p − q ) p ξ + 1 ( γ − γ p μ ) p ξ + 1 − q ξ + 1 − q μ ξ + 2 ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 , ( γ + q ) p μ ≤ γ , 2 ( p − q ) ( γ − γ p μ ) ξ + 2 ( p ξ + 2 − q ξ + 2 − p ξ + 1 + q ξ + 1 ) q ξ + 1 ( p ξ + 1 − q ξ + 1 ) ( p ξ + 2 − q ξ + 2 ) + q μ ξ + 2 ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 − μ ξ + 1 ( p − q ) p ξ + 1 ( γ − γ p μ ) p ξ + 1 − q ξ + 1 , ( γ + q ) p μ > γ

and

ν 2 ( γ , p μ , ξ ) = ∫ 0 p μ ( 1 − λ ) ξ | ( q λ − ( γ − γ p μ ) | 0 d p , q λ = ( p − q ) μ ∑ n =0 ∞ q n p n γ − γ p μ − q n + 1 p n μ 1 − q n p n μ ξ , ( γ + q ) p μ ≤ γ , 2( p − q ) ( γ − γ p μ ) 2 ∑ n =0 ∞ q n − 1 p n + 1 1 − q n p n + 1   1 − q n − 1 p n + 1 ( γ − γ p μ ) ξ − ( p − q ) μ ∑ n =0 ∞ q n p n γ − γ p μ − q n + 1 p n μ 1 − q n p n μ ξ , ( γ + q ) p μ > γ .

Proof

If ( γ + q ) p μ ≤ γ , then it follows from Lemma 2.1 that

∫ 0 p μ λ ξ | ( q λ − ( γ − γ p μ ) | 0 d p , q λ = ∫ 0 p μ [ ( γ − γ p μ ) λ ξ − q λ ξ + 1 ] 0 d p , q λ = μ ξ + 1 ( p − q ) p ξ + 1 ( γ − γ p μ ) p ξ + 1 − q ξ + 1 − q μ ξ + 2 ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 .

If ( γ + q ) p μ > γ , then from Lemma 2.1 we get

∫ 0 p μ λ ξ | q λ − ( γ − γ p μ ) | 0 d p , q λ = ∫ 0 γ − γ p μ q [ ( γ − γ p μ ) λ ξ − q λ ξ + 1 ] 0 d p , q λ + ∫ γ − γ p μ q p μ [ q λ ξ + 1 − ( γ − γ p μ ) λ ξ ] 0 d p , q λ = 2 ∫ 0 γ − γ p μ q [ ( γ − γ p μ ) λ ξ − q λ ξ + 1 ] 0 d p , q λ + ∫ 0 p μ [ q λ ξ + 1 − ( γ − γ p μ ) λ ξ ] 0 d p , q λ = 2 ( p − q ) ( γ − γ p μ ) ξ + 2 ( p ξ + 2 − q ξ + 2 − p ξ + 1 + q ξ + 1 ) q ξ + 1 ( p ξ + 1 − q ξ + 1 ) ( p ξ + 2 − q ξ + 2 ) + q μ ξ + 2 ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 − μ ξ + 1 ( p − q ) p ξ + 1 ( γ − γ p μ ) p ξ + 1 − q ξ + 1 .

Similarly, we also get

∫ 0 p μ ( 1 − λ ) ξ | ( q λ − ( γ − γ p μ ) | 0 d p , q λ = ( p − q ) μ ∑ n = 0 ∞ q n p n γ − γ p μ − q n + 1 p n μ 1 − q n p n μ ξ , ( γ + q ) p μ ≤ γ , 2 ( p − q ) ( γ − γ p μ ) 2 ∑ n = 0 ∞ q n − 1 p n + 1 1 − q n p n + 1 1 − q n − 1 p n + 1 ( γ − γ p μ ) ξ − ( p − q ) μ ∑ n = 0 ∞ q n p n γ − γ p μ − q n + 1 p n μ 1 − q n p n μ ξ , γ + q p μ > γ ,

which completes the proof of Lemma 2.3.□

The following Lemmas 2.4–2.9 can be obtained by using the definition of q-integrals, we omit the details of their proofs.

Lemma 2.4

Let 0 < q < p ≤ 1 , γ , μ ∈ [ 0 , 1 ] and ξ ∈ [ 0 , ∞ ) . Then we have

ν 3 ( γ , p μ , ξ ) = ∫ 0 1 λ ξ | q λ − ( 1 − γ p μ ) | 0 d p , q λ = ( p − q ) p ξ + 1 ( 1 − γ p μ ) p ξ + 1 − q ξ + 1 − q ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 , γ p μ + q ≤ 1 , 2 ( p − q ) ( 1 − γ p μ ) ξ + 2 p ξ + 2 − q ξ + 2 − p ξ + 1 + q ξ + 1 q ξ + 1 ( p ξ + 1 − q ξ + 1 ) ( p ξ + 2 − q ξ + 2 ) + q ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 − ( p − q ) p ξ + 1 ( 1 − γ p μ ) p ξ + 1 − q ξ + 1 , γ p μ + q > 1

and

ν 4 ( γ , p μ , ξ ) = ∫ 0 1 ( 1 − λ ) ξ | q λ − ( 1 − γ p μ ) | 0 d p , q λ = ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − γ p μ − q n + 1 p n + 1 1 − q n p n + 1 ξ , γ p μ + q ≤ 1 , 2 ( p − q ) ( 1 − γ p μ ) 2 ∑ n = 0 ∞ q n − 1 p n + 1 1 − q n p n + 1 1 − q n − 1 p n + 1 ( 1 − γ p μ ) ξ − ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − γ p μ − q n + 1 p n + 1 1 − q n p n + 1 ξ , γ p μ + q > 1 .

Lemma 2.5

Let 0 < q < p ≤ 1 , γ , μ ∈ [ 0 , 1 ] and ξ ∈ [ 0 , ∞ ) . Then one has

ν 5 ( γ , p μ , ξ ) = ∫ 0 p μ λ ξ | q λ − ( 1 − γ p μ ) | 0 d p , q λ = μ ξ + 1 ( p − q ) p ξ + 1 ( 1 − γ p μ ) p ξ + 1 − q ξ + 1 − q μ ξ + 2 ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 , ( γ + q ) p μ ≤ 1 , 2 ( p − q ) ( 1 − γ p μ ) ξ + 2 ( p ξ + 2 − q ξ + 2 − p ξ + 1 + q ξ + 1 ) q ξ + 1 ( p ξ + 1 − q ξ + 1 ) ( p ξ + 2 − q ξ + 2 ) + q μ ξ + 2 ( p − q ) p ξ + 2 p ξ + 2 − q ξ + 2 − μ ξ + 1 ( p − q ) p ξ + 1 ( 1 − γ p μ ) p ξ + 1 − q ξ + 1 , ( γ + q ) p μ > 1

and

ν 6 ( γ , p μ , ξ ) = ∫ 0 p μ ( 1 − λ ) ξ | q λ − ( 1 − γ μ ) | 0 d p , q λ = ( p − q ) μ ∑ n = 0 ∞ q n p n ( 1 − γ p μ − q n + 1 p n μ ) 1 − q n p n μ ξ , ( γ + q ) p μ ≤ 1 , 2 ( p − q ) ( 1 − γ μ ) 2 ∑ n = 0 ∞ q n − 1 p n + 1 1 − q n p n + 1 1 − q n − 1 p n + 1 ( 1 − γ p μ ) ξ − ( p − q ) μ ∑ n = 0 ∞ q n p n 1 − γ p μ − q n + 1 p n μ 1 − q n p n μ ) ξ , ( γ + q ) p μ > 1 .

Lemma 2.6

Let 0 < q < p ≤ 1 and γ , μ ∈ [ 0 , 1 ] . Then we have

ν 7 ( γ , p μ ) = ∫ 0 p μ | q λ − ( γ − γ p μ ) | 0 d p , q λ = γ p μ ( 1 − p μ ) − q μ 2 p 2 [ 2 ] p , q , ( γ + q ) p μ ≤ γ , 2 ( γ − γ p μ ) 2 ( [ 2 ] p , q − 1 ) q [ 2 ] p , q + q μ 2 p 2 [ 2 ] p , q − γ p μ ( 1 − p μ ) , ( γ + q ) p μ > γ .

Lemma 2.7

Let 0 < q < p ≤ 1 and γ , μ ∈ [ 0 , 1 ] . Then we get

ν 8 ( γ , p μ ) = ∫ 0 1 | q λ − ( 1 − γ p μ ) | 0 d p , q λ = p [ 2 ] p , q − γ p μ , γ p μ + q ≤ 1 , 2 ( 1 − γ p μ ) 2 ( [ 2 ] p , q − 1 ) q [ 2 ] p , q + γ p μ − p [ 2 ] p , q , γ p μ + q > 1 .

Lemma 2.8

Let 0 < q < p ≤ 1 and γ , μ ∈ [ 0 , 1 ] . Then one has

ν 9 ( γ , p μ ) = ∫ 0 p μ | q λ − ( 1 − γ p μ ) | 0 d p , q λ = p μ ( 1 − γ p μ ) − q μ 2 p 2 [ 2 ] p , q , γ + q p μ ≤ 1 , 2 ( 1 − γ p μ ) 2 ( [ 2 ] p , q − 1 ) q [ 2 ] p , q + q μ 2 p 2 [ 2 ] p , q − p μ ( 1 − γ p μ ) x, γ + q p μ > 1 .

Lemma 2.9

Let 0 < q < p ≤ 1 , γ , μ ∈ [ 0 , 1 ] and σ ∈ [ 1 , ∞ ) . Then we get

ν 10 ( γ , p μ ) = ∫ 0 1 | q λ − (1 − γ p μ ) | σ d p , q 0 λ = ( p − q ) ∑ n =0 ∞ q n p n + 1 1 − γ p μ − q n + 1 p n + 1 σ , 0 ≤ γ p μ ≤ 1 − q , ( p − q )(1 − γ p μ ) σ + 1 ∑ n =0 ∞ q n − 1 p n + 1 1 − q n p n + 1 σ + ( p − q ) ∑ n =0 ∞ q n p n + 1 q n + 1 p n + 1 − 1 + γ p μ σ − ( p − q )(1 − γ p μ ) σ + 1 ∑ n =0 ∞ q n − 1 p n + 1 q n p n + 1 − 1 σ , 1 − q < γ p μ ≤ 1 .

3 Main results

Theorem 3.1

Let 0 ≤ ϕ < ψ < ∞ , 0 < q < p ≤ 1 , α , m ∈ ( 0 , 1 ] and g : J ⊃ 0 , ∞ ) → ℝ be a ( p , q ) -differentiable function on J° (the interior of J) such that D p , q ϕ g is continuous and integrable on [ 0, ψ m ] and | ϕ D p , q g | is ( α , m ) -convex on [ 0, ψ m ] . Then the inequality

γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + (1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x ≤ min [ H 1 ( γ , p μ , α , m ) , H 2 ( γ , p μ , α , m ) ]

holds for all γ , μ ∈ [ 0 , 1 ] , where

H 1 ( γ , p μ , α , m ) = ( ψ − ϕ ) { [ ν 1 ( γ , p μ , α ) + ν 3 ( γ , p μ , α ) − ν 5 ( γ , p μ , α ) ] | ϕ D p , q g ( ψ ) | + m [ ν 7 ( γ , p μ ) + ν 8 ( γ , p μ ) − ν 9 ( γ , p μ ) − ν 1 ( γ , p μ , α ) − ν 3 ( γ , p μ , α ) + ν 5 ( γ , p μ , α ) ] | ϕ D p , q g ϕ m | ,

H 2 ( γ , p μ , α , m ) = ( ψ − ϕ ) { [ ν 2 ( γ , p μ , α ) + ν 4 ( γ , p μ , α ) − ν 6 ( γ , p μ , α ) ] | ϕ D p , q g ( ϕ ) | + m [ ν 7 ( γ , p μ ) + ν 8 ( γ , p μ ) − ν 9 ( γ , p μ ) − ν 2 ( γ , p μ , α ) − ν 4 ( γ , p μ , α ) + ν 6 ( γ , p μ , α ) ] | ϕ D p , q g ψ m | .

Proof

From Lemma 2.1, the property of the modulus and the ( α , m ) -convexity of | D p , q ϕ g | we have

γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + (1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + (1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) ∫ 0 p μ | q λ + γ p μ − γ | | ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) | 0 d p , q λ + ∫ p μ 1 | q λ + γ p μ − 1 | | ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) | 0 d p , q λ ≤ ( ψ − ϕ ) ∫ 0 p μ | q λ − ( γ − γ p μ ) | λ α | ϕ D p , q g ( ψ ) | + m ( 1 − λ α ) | ϕ D p , q g ϕ m | 0 d p , q λ + ∫ p μ 1 | q λ − ( 1 − γ p μ ) | λ α | ϕ D p , q g ( ψ ) | + m (1 − λ α ) | ϕ D p , q g ϕ m | 0 d p , q λ

= ( ψ − ϕ ) ∫ 0 p μ λ α | q λ − ( γ − γ p μ ) | 0 d p , q λ + ∫ 0 1 λ α | q λ − ( 1 − γ p μ ) | 0 d p , q λ − ∫ 0 p μ λ α | q λ − ( 1 − γ p μ ) | 0 d p , q λ | ϕ D p , q g ( ψ ) | + m ∫ 0 p μ | q λ − ( γ − γ p μ ) | 0 d p , q λ

+ ∫ 0 1 | q λ − ( 1 − γ p μ ) | 0 d p , q λ − ∫ 0 p μ | q λ − ( 1 − γ p μ ) | 0 d p , q λ − ∫ 0 p μ λ α | q λ − ( γ − γ p μ ) | 0 d p , q λ − ∫ 0 1 λ α | q λ − ( 1 − γ p μ ) | 0 d p , q λ + ∫ 0 p μ λ α | q λ − ( 1 − γ p μ ) | 0 d p , q λ | ϕ D p , q g ϕ m |

≤ ( ψ − ϕ ) ∫ 0 p μ ( 1 − λ ) α | q λ − ( γ − γ p μ ) | 0 d p , q λ + ∫ 0 1 ( 1 − λ ) α × | q λ − ( 1 − γ p μ ) | 0 d p , q λ − ∫ 0 p μ ( 1 − λ ) α | q λ − ( 1 − γ p μ ) | 0 d p , q λ × | ϕ D p , q g ( ϕ ) | + m ∫ 0 p μ | q λ − ( γ − γ p μ ) | 0 d p , q λ + ∫ 0 1 | q λ − ( 1 − γ p μ ) | 0 d p , q λ − ∫ 0 p μ | q λ − ( 1 − γ p μ ) | 0 d p , q λ − ∫ 0 p μ ( 1 − λ ) α | q λ − ( γ − γ p μ ) | 0 d p , q λ − ∫ 0 1 ( 1 − λ ) α × | q λ − ( 1 − γ p μ ) | 0 d p , q λ + ∫ 0 p μ ( 1 − λ ) α | q λ − ( 1 − γ p μ ) | 0 d p , q λ | ϕ D p , q g ψ m | .

Using Lemmas 2.3–2.8, we get the desired result.□

Corollary 3.1

Let μ = 1 / [ 2 ] p , q . Then Theorem 3.1 leads to

γ q g ( ϕ ) + p g ( ψ ) [ 2 ] p , q + ( 1 − γ ) g q ϕ + p ψ [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ min H 1 γ , p [ 2 ] p , q , α , m , H 2 γ , p [ 2 ] p , q , α , m .

Remark 3.1

  1. Let γ = 0 . Then Corollary 3.1 gives the midpoint-type integral inequality

    g q ϕ + p ψ [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ min H 1 0 , p [ 2 ] p , q , α , m , H 2 0 , p [ 2 ] p , q , α , m ,

    where

    H 1 0 , p [ 2 ] p , q , α , m = ( ϕ − ψ ) [ ( [ 2 ] p , q ) α + 2 − ( p α + 2 + q α + 2 ) ] ( p − q ) 2 ( [ 2 ] p , q ) α + 2 ( p α + 1 − q α + 1 ) ( p α + 2 − q α + 2 ) | ϕ D p , q g ( ψ ) | + m 2 q p 2 ( [ 2 ] p , q ) 3 − [ ( [ 2 ] p , q ) α + 2 − ( p α + 2 + q α + 2 ) ] ( p − q ) 2 ( [ 2 ] p , q ) α + 2 ( p α + 1 − q α + 1 ) ( p α + 2 − q α + 2 ) | ϕ D p , q g ϕ m | ,

    H 2 0 , p [ 2 ] p , q , α , m = ( ψ − ϕ ) ν 2 0 , p [ 2 ] p , q , α + ν 4 0 , p [ 2 ] p , q , α − ν 6 0 , p [ 2 ] p , q , α | ϕ D p , q g ( ϕ ) | + m 2 q p 2 ( [ 2 ] p , q ) 3 − ν 2 0 , p [ 2 ] p , q , α − ν 4 0 , p [ 2 ] p , q , α + ν 6 0 , p [ 2 ] p , q , α | ϕ D p , q g ψ m | .

    In particular, if α = 1 = m , then we obtain

    (3.1) g q ϕ + p ψ [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) q 3 p 3 ( [ 2 ] p , q ) 3 ( p 2 + p q + q 2 ) | ϕ D p , q g ( ψ ) | + 2 p 4 + 2 p 2 q 2 + 2 p 3 q − 3 p 3 ( [ 2 ] p , q ) 3 ( p 2 + p q + q 2 ) | ϕ D p , q g ( ϕ ) | ,

    which was proposed by Kunt et al. in [61].

    Let p = 1 . Then equation (3.1) becomes Theorem 13 of [65].

  2. Taking γ = 1 / 3 and α = 1 = m in Corollary 3.1, we get the Simpson-type integral inequality

    1 3 q g ( ϕ ) + p g ( ψ ) [ 2 ] p , q + 2 g q ϕ + p ψ [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ min H 1 1 3 , p [ 2 ] p , q , 1 , 1 , H 2 1 3 , p [ 2 ] p , q , 1 , 1 .

    If q → 1 − and p = 1 , then we obtain

    1 3 g ( ϕ ) + g ( ψ ) 2 + 2 g ϕ + ψ 2 − 1 ( ψ − ϕ ) ∫ a ψ g ( x ) d x ≤ 5 ( ψ − ϕ ) 72 [ | g ′ ( ψ ) | + | g ′ ( ϕ ) | ] ,

    which was proposed by Alomari et al. in [67].

  3. Let γ = 1 / 2 and α = 1 = m . Then we get the average of midpoint and trapezoid-type integral inequality

    1 2 q g ( ϕ ) + p g ( ψ ) [ 2 ] p , q + g q ϕ + p ψ [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ min H 1 1 2 , p [ 2 ] p , q , 1 , 1 , H 2 1 2 , p [ 2 ] p , q , 1 , 1 .

    If q → 1 − and p = 1 , then we obtain

    1 2 g ( ϕ ) + g ( ψ ) 2 + g ϕ + ψ 2 − 1 ( ψ − ϕ ) ∫ ϕ ψ g ( x ) d x ≤ ψ − ϕ 16 [ | g ′ ( ψ ) c| + | g ′ ( ϕ ) | ] ,

    which was proposed by Xi et al. in [68].

  4. Let γ = 1 . Then we get the trapezoid-type integral inequality

q g ( ϕ ) + p g ( ψ ) [ 2 ] p , q − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ min H 1 1 , p [ 2 ] p , q , α , m , H 2 1 , p [ 2 ] p , q , α , m .

In particular, if α = p = m = 1 , then we obtain

q g ( ϕ ) + g ( ψ ) [2] q − 1 ψ − ϕ ∫ ϕ ψ g ( x ) ϕ d q x ≤ ( ψ − ϕ ) q 2 1 + 4 q + q 2 ([2] q ) 4 [3] q | ϕ D q g ( ψ ) | + 1 + 3 q 2 + 2 q 3 ([2] q ) 4 [3] q ϕ D q g ( ϕ ) | ,

which was proposed by Sudsutad et al. in [55].

Theorem 3.2

Let 0 ≤ ϕ < ψ < ∞ , r > 1 , α , m ∈ ( 0 , 1 ] , 0 < q < p ≤ 1 and g : J ⊃ [ 0 , ∞ ) → ℝ be a ( p , q ) -differentiable function on J° (the interior of J) such that D p , q ϕ g is continuous and integrable on [ 0 , ψ m ] and | ϕ D p , q g | r is ( α , m ) -convex on [ 0 , ψ m ] . Then the inequality

γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) min [ T 1 ( γ , p μ , α , m , r ) , T 2 ( γ , p μ , α , m , r ) ]

holds for all γ , μ ∈ [ 0 , 1 ] , where

T 1 ( γ , p μ , α , m , r ) = ν 8 1 − 1 r ( γ , p μ ) [ ν 3 ( γ , p μ , α ) | ϕ D p , q g ( ψ ) | r + m ( ν 8 ( γ , p μ ) − ν 3 ( γ , p μ , α ) ) | D p , q ϕ g ϕ m | r 1 r + ( 1 − γ ) ( p μ ) 1 − 1 r Γ 1 ( p μ , α ) | ϕ D p , q g ( ψ ) | r + m Γ 2 ( p μ , α ) | ϕ D p , q g ϕ m | r 1 r , T 2 ( γ , p μ , α , m , r ) = ν 8 1 − 1 r ( γ , p μ ) ν 4 ( γ , p μ , α ) | ϕ D p , q g ( ϕ ) | r + m ( ν 8 ( γ , p μ ) − ν 4 ( γ , p μ , α ) ) | ϕ D p , q g ψ m | r 1 r + p μ 1 − 1 r ( 1 − γ ) Γ 3 ( p μ , α ) | ϕ D p , q g ( ϕ ) | r + m Γ 4 ( p μ , α ) | ϕ D p , q g ψ m | r 1 r

and

(3.2) Γ 1 ( p μ , α ) = ∫ 0 p μ λ α d p , q 0 λ = μ α + 1 ( p − q ) p α + 1 p α + 1 − q α + 1 ,

(3.3) Γ 2 ( p μ , α ) = ∫ 0 p μ ( 1 − λ α ) 0 d p , q λ = p μ − μ α + 1 ( p − q ) p α + 1 p α + 1 − q α + 1 ,

(3.4) Γ 3 ( p μ , α ) = ∫ 0 p μ ( 1 − λ ) α d p , q 0 λ = ( p − q ) μ ∑ n = 0 ∞ q n p n 1 − q n p n μ α ,

(3.5) Γ 4 ( p μ , α ) = ∫ 0 p μ ( 1 − ( 1 − λ ) α ) 0 d p , q λ = p μ − ( p − q ) μ ∑ n = 0 ∞ q n p n 1 − q n p n μ α

and ν 3 ( γ , p μ , α ) , ν 4 ( γ , p μ , α ) and ν 8 ( γ , p μ ) are defined in Lemmas 2.4 and 2.7, respectively.

Proof

Using Lemma 2.1 and the power mean inequality, we have

(3.6) γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) ∫ 0 1 | q λ − ( 1 − γ p μ ) | 0 d p , q λ 1 − 1 r × ∫ 0 1 | q λ − ( 1 − γ p μ ) | | D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ 1 r + ( 1 − γ ) ∫ 0 p μ 1 0 d p , q λ 1 − 1 r ∫ 0 p μ | ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ 1 r .

Utilizing the ( α , m ) -convexity of | D p , q ϕ g | r , we get

(3.7) ∫ 0 1 | q λ − ( 1 − γ p μ ) | | D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ ≤ ∫ 0 1 | q λ − ( 1 − γ p μ ) | λ α | D p , q ϕ g ( ψ ) | r + m ( 1 − λ α ) D p , q ϕ g ϕ m r d p , q 0 λ = ∫ 0 1 λ α | q λ − ( 1 − γ p μ ) | d p , q 0 λ | D p , q ϕ g ( ψ ) | r + m ∫ 0 1 | q λ − ( 1 − γ p μ ) | d p , q 0 λ − ∫ 0 1 λ α | q λ − ( 1 − γ p μ ) | d p , q 0 λ D p , q ϕ g ϕ m r ,

(3.8) ∫ 0 p μ | D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ ≤ ∫ 0 p μ λ α | D p , q ϕ g ( ψ ) | r + m ( 1 − λ α ) D p , q ϕ g ϕ m r d p , q 0 λ = ∫ 0 p μ λ α 0 d p , q λ | D p , q ϕ g ( ψ ) | r + m ∫ 0 p μ ( 1 − λ α ) 0 d p , q λ D p , q ϕ g ϕ m r .

Using (3.7) and (3.8) in (3.6), we get

(3.9) γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) d p , q ϕ x ≤ ( ψ − ϕ ) ∫ 0 1 | q λ − ( 1 − γ p μ ) | d p , q 0 λ 1 − 1 r ∫ 0 1 λ α | q λ − ( 1 − γ p μ ) | d p , q 0 λ | D p , q ϕ g ( ψ ) | r + m ∫ 0 1 | q λ − ( 1 − γ p μ ) | d p , q 0 λ − ∫ 0 1 λ α | q λ − ( 1 − γ p μ ) | d p , q 0 λ D p , q ϕ g ϕ m r 1 r + ( 1 − γ ) ( p μ ) 1 − 1 r ∫ 0 p μ λ α 0 d p , q λ | D p , q ϕ g ( ψ ) | r + m ∫ 0 p μ ( 1 − λ α ) d p , q 0 λ | D p , q ϕ g ϕ m | r 1 r .

Similarly, we get

(3.10) ∫ 0 1 | q λ − ( 1 − γ p μ ) | | D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ ≤ ∫ 0 1 | q λ − ( 1 − γ p μ ) | ( 1 − λ ) α | D p , q ϕ g ( ϕ ) | r + m ( 1 − ( 1 − λ ) α ) | D p , q ϕ g ψ m | r d p , q 0 λ = ∫ 0 1 ( 1 − λ ) α | q λ − ( 1 − γ p μ ) | 0 d p , q λ | D p , q ϕ g ( ϕ ) | r + m ∫ 0 1 | q λ − ( 1 − γ p μ ) | d p , q 0 λ − ∫ 0 1 ( 1 − λ ) α | q λ − ( 1 − γ p μ ) | d p , q 0 λ D p , q ϕ g ψ m r .

(3.11) ∫ 0 p μ D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) r d p , q 0 λ ≤ ∫ 0 p μ ( 1 − λ ) α D p , q ϕ g ( ϕ ) r + m ( 1 − ( 1 − λ ) α ) D p , q ϕ g ψ m D p , q g ψ m r d p , q 0 λ = ∫ 0 p μ ( 1 − λ ) α d p , q 0 λ | D p , q ϕ g ( ϕ ) | r + m ∫ 0 p μ ( 1 − ( 1 − λ ) α ) d p , q 0 λ | D p , q ϕ g ψ m | r .

Using (3.10) and (3.11) in (3.6), we get

(3.12) γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ b + ( 1 − p μ ) a ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) ∫ 0 1 | q λ − ( 1 − γ p μ ) | 0 d p , q λ 1 − 1 r ∫ 0 1 ( 1 − λ ) α | q λ − ( 1 − γ p μ ) | 0 d p , q λ | ϕ D p , q g ( ϕ ) | r + m ∫ 0 1 | q λ − ( 1 − γ p μ ) | 0 d p , q λ − ∫ 0 1 ( 1 − λ ) α | q λ − ( 1 − γ p μ ) | 0 d p , q λ D p , q ϕ g ψ m r 1 r + ( 1 − γ ) ( p μ ) 1 − 1 r ∫ 0 p μ ( 1 − λ )  0 α d p , q λ | ϕ D p , q g ( ϕ ) | r + m ∫ 0 p μ ( 1 − ( 1 − λ ) α ) 0 d p , q λ D p , q ϕ g ψ m r 1 r .

Therefore, the desired result follows from (3.9) and (3.12) together with Lemmas 2.4 and 2.7.□

Theorem 3.3

Let 0 ≤ ϕ < ψ < ∞ , 0 < q < p ≤ 1 , r , s > 1 with r − 1 + s − 1 = 1 , α , m ∈ ( 0 , 1 ] and g : J ⊃ [ 0 , ∞ ) → ℝ be a ( p , q ) -differentiable function on J° (the interior of J) such that D p , q ϕ g is continuous and integrable on [ 0, ψ m ] and | D p , q ϕ g | r is ( α , m ) -convex on [ 0, ψ m ] . Then the inequality

γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) min [ K 1 ( γ , p μ , α , m ) , K 2 ( γ , p μ , α , m ) ]

holds for all γ , μ ∈ [ 0 , 1 ] , where

K 1 ( γ , p μ , α , m ) = ν 10 1 s ( γ , p μ ) η 2 ( α ) | ϕ D p , q g ( ψ ) | r + m ( 1 − η 2 ( α ) ) | ϕ D p , q g ϕ m | r 1 r + ( 1 − γ ) ( p μ ) 1 s Γ 1 ( p μ , α ) | ϕ D p , q g ( ψ ) | r + m Γ 2 ( p μ , α ) | ϕ D p , q g ϕ m | r 1 r ,

K 2 ( γ , p μ , α , m ) = ν 10 1 s ( γ , p μ ) η 3 ( α ) | ϕ D p , q g ( ϕ ) | r + m ( 1 − η 3 ( α ) ) | ϕ D p , q g ψ m | r 1 r + (1 − γ ) ( p μ ) 1 s Γ 3 ( p μ , α ) | ϕ D p , q g ( ϕ ) | r + m Γ 4 ( p μ , α ) | ϕ D p , q g ψ m | r 1 r ,

η 2 ( α ) = ∫ 0 1 λ α 0 d p , q λ = p − q p 1 + α − q 1 + α , η 3 ( α ) = ∫ 0 1 (1 − λ )  0 α d p , q λ = ( p − q ) ∑ n =0 ∞ q n p n + 1 1 − q n p n + 1 α

and Γ 1 ( p μ , α ) , Γ 2 ( p μ , α ) , Γ 3 ( p μ , α ) and Γ 4 ( p μ , α ) are defined in Theorem 3.2.

Proof

Using Lemma 2.1 and the Hölder inequality, we have

(3.13) γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) ∫ 0 1 | q λ − ( 1 − γ p μ ) | s d p , q 0 λ 1 s ∫ 0 1 | ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ 1 r + ( 1 − γ ) ∫ 0 p μ 1 s d p , q 0 λ 1 s ∫ 0 p μ | ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ 1 r .

Utilizing the ( α , m ) -convexity of | ϕ D p , q g | r , we get

(3.14) ∫ 0 1 | ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ ≤ ∫ 0 1 λ α | ϕ D p , q g ( ψ ) | r + m ( 1 − λ α ) | ϕ D p , q g ϕ m | r d p , q 0 λ = ∫ 0 1 λ α d p , q 0 λ | ϕ D p , q g ( ψ ) | r + m ∫ 0 1 ( 1 − λ α ) d p , q 0 λ | ϕ D p , q g ϕ m | r

and

(3.15) ∫ 0 p μ | D p , q ϕ g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ ≤ ∫ 0 p μ λ α D p , q ϕ g ( ψ ) r + m ( 1 − λ α ) D p , q ϕ g ϕ m r d p , q 0 λ = ∫ 0 p μ λ α d p , q 0 λ | D p , q ϕ g ( ψ ) | r + m ∫ 0 p μ ( 1 − λ α ) d p , q 0 λ D p , q ϕ g ϕ m r .

Using (3.14) and (3.15) in (3.13), we get

(3.16) γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) ∫ 0 1 | q λ − ( 1 − γ p μ ) | s d p , q 0 λ 1 s ∫ 0 1 λ α d p , q 0 λ | ϕ D p , q g ( ψ ) | r + m ∫ 0 1 ( 1 − λ α ) d p , q 0 λ | ϕ D p , q g ϕ m | r 1 r + ( 1 − γ ) p μ 1 s ∫ 0 p μ λ α d p , q 0 λ | D p , q ϕ g ( ψ ) | r + m ∫ 0 p μ ( 1 − λ α ) d p , q 0 λ | D p , q ϕ g ϕ m | r 1 r .

Similarly, we get

(3.17) ∫ 0 1 ∣ ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ )  0 r d p , q λ ≤ ∫ 0 1 ( 1 − λ ) α ∣ ϕ D p , q g ( ϕ ) r + m ( 1 − ( 1 − λ ) α ) ϕ D p , q g ψ m r 0 d p , q λ = ∫ 0 1 ( 1 − λ )   0 α d p , q λ ∣ ϕ D p , q g ( ϕ ) r + m ∫ 0 1 ( 1 − ( 1 − λ ) α )  0 d p , q λ ϕ D p , q g ψ m r

and

(3.18) ∫ 0 p μ | ϕ D p , q g ( λ ψ + ( 1 − λ ) ϕ ) | r d p , q 0 λ ≤ ∫ 0 p μ ( 1 − λ ) α | D p , q ϕ g ( ϕ ) | r + m ( 1 − ( 1 − λ ) α ) | D p , q ϕ f ψ m | r d p , q 0 λ = ∫ 0 p μ ( 1 − λ ) α d p , q 0 λ | D p , q ϕ g ( ϕ ) | r + m ∫ 0 p μ ( 1 − ( 1 − λ ) α ) d p , q 0 λ | D p , q ϕ g ψ m | r .

Using (3.17) and (3.18) in (3.13), we get

(3.19) γ [ μ f ( ψ ) + 1 − μ g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x ≤ ( ψ − ϕ ) ∫ 0 1 | q λ − ( 1 − γ p μ ) | s d p , q 0 λ 1 s ∫ 0 1 ( 1 − λ )   0 α d p , q λ | ϕ D p , q g ( ϕ ) | r + m ∫ 0 1 ( 1 − ( 1 − λ ) α ) d p , q 0 λ | D p , q ϕ g ψ m | r 1 r + ( 1 − γ ) ( p μ ) 1 s ∫ 0 p μ ( 1 − λ ) α d p , q 0 λ | ϕ D p , q g ( ϕ ) | r + m ∫ 0 p μ ( 1 − ( 1 − λ ) α ) d p , q 0 λ | ϕ D p , q g ψ m | r 1 r .

Therefore, Theorem 3.3 follows from (3.2)–(3.5), (3.16) and (3.19) together with Lemmas 2.2 and 2.9.□

Remark 3.2

If we put γ = 0 , 1 3 , 1 2 , 1 and μ = 1 [ 2 ] p , q in Theorems 3.2 and 3.3, then we can get the midpoint-type integral inequality, the Simpson-type integral inequality, average of midpoint and trapezoid-type integral inequality and the trapezoid-type integral inequality, respectively.

Next, we establish the ( p , q ) -integral inequalities involving the product of two ( α , m ) -convex functions.

Theorem 3.4

Let 0 ≤ ϕ < ψ < ∞ , 0 < q < p ≤ 1 , α 1 , α 2 , m ∈ ( 0 , 1 ] and f , g : J ⊃ [ 0 , ∞ ) → ℝ be continuous and integrable on [ 0, ψ m ] such that f and g are ( α 1 , m ) -convex and ( α 2 , m ) -convex on [ 0, ψ m ] , respectively. Then the inequality

1 ψ − ϕ ∫ ϕ ψ f ( x ) g ( x ) d p , q ϕ x ≤ min { L 1 ( α 1 , α 2 , m ) , L 2 ( α 1 , α 2 , m ) }

holds, where

L 1 ( α 1 , α 2 , m ) = m 2 p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 − p − q p α 1 + 1 − q α 1 + 1 − p − q p α 2 + 1 − q α 2 + 1 + 1 f ϕ m g ϕ m + p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 f ( ψ ) g ( ψ ) + m p − q p α 2 + 1 − q α 2 + 1 − p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 f ϕ m g ( ψ ) + m p − q p α 1 + 1 − q α 1 + 1 − p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 f ( ψ ) g ϕ m ,

L 2 ( α 1 , α 2 , m ) = m 2 [ Λ ( α 1 , α 2 ) − Λ ( α 1 ) − Λ ( α 2 ) + 1 ] f ψ m g ψ m + Λ ( α 1 , α 2 ) f ( ϕ ) g ( ϕ ) + m [ Λ ( α 1 ) − Λ ( α 1 , α 2 ) ] f ( ϕ ) g ψ m + m [ Λ ( α 2 ) − Λ ( α 1 , α 2 ) ] f ψ m g ( ϕ ) ,

Λ ( α 1 , α 2 ) = ∫ 0 1 ( 1 − λ ) α 1 + α 2 d p , q 0 λ = ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − q n p n + 1 α 1 + α 2

and

Λ ( α i ) = ∫ 0 1 ( 1 − λ ) α i d p , q 0 λ = ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − q n p n + 1 α i ( i = 1 , 2 ) .

Proof

Let λ ∈ [ 0 , 1 ] . Then it follows from the ( α 1 , m ) -convexity of f and the ( α 2 , m ) -convexity of g that

(3.20) f ( λ ψ + ( 1 − λ ) ϕ ) ≤ λ α 1 f ( ψ ) + m ( 1 − λ α 1 ) f ϕ m

and

(3.21) g ( λ ψ + ( 1 − λ ) ϕ ) ≤ λ α 2 g ( ψ ) + m ( 1 − λ α 2 ) g ϕ m .

Multiplying (3.20) with (3.21), we get

(3.22) f ( λ ψ + ( 1 − λ ) ϕ ) g ( λ ψ + ( 1 − λ ) ϕ ) ≤ λ α 1 + α 2 f ( ψ ) g ( ψ ) + m 2 ( 1 − λ α 1 ) ( 1 − λ α 2 ) f ϕ m g ϕ m + m λ α 2 ( 1 − λ α 1 ) f ϕ m g ( ψ ) + m λ α 1 ( 1 − λ α 2 ) f ( ψ ) g ϕ m .

Taking the ( p , q ) -integral for (3.22) with respect to λ on ( 0 , 1 ) and by using Lemma 2.2, we get

(3.23) ∫ 0 1 f ( λ ψ + ( 1 − λ ) ϕ ) g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ ≤ m 2 p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 − p − q p α 1 + 1 − q α 1 + 1 − p − q p α 2 + 1 − q α 2 + 1 + 1 f ϕ m g ϕ m + p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 f ( ψ ) g ( ψ ) + m p − q p α 2 + 1 − q α 2 + 1 − p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 f ϕ m g ( ψ ) + m p − q p α 1 + 1 − q α 1 + 1 − p − q p α 1 + α 2 + 1 − q α 1 + α 2 + 1 f ( ψ ) g ϕ m .

Similarly, we have

(3.24) ∫ 0 1 f ( λ ψ + ( 1 − λ ) ϕ ) g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ ≤ m 2 ∫ 0 1 ( 1 − λ ) α 1 + α 2 d p , q 0 λ − ∫ 0 1 ( 1 − λ ) α 1 d p , q 0 λ − ∫ 0 1 ( 1 − λ ) α 2 d p , q 0 λ + 1 f ψ m g ψ m + ∫ 0 1 ( 1 − λ ) α 1 + α 2 d p , q 0 λ f ( ϕ ) g ( ϕ ) + m ∫ 0 1 ( 1 − λ ) α 1 d p , q 0 λ − ∫ 0 1 ( 1 − λ ) α 1 + α 2 d p , q 0 λ f ( ϕ ) g ψ m + m ∫ 0 1 ( 1 − λ ) α 2 d p , q 0 λ − ∫ 0 1 ( 1 − λ ) α 1 + α 2 d p , q 0 λ f ψ m g ( ϕ ) .

Some simple calculations lead to

(3.25) ∫ 0 1 f ( λ ψ + ( 1 − λ ) ϕ ) g ( λ ψ + ( 1 − λ ) ϕ ) 0 d p , q λ = 1 ψ − ϕ ∫ ϕ ψ f ( x ) g ( x ) ϕ d p , q x .

Therefore, the desired result follows easily from (3.23) to (3.25).□

Corollary 3.2

If we choose α 1 = α 2 = α in Theorem 3.4, then we obtain

1 ψ − ϕ ∫ ϕ ψ f ( x ) g ( x ) ϕ d p , q x ≤ min { L 1 ( α , m ) , L 2 ( α , m ) } ,

where

L 1 ( α , m ) = m 2 p − q p 2 α + 1 − q 2 α + 1 − 2 ( p − q ) p α + 1 − q α + 1 + 1 f ϕ m g ϕ m + p − q p 2 α + 1 − q 2 α + 1 f ( ψ ) g ( ψ ) + m q α + 1 ( p − q ) ( p α − q α ) ( p α + 1 − q α + 1 ) ( p 2 α + 1 − q 2 α + 1 ) f ϕ m g ( ψ ) + f ( ψ ) g ϕ m

and

L 2 ( α , m ) = m 2 ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − q n p n + 1 2 α − 2 ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − q n p n + 1 α + 1 f ψ m g ψ m + ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − q n p n + 1 2 α f ( ϕ ) g ( ϕ ) + m ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − q n p n + 1 α − ( p − q ) ∑ n = 0 ∞ q n p n + 1 1 − q n p n + 1 2 α f ϕ g ψ m + f ψ m g ϕ .

In particular, the special case of p = α = m = 1 for Corollary 3.2 was proved by Sudsutad et al. in [55].

4 Examples

Example 4.1

Let g : J ⊃ [ 0 , ∞ ) → ℝ be defined by g ( x ) = 4 x + 1 . Then it is ( 1 , 1 3 ) -differentiable function on J° (the interior of J) and D 1 , 1 3 1 g is continuous and integrable on [0, 10] , 0 ≤ 1 < 5 < ∞ and 0 < 1 3 < 1 ≤ 1 . If | ϕ D 1 , 1 3 g | is ( 1 , 1 2 ) -convex on [ 0 , 10 ] with γ = 0 and μ = 3 4 , then all the assumptions of Theorem 3.1 are satisfied.

We clearly see that

(4.1) γ [ p μ g ( ψ ) + ( 1 − p μ ) g ( ϕ ) ] + ( 1 − γ ) g ( p μ ψ + ( 1 − p μ ) ϕ ) − 1 p ( ψ − ϕ ) ∫ ϕ p ψ + ( 1 − p ) ϕ g ( x ) ϕ d p , q x = g ( 4 ) − 1 4 ∫ 1 5 ( 4 x + 1 ) 1 d 1 , 1 3 d x = 17 − 68 4 = 0 ,

where

∫ 1 5 ( 4 x + 1 ) 1 d 1 , 1 3 d x = 68 .

On the other hand,

(4.2) ν 1 ( γ , p μ , α ) = ν 1 0 , 3 4 , 1 2 ≈ 0.1157 , ( γ + q ) p μ > γ , ν 2 ( γ , p μ , α ) = ν 2 0 , 3 4 , 1 2 ≈ 0.0625 , ( γ + q ) p μ > γ , ν 3 ( γ , p μ , α ) = ν 3 0 , 3 4 , 1 2 ≈ 0.5881 , γ p μ + q ≤ 1 , ν 4 ( γ , p μ , α ) = ν 4 0 , 3 4 , 1 2 = 0 , γ p μ + q ≤ 1 , ν 5 ( γ , p μ , α ) = ν 5 0 , 3 4 , 1 2 ≈ 0.4205 , ( γ + q ) p μ ≤ 1 , ν 6 ( γ , p μ , α ) = ν 6 0 , 3 4 , 1 2 ≈ 0.1875 , ( γ + q ) p μ ≤ 1 , ν 7 ( γ , p μ ) = ν 7 0 , 3 4 ≈ 0.1406 , ( γ + q ) p μ ≥ γ , ν 8 ( γ , p μ ) = ν 8 0 , 3 4 ≈ 0.75 , γ p μ + q ≤ 1 , ν 9 ( γ , p μ ) = ν 9 0 , 3 4 ≈ 0.6094 , ( γ + q ) p μ ≤ 1 , ν 10 ( γ , p μ ) = ν 10 0 , 3 4 ≈ 0.2963 , 0 ≤ γ p μ ≤ 1 − q .

Also, we have

(4.3) | D p , q ϕ g ( ψ ) | = | D 1 , 1 3 0 ( 4 ψ + 1 ) | = 4 , | D p , q ϕ g ϕ m | = | D 1 , 1 3 0 ( 8 ϕ + 1 ) | = 3 , | D p , q ϕ g ( ϕ ) | = | D 1 , 1 3 0 ( 4 ϕ + 1 ) | = 0 , | D p , q ϕ g ψ m | = | D 1 , 1 3 0 ( 8 ψ + 1 ) | = 8 .

Observe that

(4.4) H 1 ( γ , p μ , α , m ) = ( ψ − ϕ ) { [ ν 1 ( γ , p μ , α ) + ν 3 ( γ , p μ , α ) − ν 5 ( γ , p μ , α ) ] | ϕ D p , q g ( ψ ) | + m ν 7 ( γ , p μ ) + ν 8 ( γ , p μ ) − ν 9 ( γ , p μ ) − ν 1 ( γ , p μ , α ) − ν 3 ( γ , p μ , α ) + ν 5 ( γ , p μ , α ) | ϕ D p , q g ϕ m | .

Substituting (4.2) and (4.3) in (4.4), and simple computations yield

(4.5) H 1 0 , 3 4 , 1 2 , 1 2 ≈ 4.3167 .

Analogously, we have

H 2 ( γ , p μ , α , m ) = ( ψ − ϕ ) { [ ν 2 ( γ , p μ , α ) + ν 4 ( γ , p μ , α ) − ν 6 ( γ , p μ , α ) ] | ϕ D p , q g ( ϕ ) | + m [ ν 7 ( γ , p μ ) + ν 8 ( γ , p μ ) − ν 9 ( γ , p μ ) − ν 2 ( γ , p μ , α ) − ν 4 ( γ , p μ , α ) + ν 6 ( γ , p μ , α ) ] | ϕ D p , q g ψ m | .

After simplification, we have

(4.6) H 2 0 , 3 4 , 1 2 , 1 2 ≈ 6.5 .

From (4.5) and (4.6), we get

(4.7) min [ H 1 ( γ , p μ , α , m ) , H 2 ( γ , p μ , α , m ) ] = min { 4.3167 , 6.5 } ≈ 4.3167 ,

which shows that the following implications hold in (4.1) and (4.7)

0 < 4.3167 .

Example 4.2

Let f , g : J ⊃ [ 0 , ∞ ) → ℝ be defined by f ( x ) = g ( x ) = x . Then these functions are ( p , q ) -differentiable functions on J° (the interior of J) and continuous and integrable on [0 , 10] with 0 ≤ 1 < 5 < ∞ . If f and g are ( 1 , 1 2 ) -convex on [ 0 , 10 ] with α = 1 and m = 1 2 , then all assumptions of Corollary 3.2 are satisfied.

Clearly,

(4.8) 1 ψ − ϕ ∫ ϕ ψ f ( x ) g ( x ) ϕ d p , q x = 1 4 ∫ 1 5 x 1 2 d p , q x

follows from Definition 1.3.

On the other hand,

(4.9) L 1 1 , 1 2 = [ 2 ] p , q + ( p 2 + p q + q 2 ) [ [ 2 ] p , q − 2 ] + 25 ( [ 2 ] p , q ) + 10 q 2 [ 2 ] p , q ( p 2 + p q + q 2 )

and

(4.10) L 2 1 , 1 2 = 25 ( [ 2 ] p , q + ( p 2 + p q + q 2 ) [ [ 2 ] p , q − 2 ] ) + ( [ 2 ] p , q ) + 10 q 2 [ 2 ] p , q ( p 2 + p q + q 2 ) .

From (4.8) and (4.10), we get

(4.11) min L 1 1, 1 2 , L 2 1, 1 2 = L 1 1, 1 2 ,

which shows that the following implications hold in (4.8) and (4.11)

1 4 ∫ 1 5 x  1 2 d p , q x < L 1 1 , 1 2

for every 0 < q < p ≤ 1 .

Remark 4.1

Similar technique can be applied to Theorems 3.2 and 3.3 to get the immediate consequences.

Acknowledgments

The authors express their gratitude to the referees for very helpful and detailed comments and suggestions, which have significantly improved the presentation of this paper. The research was supported by the Natural Science Foundation of China under grant numbers 11701176, 61673169, 11301127, 11626101 and 11601485.

  1. Conflict of interest: The authors declare that they have no competing interests.

  2. Author contributions: All authors contributed equally to the manuscript, and they read and approved the final manuscript.

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Received: 2019-09-05
Revised: 2020-08-28
Accepted: 2020-11-12
Published Online: 2020-12-31

© 2020 Humaira Kalsoom et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
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  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
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  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
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  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
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https://doi.org/10.1515/math-2020-0114
Keywords for this article
(p,q)-quantum calculus; Hermite-Hadamard inequality; Simpson’s type inequality; (α,m)-convex functions
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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