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Refinements of quantum Hermite-Hadamard-type inequalities

  • Hüseyin Budak , Sundas Khan , Muhammad Aamir Ali and Yu-Ming Chu EMAIL logo
Published/Copyright: July 30, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, we first obtain two new quantum Hermite-Hadamard-type inequalities for newly defined quantum integral. Then we establish several refinements of quantum Hermite-Hadamard inequalities.

Keywords: Hermite-Hadamard inequality; q-integral; quantum calculus; convex function
MSC 2010: 26D10; 26D15; 26A51

1 Introduction

The Hermite-Hadamard inequality discovered by C. Hermite and J. Hadamard (see, e.g., [1], [2, p. 137]) is one of the most well-established inequalities in the theory of convex functions with a geometrical interpretation and many applications. These inequalities state that if f : I → R is a convex function on the interval I of real numbers and ω 1 , ω 2 ∈ I with ω 1 < ω 2 , then

(1) f ω 1 + ω 2 2 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 f ( ϰ ) d ϰ ≤ f ( ω 1 ) + f ( ω 2 ) 2 .

Both inequalities hold in the reversed direction if f is concave. We note that the Hermite-Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen’s inequality. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years, and a remarkable variety of refinements and generalizations have been studied.

On the other hand, quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. In the field of q -analysis, many studies have recently been carried out. It has applications in numerous areas of mathematics such as combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, and in fields of other sciences such as mechanics, theory of relativity, and quantum theory [3,4,5, 6,7]. Apparently, Euler invented this important branch of mathematics when he used the q parameter in Newton’s work on infinite series. Later, the q -calculus was first given by Jackson [3]. In 1908–1909, the general form of the q -integral and q -difference operator was defined by Jackson [6]. In 1969, for the first time Agarwal [8] defined the q -fractional derivative. In 1966–1967, Al-Salam [9] introduced a q -analog of the q -fractional integral and q -Riemann-Liouville fractional. In 2004, Rajkovic gave a definition of the Riemann-type q -integral, which was generalized to the Jackson q -integral. In 2013, Tariboon introduced the D q ω 1 -difference operator [10].

In recent years, because of the importance of convexity in numerous fields of applied and pure mathematics, it has been significantly investigated. The theory of convexity and inequalities are strongly connected to each other, therefore, various inequalities can be established in the literature which are proved for convex, generalized convex, and differentiable convex functions of single and double variables, see, for example, [10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29].

The general structure of this paper consists of five main sections including Introduction. In Section 2, we give some necessary important notations for concept q -calculus and we also mention some related works in the literature. In Section 3 we present some new Hermite-Hadamard-type inequalities for q ω 2 integrals. Some refinements of quantum Hermite-Hadamard-type inequalities are proved in Section 4. We also examine the relation between our results and inequalities presented in the earlier works. Finally, in Section 5, some conclusions and further directions of research are discussed. We note that the opinion and technique of this work may inspire new research in this area.

2 Preliminaries of q -calculus and some inequalities

In this section, we present some required definitions and related inequalities about q -calculus.

We have to give the following notation which will be used many times in the following sections (see [7]):

[ n ] q = q n − 1 q − 1 .

Definition 1

[30] For a function F : [ ω 1 , ω 2 ] → R , the q ω 1 -derivative of F at ϰ ∈ [ ω 1 , ω 2 ] is characterized by the expression:

(2) D q ω 1 F ( ϰ ) = F ( ϰ ) − F ( q ϰ + ( 1 − q ) ω 1 ) ( 1 − q ) ( ϰ − ω 1 ) , ϰ ≠ ω 1 .

If ϰ = ω 1 , we define D q ω 1 F ( ω 1 ) = lim ϰ → ω 1 D q F ( ϰ ) if it exists and it is finite.

Definition 2

[18] For a function F : [ ω 1 , ω 2 ] → R , the q ω 2 -derivative of F at ϰ ∈ [ ω 1 , ω 2 ] is characterized by the expression:

D q ω 2 F ( ϰ ) = F ( q ϰ + ( 1 − q ) ω 2 ) − F ( ϰ ) ( 1 − q ) ( ω 2 − ϰ ) , ϰ ≠ ω 2 .

If ϰ = ω 2 , we define D q ω 2 F ( ω 2 ) = lim ϰ → ω 2 D q ω 2 F ( ϰ ) if it exists and it is finite.

Definition 3

[30] Let F : [ ω 1 , ω 2 ] → R be a function. Then, the q ω 1 -definite integral on [ ω 1 , ω 2 ] is defined as

∫ ω 1 ω 2 F ( ϰ ) d q ω 1 ϰ = ( 1 − q ) ( ω 2 − ω 1 ) ∑ n = 0 ∞ q n F ( q n ω 2 + ( 1 − q n ) ω 1 ) = ( ω 2 − ω 1 ) ∫ 0 1 F ( ( 1 − t ) ω 1 + t ω 2 ) d q t .

In [10], Alp et al. proved the following q ω 1 -Hermite-Hadamard inequalities for convex functions in the setting of quantum calculus:

Theorem 1

Let F : [ ω 1 , ω 2 ] → R be a convex differentiable function on [ ω 1 , ω 2 ] and 0 < q < 1 . Then q -Hermite-Hadamard inequalities are as follows:

(3) F q ω 1 + ω 2 [ 2 ] q ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 1 ϰ ≤ q F ( ω 1 ) + F ( ω 2 ) [ 2 ] q .

On the other hand, Bermudo et al. gave the following new definition and were able to prove the corresponding Hermite-Hadamard-type inequalities:

Definition 4

[18] Let F : [ ω 1 , ω 2 ] → R be a function. Then, the q ω 2 -definite integral on [ ω 1 , ω 2 ] is defined as

∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ = ( 1 − q ) ( ω 2 − ω 1 ) ∑ n = 0 ∞ q n F ( q n ω 1 + ( 1 − q n ) ω 2 ) = ( ω 2 − ω 1 ) ∫ 0 1 F ( t ω 1 + ( 1 − t ) ω 2 ) d q t .

Theorem 2

[31] We have the following identities for q ω 2 -integrals:

  1. D q ω 2 ∫ ϰ ω 2 F ( t ) d q ω 2 t = − F ( ϰ ) .

  2. ∫ ϰ ω 2 D q ω 2 F ( t ) d q ω 2 t = F ( ω 2 ) − F ( ϰ ) .

  3. ∫ ϰ ω 2 [ F ( t ) + G ( t ) ] d q ω 2 t = ∫ ϰ ω 2 F ( t ) d q ω 2 t + ∫ ϰ ω 2 G ( t ) d q ω 2 t .

  4. ∫ ω 1 ω 2 G ( x ) D q ω 2 F ( ϰ ) d q ω 2 ϰ = [ F ( ϰ ) G ( ϰ ) ] ∣ ω 1 ω 2 − ∫ ω 1 ω 2 F ( q ϰ + ( 1 − q ) ω 2 ) D q ω 2 G ( ϰ ) d q ω 2 ϰ .

Theorem 3

[18] Let F : [ ω 1 , ω 2 ] → R be a convex function on [ ω 1 , ω 2 ] and 0 < q < 1 . Then, q -Hermite-Hadamard inequalities are as follows:

(4) F ω 1 + q ω 2 [ 2 ] q ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≤ F ( ω 1 ) + q F ( ω 2 ) [ 2 ] q .

3 New Hermite-Hadamard-type inequalities for q ω 2 -integrals

In this section, we prove two new quantum Hermite-Hadamard inequalities for q ω 2 -integrals.

Theorem 4

If F : [ ω 1 , ω 2 ] → R is a convex differentiable function on [ ω 1 , ω 2 ] and 0 < q < 1 , then we have

(5) F q ω 1 + ω 2 [ 2 ] q − ( 1 − q ) ( ω 2 − ω 1 ) [ 2 ] q F ′ q ω 1 + ω 2 [ 2 ] q ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≤ F ( ω 1 ) + q F ( ω 2 ) [ 2 ] q .

Proof

We can write the equation of the tangent line for the function F at the point q ω 1 + ω 2 [ 2 ] q , F q ω 1 + ω 2 [ 2 ] q as follows:

h ( ϰ ) = F q ω 1 + ω 2 [ 2 ] q + F ′ q ω 1 + ω 2 [ 2 ] q ϰ − q ω 1 + ω 2 [ 2 ] q

for all ϰ ∈ [ ω 1 , ω 2 ] . Since F is a convex function, it is clear that h ( ϰ ) ≤ F ( ϰ ) for all ϰ ∈ [ ω 1 , ω 2 ] . Thus, we have

∫ ω 1 ω 2 h ( ϰ ) d q ω 2 ϰ ≤ ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ .

By Definition 4, we get

∫ ω 1 ω 2 h ( ϰ ) d q ω 2 ϰ = ∫ ω 1 ω 2 F q ω 1 + ω 2 [ 2 ] q + F ′ q ω 1 + ω 2 [ 2 ] q ϰ − q ω 1 + ω 2 [ 2 ] q d q ω 2 ϰ = ( ω 2 − ω 1 ) F q ω 1 + ω 2 [ 2 ] q + F ′ q ω 1 + ω 2 [ 2 ] q ∫ ω 1 ω 2 ϰ − q ω 1 + ω 2 [ 2 ] q d q ω 2 ϰ = ( ω 2 − ω 1 ) F q ω 1 + ω 2 [ 2 ] q + F ′ q ω 1 + ω 2 [ 2 ] q ( 1 − q ) ( ω 2 − ω 1 ) ∑ n = 0 ∞ q n ( q n ω 1 + ( 1 − q n ) ω 2 ) − ( ω 2 − ω 1 ) q ω 1 + ω 2 [ 2 ] q = ( ω 2 − ω 1 ) F q ω 1 + ω 2 [ 2 ] q − ( 1 − q ) ( ω 2 − ω 1 ) 2 [ 2 ] q F ′ q ω 1 + ω 2 [ 2 ] q ,

which gives the first inequality in (5). The second inequality is the same as in Theorem 3.□

Remark 1

If we take the limit q → 1 − in Theorem 4, then the inequalities (5) reduce to (1).

Theorem 5

If F : [ ω 1 , ω 2 ] → R is a convex differentiable function on [ ω 1 , ω 2 ] and 0 < q < 1 , then we have

(6) F ω 1 + ω 2 2 − ( 1 − q ) ( ω 2 − ω 1 ) 2 [ 2 ] q F ′ ω 1 + ω 2 2 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≤ F ( ω 1 ) + q F ( ω 2 ) [ 2 ] q .

Proof

Similar way as in Theorem 4, we can write tangent line for the function F at the point ω 1 + ω 2 2 , F ω 1 + ω 2 2 as follows:

k ( ϰ ) = F ω 1 + ω 2 2 + F ′ ω 1 + ω 2 2 ϰ − ω 1 + ω 2 2

for all ϰ ∈ [ ω 1 , ω 2 ] . Since F is a convex function, we get

∫ ω 1 ω 2 k ( ϰ ) d q ω 2 ϰ ≤ ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ .

By Definition 4, we get

∫ ω 1 ω 2 k ( ϰ ) d q ω 2 ϰ = ∫ ω 1 ω 2 F ω 1 + ω 2 2 + F ′ ω 1 + ω 2 2 ϰ − ω 1 + ω 2 2 d q ω 2 ϰ = ( ω 2 − ω 1 ) F ω 1 + ω 2 2 + F ′ ω 1 + ω 2 2 ∫ ω 1 ω 2 ϰ − ω 1 + ω 2 2 d q ω 2 ϰ = ( ω 2 − ω 1 ) F ω 1 + ω 2 2 + F ′ ω 1 + ω 2 2 ( 1 − q ) ( ω 2 − ω 1 ) ∑ n = 0 ∞ q n ( q n ω 1 + ( 1 − q n ) ω 2 ) − ( ω 2 − ω 1 ) ω 1 + ω 2 2 = ( ω 2 − ω 1 ) F ω 1 + ω 2 2 − ( 1 − q ) ( ω 2 − ω 1 ) 2 2 [ 2 ] q F ′ ω 1 + ω 2 2 .

This gives the first inequality in (6). The second inequality is the same as in Theorem 3.□

Remark 2

If we take the limit q → 1 − in Theorem 5, then the inequalities (6) reduce to (1).

Lemma 1

Let F : [ ω 1 , ω 2 ] → R be a convex continuous function on [ ω 1 , ω 2 ] and 0 < q < 1 . Then we have

F 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y ≤ 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y .

Proof

Lemma 1 follows directly from Definition 3 and quantum Jensen’s inequality.□

4 Main results

In this section, we present the refinements of quantum Hermite-Hadamard inequalities for q ω 2 -integrals.

Theorem 6

Let F : [ ω 1 , ω 2 ] → R be a convex continuous function on [ ω 1 , ω 2 ] and 0 < q < 1 . Then we have

(7) F ω 1 + q ω 2 [ 2 ] q ≤ 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≤ F ( ω 1 ) + q F ( ω 2 ) [ 2 ] q

for all t ∈ [ 0 , 1 ] .

Proof

Since F is convex on [ ω 1 , ω 2 ] , it follows that

(8) F ( t ϰ + ( 1 − t ) y ) ≤ t F ( ϰ ) + ( 1 − t ) F ( y )

for all ϰ , y ∈ [ ω 1 , ω 2 ] and t ∈ [ 0 , 1 ] . Taking double q ω 2 -integration on both sides of (8) on [ ω 1 , ω 2 ] × [ ω 1 , ω 2 ] , we obtain

∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y ≤ ∫ ω 1 ω 2 ∫ ω 1 ω 2 [ t F ( ϰ ) + ( 1 − t ) F ( y ) ] d q ω 2 ϰ d q ω 2 y = t ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ d q ω 2 y + ( 1 − t ) ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( y ) d q ω 2 ϰ d q ω 2 y = t ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ + ( 1 − t ) ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( y ) d q ω 2 y = ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ,

which proves the second part of (7). On the other hand, by Lemma 1, we have

(9) F 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y ≤ 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y

and we also get

(10) 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y = ω 1 + q ω 2 [ 2 ] q .

By (9) and (10), we have the first part of (7).□

Remark 3

If we take the limit q → 1 − in Theorem 6, then the inequalities (7) reduce to the following inequalities:

F ω 1 + ω 2 2 ≤ 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d ϰ d y ≤ 1 ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d ϰ ≤ F ( ω 1 ) + F ( ω 2 ) 2 ,

which are given by Dragomir in [32].

Corollary 1

Under assumptions of Theorem 6 with t = 1 2 , we have the inequalities

(11) F q ω 2 + ω 1 [ 2 ] q ≤ 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ϰ + y 2 d q ω 2 ϰ d q ω 2 y ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≤ q F ( ω 2 ) + F ( ω 1 ) [ 2 ] q .

Remark 4

If we take the limit q → 1 − in Corollary 1, then (11) reduces to the inequalities

F ω 1 + ω 2 2 ≤ 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ϰ + y 2 d ϰ d y ≤ 1 ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d ϰ ≤ F ( ω 1 ) + F ( ω 2 ) 2 ,

which are given in [33].

Theorem 7

Let F : [ ω 1 , ω 2 ] → R be a convex continuous function on [ ω 1 , ω 2 ] and 0 < q < 1 . Then we have

(12) 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F q ϰ + y [ 2 ] q d q ω 2 ϰ d q ω 2 y ≤ 1 ( ω 2 − ω 1 ) 2 ∫ 0 1 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y d q ω 2 t ≤ 1 ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ .

Proof

Let us consider the mapping ϕ : [ 0 , 1 ] → R defined by

ϕ ( t ) = 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y .

For all t 1 , t 2 ∈ [ 0 , 1 ] and α , β ≥ 0 with α + β = 1 , by convexity of F , we have

ϕ ( α t 1 + β t 2 ) = 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( ( α t 1 + β t 2 ) ϰ + ( 1 − ( α t 1 + β t 2 ) ) y ) d q ω 2 ϰ d q ω 2 y = 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( α ( t 1 ϰ + ( 1 − t 1 ) y ) + β ( t 2 ϰ + ( 1 − t 2 ) y ) ) d q ω 2 ϰ d q ω 2 y ≤ α ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t 1 ϰ + ( 1 − t 1 ) y ) d q ω 2 ϰ d q ω 2 y + β ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t 2 ϰ + ( 1 − t 2 ) y ) d q ω 2 ϰ d q ω 2 y = α ϕ ( t 1 ) + β ϕ ( t 2 ) ,

which shows that ϕ is convex on [ 0 , 1 ] . By applying Theorem 3 for the convex function ϕ on [ 0 , 1 ] , we have the inequalities

ϕ q [ 2 ] q ≤ ∫ 0 1 ϕ ( t ) d q 1 t ≤ ϕ ( 0 ) + q ϕ ( 1 ) [ 2 ] q .

That is,

1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F q ϰ + y [ 2 ] q d q ω 2 ϰ d q ω 2 y ≤ 1 ( ω 2 − ω 1 ) 2 ∫ 0 1 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y d q 1 t ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ .

This completes the proof.□

Remark 5

If we take the limit q → 1 − in Theorem 7, then the inequalities (12) reduce to the inequalities

1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ϰ + y 2 d ϰ d y ≤ 1 ( ω 2 − ω 1 ) 2 ∫ 0 1 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d ϰ d y d t ≤ 1 ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d ϰ ,

which are given in [33].

Theorem 8

Let F : [ ω 1 , ω 2 ] → R be a q -differentiable convex continuous function and 0 < q < 1 . Then the inequalities

(13) 0 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ − 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y ≤ t q F ( ω 1 ) + F ( ω 2 ) [ 2 ] q − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( q ϰ + ( 1 − q ) ω 2 ) d q ω 2 ϰ

are valid for all t ∈ [ 0 , 1 ] .

Proof

Since F is convex on [ ω 1 , ω 2 ] , it follows that

(14) F ( t ϰ + ( 1 − t ) y ) ≤ t F ( ϰ ) + ( 1 − t ) F ( y )

for all ϰ , y ∈ [ ω 1 , ω 2 ] and t ∈ [ 0 , 1 ] . Taking double q ω 2 -integration on both sides of (14) on [ ω 1 , ω 2 ] × [ ω 1 , ω 2 ] , we obtain

∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y ≤ ∫ ω 1 ω 2 ∫ ω 1 ω 2 [ t F ( ϰ ) + ( 1 − t ) F ( y ) ] d q ω 2 ϰ d q ω 2 y = ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ,

which gives the first part of (13). On the other hand, since F is q -differentiable convex on [ ω 1 , ω 2 ] and F ′ ≥ D q ω 2 F , we have

F ( t ϰ + ( 1 − t ) y ) − F ( y ) ≥ t ( ϰ − y ) D q ω 2 F ( y )

for all ϰ , y ∈ [ ω 1 , ω 2 ] and t ∈ [ 0 , 1 ] . Taking double q ω 2 -integration on both sides of the above inequality on [ ω 1 , ω 2 ] × [ ω 1 , ω 2 ] , we obtain

(15) ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y − ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≥ t ∫ ω 1 ω 2 ∫ ω 1 ω 2 ( ϰ − y ) D q ω 2 F ( y ) d q ω 2 ϰ d q ω 2 y .

By (iv) of Theorem 2, we have

∫ ω 1 ω 2 ∫ ω 1 ω 2 ( ϰ − y ) F D q ω 2 F ( y ) d q ω 2 ϰ d q ω 2 y = ( ω 2 − ω 1 ) ∫ ω 1 ω 2 a + q b [ 2 ] q − y D q ω 2 F ( y ) d q ω 2 y = ( ω 2 − ω 1 ) a + q b [ 2 ] q − y F ( y ) ω 1 ω 2 − ∫ ω 1 ω 2 F ( q y + ( 1 − q ) b ) D q ω 2 a + q b 1 + q − y d q ω 2 y = ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ − ( ω 2 − ω 1 ) 2 q F ( ω 1 ) + F ( ω 2 ) [ 2 ] q .

It follows from (15) that

( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ − ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d q ω 2 ϰ d q ω 2 y ≤ t ( ω 2 − ω 1 ) 2 q F ( ω 1 ) + F ( ω 2 ) [ 2 ] q − ( ω 2 − ω 1 ) ∫ ω 1 ω 2 F ( q ϰ + ( q − 1 ) ω 2 ) d q ω 2 ϰ ,

for all t ∈ [ 0 , 1 ] , which is the second part of (13).□

Remark 6

If we take the limit q → 1 − in Theorem 8, then the inequalities (13) reduce to the inequalities

0 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d ϰ − 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ( t ϰ + ( 1 − t ) y ) d ϰ d y ≤ t F ( ω 1 ) + F ( ω 2 ) 2 − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d ϰ ,

which can be seen in [33,34].

Corollary 2

Under assumptions of Theorem 8 with t = 1 2 , we have the inequalities

(16) 0 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ − 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ϰ + y 2 d q ω 2 ϰ d q ω 2 y ≤ 1 2 q F ( ω 1 ) + F ( ω 2 ) [ 2 ] q − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( q ϰ + ( 1 − q ) ω 2 ) d q ω 2 ϰ .

Remark 7

If we take the limit q → 1 − in Corollary 2, then the inequalities (16) reduce to the inequalities

0 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d ϰ − 1 ( ω 2 − ω 1 ) 2 ∫ ω 1 ω 2 ∫ ω 1 ω 2 F ϰ + y 2 d ϰ d y ≤ 1 2 F ( ω 1 ) + F ( ω 2 ) 2 − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d ϰ ,

which are given in [33].

Theorem 9

Let F : [ ω 1 , ω 2 ] → R be a q -differentiable convex continuous function which is defined at the point ω 1 + q ω 2 [ 2 ] q ∈ ( ω 1 , ω 2 ) and 0 < q < 1 . Then the following inequalities hold:

(17) 0 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F t ϰ + ( 1 − t ) q ω 2 + ω 1 [ 2 ] q d q ω 2 ϰ ≤ ( 1 − t ) q F ( ω 1 ) + F ( ω 2 ) [ 2 ] q − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( q ϰ + ( 1 − q ) ω 2 ) d q ω 2 ϰ .

Proof

Since F is convex on [ ω 1 , ω 2 ] , by Theorem 3, it follows that

1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ = t ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ + 1 − t ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≥ t ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ + ( 1 − t ) F ω 1 + q ω 2 [ 2 ] q ≥ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F t ϰ + ( 1 − t ) ω 1 + q ω 2 [ 2 ] q d q ω 2 ϰ

for all t ∈ [ 0 , 1 ] , which becomes the first part of (17). On the other hand, since F is q -differentiable convex on [ ω 1 , ω 2 ] , we have

F t ϰ + ( 1 − t ) ω 1 + q ω 2 [ 2 ] q − F ( ϰ ) ≥ ( 1 − t ) ω 1 + q ω 2 [ 2 ] q − ϰ D q ω 2 F ( ϰ ) .

Taking the q ω 2 -integration on the above inequality on [ ω 1 , ω 2 ] , we obtain

(18) 1 ω 2 − ω 1 ∫ ω 1 ω 2 F t ϰ + ( 1 − t ) ω 1 + q ω 2 [ 2 ] q d q ω 2 ϰ − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ ≥ 1 ω 2 − ω 1 ∫ ω 1 ω 2 ( 1 − t ) ω 1 + q ω 2 [ 2 ] q − ϰ D q ω 2 F ( ϰ ) d q ω 2 ϰ .

We also have

(19) ∫ ω 1 ω 2 ω 1 + q ω 2 [ 2 ] q − ϰ ω 2 D q F ( ϰ ) d q ω 2 ϰ = ∫ ω 1 ω 2 F ( q ϰ + ( 1 − q ) ω 2 ) ω 2 d q ϰ − ( ω 2 − ω 1 ) q F ( ω 1 ) + F ( ω 2 ) [ 2 ] q .

From (18) and (19), we get the second part of (17).□

Theorem 10

Let F : [ ω 1 , ω 2 ] → R be a q -differentiable convex continuous function which is defined at the point q ω 1 + ω 2 [ 2 ] q ∈ ( ω 1 , ω 2 ) and 0 < q < 1 . Then the inequalities

(20) ( 1 − t ) ( 1 − q ) ( ω 2 − ω 1 ) [ 2 ] q F ′ q ω 1 + ω 2 [ 2 ] q ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F t ϰ + ( 1 − t ) q ω 1 + ω 2 [ 2 ] q d q ω 2 ϰ ≤ ( 1 − t ) q F ( ω 2 ) + F ( ω 1 ) [ 2 ] q − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( q ϰ + ( 1 − q ) ω 2 ) d q ω 2 ϰ

are valid for all t ∈ [ 0 , 1 ] .

Proof

The proof of this theorem follows a similar procedure to that in Theorem 9 by using Theorem 4.□

Theorem 11

Let F : [ ω 1 , ω 2 ] → R be a q -differentiable convex continuous function which is defined at the point ω 1 + ω 2 2 ∈ ( ω 1 , ω 2 ) and 0 < q < 1 . Then the inequalities

(21) ( 1 − t ) ( 1 − q ) ( ω 2 − ω 1 ) 2 ( [ 2 ] q ) F ′ ω 1 + ω 2 2 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d q ω 2 ϰ − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F t ϰ + ( 1 − t ) ω 1 + ω 2 2 d q ω 2 ϰ ≤ ( 1 − t ) F ( ω 1 ) + F ( ω 2 ) 2 − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( q ϰ + ( 1 − q ) ω 2 ) d q ω 2 ϰ

are valid for all t ∈ [ 0 , 1 ] .

Proof

The proof of this theorem follows a similar procedure to that in Theorem 9 by using Theorem 5.□

Remark 8

If we take the limit q → 1 − , then the inequalities (17), (20) and (21) reduce to

0 ≤ 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d ϰ − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F t ϰ + ( 1 − t ) ω 1 + ω 2 2 d ϰ ≤ ( 1 − t ) F ( ω 1 ) + F ( ω 2 ) 2 − 1 ω 2 − ω 1 ∫ ω 1 ω 2 F ( ϰ ) d ϰ ,

which are given in [33].

Acknowledgement

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

  1. Funding information: This work was supported by the Natural Science Foundation of China (Grant No. 11971241).

  2. Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.

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Received: 2020-07-09
Revised: 2021-01-07
Accepted: 2021-01-07
Published Online: 2021-07-30

© 2021 Hüseyin Budak et al., published by De Gruyter

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

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https://doi.org/10.1515/math-2021-0029
Keywords for this article
Hermite-Hadamard inequality; q-integral; quantum calculus; convex function
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  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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