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New extensions related to Fejér-type inequalities for GA-convex functions

  • Muhammad Amer Latif EMAIL logo
Published/Copyright: August 2, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

In this study, some mappings related to the Fejér-type inequalities for G A -convex functions are defined over the interval [ 0 , 1 ] . Some Fejér-type inequalities for G A -convex functions are proved using these mappings. Properties of these mappings are considered and consequently we obtain refinements of some known results.

Keywords: Hermite-Hadamard inequality; convex function; harmonic GA-convex function; Fejér inequality
MSC 2010: 26D15; 26D20; 26D07

1 Introduction

For convex functions, the following double inequality has great significance in the literature and is known as Hermite-Hadamard’s inequality [1,2]:

Let φ : I ⟶ R , ∅ ≠ I ⊆ R , ν 1 , ν 2 ∈ I with ν 1 < ν 2 be a convex function, then

(1.1) φ ν 1 + ν 2 2 ≤ 1 ν 2 − ν 1 ∫ ν 1 ν 2 φ ( ξ ) d ξ ≤ φ ( ν 1 ) + φ ( ν 2 ) 2 ,

the inequality (1.1) holds in reversed direction if φ is concave.

Fejér [3] established the following double inequality as a weighted generalization of (1.1):

(1.2) φ ν 1 + ν 2 2 ∫ ν 1 ν 2 ϑ ( ξ ) d ξ ≤ ∫ ν 1 ν 2 φ ( ξ ) ϑ ( ξ ) d ξ ≤ φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) d ξ ,

where φ : I ⟶ R and ∅ ≠ I ⊆ R , ν 1 , ν 2 ∈ I with ν 1 < ν 2 is any convex function and ϑ : [ ν 1 , ν 2 ] → R is non-negative integrable and symmetric about ξ = ν 1 + ν 2 2 .

Inequalities (1.1) and (1.2) have many extensions and generalizations. We refer the readers to [4,5], the studies carried out by Ardic et al. which deal with the Ostrowski-type inequalities for G G -convex and G A -convex functions and some other important inequalities for inequalities via G G -convexity and G A -convexity, respectively. Dragomir and Latif discussed in their studies some important Fejér-type integral inequalities related to the geometrically-arithmetically convex functions and their applications in [6–17]. In [18,19], Kunt and İşcan have proven very interesting results of Hermite-Hadamard and Fejér-type inequalities for G A -convex for G A - s -convex functions. The author has provided results of Hermite-Hadamard-type and weighted Hermite-Hadamard-type using differentiable G A -convex, coordinated G A -convex, and geometrically-quasi-convex mappings, and the notion of geometrically symmetric mappings in [20–22]. Further research studies have been accomplished concomitant to the Hermite-Hadamard and Fejér-type results using the notions of geometrically-arithmetically, geometrically quasi convex functions, and a no-negative integrable geometrically symmetric function in [23,24,25]. Here we also mention the contributions of the mathematicians, for instance, Noor et al. [26] who proved inequalities for geometrically-arithmetically h -convex functions, Obeidat and Latif [27] obtained the results of weighted Hermite-Hadamard-type inequalities using geometrically quasi-convex functions and a no-negative integrable geometrically symmetric function, Qi and Xi [28] accomplished some new Hermite-Hadamard-type inequalities for geometrically quasi-convex functions, and Zhang et al. [29] who proved Hermite-Hadamard-type inequalities for differentiable G A -convex functions.

Many important results that characterize the properties of the mappings related to Inequalities (1.1) and (1.2), and inequalities that provide refinements of Inequalities (1.1) and (1.2) are discussed by a number of researchers. Dragomir et al. [30] considered inequalities of Hadamard’s type for Lipschitzian mappings. Dragomir et al. [31] gave refinements of Hadamard’s inequalities. In another study, Dragomir [32] proved Hadamard’s inequality for convex functions by defining some functionals. Dragomir and Agarwal complemented the study carried out in [33] by defining new mappings associated with Hadamard’s inequalities for convex functions and obtained some refinements of (1.1). Dragomir [34] further investigated mappings in connection to Hadamard’s inequalities and obtained some more refinements to (1.1). Tseng et al. [35–37] generalized the results given in [30–34] and acquired new inequalities of Hermite-Hadamard-Fejér-type involving convex functions and a weight function that is non-negative integrable symmetric with respect to the mean of the closed interval. Yang and Hong [38] also proved some refinements of (1.1) by considering the properties of some functionals. The interested readers are referred to Yang and Tseng [39–42] for more results of properties of functionals in connection to (1.1) and (1.2), and results which refine and generalize the inequalities (1.1) and (1.2) [43,44].

One of the generalizations of the convex functions is geometrically-arithmetically convex functions also known as G A -convex functions which is stated as follows:

Definition 1

[45] A function φ : I ⊆ ( 0 , ∞ ) → R is considered to be G A -convex, if

(1.3) φ ( ξ ϱ λ 1 − ϱ ) ≤ ϱ φ ( ξ ) + ( 1 − ϱ ) φ ( λ )

for all ξ , λ ∈ I and ϱ ∈ [ 0 , 1 ] . A function φ : I → R is concave if the inequality in (1.3) is reversed.

We state some important facts that relate G A -convex and convex functions and use them to prove the main results.

Theorem 1

[45] If  [ ν 1 , ν 2 ] ⊂ ( 0 , ∞ ) and the function χ : [ ln ν 1 , ln ν 2 ] → R is convex (concave) on [ ln ν 1 , ln ν 2 ] , then the function φ : [ ν 1 , ν 2 ] → R , φ ( ϱ ) = χ ( ln ϱ ) is GA-convex (concave) on [ ν 1 , ν 2 ] .

Remark 1

It is obvious from Theorem 1 that if φ : [ ν 1 , ν 2 ] → R is G A -convex on [ ν 1 , ν 2 ] ⊂ ( 0 , ∞ ) , then φ ∘ exp is convex on [ ln ν 1 , ln ν 2 ] . It follows that φ ∘ exp has finite lateral derivatives on ( ln ν 1 , ln ν 2 ) and by gradient inequality for convex functions we have

φ ∘ exp ( ξ ) − φ ∘ exp ( λ ) ( ξ − λ ) ≥ φ ( exp λ ) exp ( λ ) ,

where φ ( exp λ ) ∈ [ φ − ′ ( exp λ ) , φ + ′ ( exp λ ) ] for any ξ , λ ∈ ( ln ν 1 , ln ν 2 ) .

Theorem 2

(Jensen’s inequality for GA-convex functions) [45,46] Let φ : I ⊂ ( 0 , ∞ ) → R be a GA-convex function and [ k , K ] ⊂ I ∘ . Assume also that h : Ω → R is μ -measurable, satisfying the bounds

0 < k ≤ h ( ϱ ) ≤ K < ∞ for μ -a.e. ϱ ∈ Ω

and w ≥ 0   μ -a.e. on Ω with ∫ Ω w d μ = 1 . If φ ∈ ∂ φ the subdifferential of φ and φ ∘ h , ln h ∈ L w ( Ω , μ ) , then

(1.4) φ ∘ exp ∫ Ω w ln h d μ ≤ ∫ Ω ( φ ∘ h ) w d μ .

The following inequality of Hermite-Hadamard-type for G A -convex functions holds ([26] for an extension for G A h -convex functions):

Theorem 3

[26] Let φ : I ⊆ ( 0 , ∞ ) → R be a GA-convex function and ν 1 , ν 2 ∈ I with ν 1 < ν 2 . If φ ∈ L ( [ ν 1 , ν 2 ] ) , then the following inequalities hold:

(1.5) φ ( ν 1 ν 2 ) ≤ 1 ln ν 2 − ln ν 1 ∫ ν 2 ν 1 φ ( ξ ) ξ d ξ ≤ φ ( ν 1 ) + φ ( ν 2 ) 2 .

The notion of geometrically symmetric functions was introduced in [23].

Definition 2

[23] A function ϑ : [ ν 1 , ν 2 ] ⊆ ( 0 , ∞ ) → R is geometrically symmetric with respect to ν 1 ν 2 if

ϑ ( ξ ) = ϑ ν 1 ν 2 ξ

holds for all ξ ∈ [ ν 1 , ν 2 ] .

Fejér-type inequalities using G A -convex functions and the notion of geometric symmetric functions were presented in Latif et al. [23].

Theorem 4

[23] Let φ : I ⊆ ( 0 , ∞ ) → R be a GA-convex function and ν 1 , ν 2 ∈ I with ν 1 < ν 2 . If φ ∈ L ( [ ν 1 , ν 2 ] ) and ϑ : [ ν 1 , ν 2 ] ⊆ R \ { 0 } → R is non-negative, integrable, and geometrically symmetric with respect to ν 1 ν 2 , then

(1.6) φ ( ν 1 ν 2 ) ∫ ν 2 ν 1 ϑ ( ξ ) ξ d ξ ≤ ∫ ν 2 ν 1 φ ( ξ ) ϑ ( ξ ) ξ d ξ ≤ φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 2 ν 1 ϑ ( ξ ) ξ d ξ .

Suppose that φ : I ⊆ ( 0 , ∞ ) → R is G A -convex on I and ν 1 , ν 2 ∈ I , let χ , ℱ , V : [ 0 , 1 ] → R be defined by

χ ( ϱ ) = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 1 ξ φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) d ξ ,

ℱ ( ϱ ) = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 ∫ ν 1 ν 2 1 ξ λ φ ( ξ ϱ λ 1 − ϱ ) d ξ d λ ,

and

V ( ϱ ) = 1 2 ( ln ν 2 − ln ν 1 ) ∫ ν 1 ν 2 1 ξ φ ν 2 1 + ϱ 2 ξ 1 − ϱ 2 + φ ν 1 1 + ϱ 2 ξ 1 − ϱ 2 d ξ .

Latif et al. [47] obtained the following refinements for Inequalities (1.5):

Theorem 5

[47] A function φ : I ⊆ ( 0 , ∞ ) → R as above. Then,

  1. χ is GA-convex on ( 0 , 1 ] .

  2. We have

    inf ϱ ∈ [ 0 , 1 ] χ ( ϱ ) = χ ( 0 ) = φ ( ν 1 ν 2 )

    and

    sup ϱ ∈ [ 0 , 1 ] χ ( ϱ ) = χ ( 1 ) = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 φ ( ξ ) ξ d ξ .

  3. χ increases monotonically on [ 0 , 1 ] .

The following theorem holds:

Theorem 6

[47] Let φ : [ ν 1 , ν 2 ] ⊆ ( 0 , ∞ ) → R be as above. Then,

  1. ℱ ( ϱ + 1 2 ) = ℱ ( 1 2 − ϱ ) for all ϱ in [ 0 , 1 2 ] .

  2. ℱ is GA-convex on ( 0 , 1 ] .

  3. We have

    sup ϱ ∈ [ 0 , 1 ] ℱ ( ϱ ) = ℱ ( 0 ) = ℱ ( 1 ) = 1 ( ln ν 2 − ln ν 1 ) 2 ∫ ν 1 ν 2 1 ξ φ ( ξ ) d ξ

    and

    inf ϱ ∈ [ 0 , 1 ] ℱ ( ϱ ) = ℱ 1 2 = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 ∫ ν 1 ν 2 1 ξ λ φ ( ξ λ ) d ξ d λ .

  4. The following inequality is valid:

    φ ( ξ λ ) ≤ ℱ 1 2 .

  5. ℱ decreases monotonically on [ 0 , 1 2 ] and increases monotonically on 1 2 , 1 .

  6. We have the inequality χ ( ϱ ) ≤ ℱ ( ϱ ) for all ϱ ∈ [ 0 , 1 ] .

Theorem 7

[47] Let V : [ 0 , 1 ] → R and φ : [ ν 1 , ν 2 ] ⊂ ( 0 , ∞ ) → R be as defined above. Then,

  1. V is GA-convex on ( 0 , 1 ] .

  2. The following hold:

    inf ϱ ∈ [ 0 , 1 ] V ( ϱ ) = V ( 0 ) = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 φ ( ξ ) ξ d ξ

    and

    sup ϱ ∈ [ 0 , 1 ] V ( ϱ ) = V ( 1 ) = φ ( ν 1 ) + φ ( ν 2 ) 2 .

  3. V increases monotonically on [ 0 , 1 ] .

Motivated by the studies conducted in [48,30–42], we define some new functionals involving G A -convex functions and a non-negative integrable symmetric weight functions which is with respect to the geometric mean of the end points of the closed interval in connection to Inequalities (1.5) and (1.6) to prove new Féjer-type inequalities which indeed provide refinement inequalities as well.

2 Main results

Let us now define some mappings on [ 0 , 1 ] related to (1.6) and prove some refinement inequalities.

χ 1 ( ϱ ) = 1 2 φ ν 1 1 + ϱ 2 ν 2 1 − ϱ 2 + φ ν 1 1 − ϱ 2 ν 2 1 + ϱ 2 ,

χ ( ϱ ) = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 1 ξ φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) d ξ ,

χ ϑ ( ϱ ) = ∫ ν 1 ν 2 φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) ϑ ( ξ ) ξ d ξ ,

T ( ϱ ) = 1 2 ( ln ν 2 − ln ν 1 ) ∫ ν 1 ν 2 1 ξ [ φ ( ν 1 ϱ ξ 1 − ϱ ) + φ ( ν 2 ϱ ξ 1 − ϱ ) ] d ξ ,

and

T ϑ ( ϱ ) = 1 2 ∫ ν 1 ν 2 [ φ ( ν 1 ϱ ξ 1 − ϱ ) + φ ( ν 2 ϱ ξ 1 − ϱ ) ] ϑ ( ξ ) ξ d ξ ,

where φ : [ ν 1 , ν 2 ] → R is a G A -convex function and ϑ : [ ν 1 , ν 2 ] → R is non-negative integrable and symmetric about ξ = ν 1 ν 2 .

The following result is very important to establish the results of this section.

Lemma 1

[24] Let φ : [ ν 1 , ν 2 ] → R be a GA-convex function and let ν 1 ≤ λ 1 ≤ ξ 1 ≤ ξ 2 ≤ λ 2 ≤ ν 2 with ξ 1 ξ 2 = λ 1 λ 2 . Then,

φ ( ξ 1 ) + φ ( ξ 2 ) ≤ φ ( λ 1 ) + φ ( λ 2 ) .

Proof

For λ 1 = λ 2 , the result is obvious. We observe that

ξ 1 = ( λ 1 ) ln λ 2 − ln ξ 1 ln λ 2 − ln λ 1 ( λ 2 ) ln ξ 1 − ln λ 1 ln λ 2 − ln λ 1

and

ξ 2 = ( λ 1 ) ln λ 2 − ln ξ 2 ln λ 2 − ln λ 1 ( λ 2 ) ln ξ 2 − ln λ 1 ln λ 2 − ln λ 1

are in the interval [ ν 1 , ν 2 ] , and ξ 1 ξ 2 = λ 1 λ 2 .

By applying the G A -convexity, we obtain

□ φ ( ξ 1 ) + φ ( ξ 2 ) ≤ ln λ 2 − ln ξ 1 ln λ 2 − ln λ 1 φ ( λ 1 ) + ln ξ 1 − ln λ 1 ln λ 2 − ln λ 1 φ ( λ 2 ) + ln λ 2 − ln ξ 2 ln λ 2 − ln λ 1 φ ( λ 1 ) + ln ξ 2 − ln λ 1 ln λ 2 − ln λ 1 φ ( λ 2 ) = φ ( λ 1 ) + φ ( λ 2 ) .

We first prove a result similar to results proved in [39] for G A -convex functions which provide refinement inequalities for (1.6).

Theorem 8

Let φ : [ ν 1 , ν 2 ] ⊂ ( 0 , ∞ ) → R be a GA-convex function, 0 < α < 1 , 0 < β < 1 , σ = ν 1 α ν 2 1 − α , ν 0 = ν 2 ν 1 − 1 min α 1 − β , 1 − α β and let ϑ : [ ν 1 , ν 2 ] → R be non-negative and integrable and ϑ ( σ ϱ − β ) = ϑ ( σ ϱ 1 − β ) , ϱ ∈ [ 0 , ν 0 ] . Then,

(2.1) φ ( ν 1 α ν 2 1 − α ) ∫ σ ϱ − β σ ϱ 1 − β ϑ ( ξ ) ξ d ξ ≤ 1 − β β ∫ σ ϱ − β σ φ ( ξ ) ϑ ( ξ ) ξ d ξ + β 1 − β ∫ σ σ ϱ 1 − β φ ( ξ ) ϑ ( ξ ) ξ d ξ ≤ [ α φ ( ν 2 ) + ( 1 − α ) φ ( ν 1 ) ] ∫ σ ϱ − β σ ϱ 1 − β ϑ ( ξ ) ξ d ξ .

Proof

For every ϱ ∈ [ 0 , ν 0 ] , we have the identity

(2.2) ∫ σ ϱ − β σ ϱ 1 − β ϑ ( ξ ) ξ d ξ = ∫ σ ϱ − β σ ϱ 1 − β ϑ ( ξ ) ξ d ξ + ∫ σ σ ϱ 1 − β ϑ ( ξ ) ξ d ξ = β ∫ 0 ϱ ϑ ( σ ϱ − β ) ξ d ξ + ( 1 − β ) ∫ 0 ϱ ϑ ( σ ϱ − β ) ξ d ξ = ∫ 0 ϱ ϑ ( σ ϱ − β ) ξ d ξ .

We now prove that the mapping W : [ 0 , ν 0 ] → R defined by

W ( ϱ ) = ( 1 − β ) φ ( σ ϱ − β ) + β φ ( σ ϱ 1 − β )

is G A -convex ( 0 , ν 0 ] and monotonically increasing on [ 0 , ν 0 ] .

Since the sum of two G A -convex functions is a G A -convex, hence W is a G A -convex on ( 0 , ν 0 ] . Let ϱ ∈ ( 0 , ν 0 ] , it follows from the G A -convexity of φ that

(2.3) W ( ϱ ) = ( 1 − β ) φ ( σ ϱ − β ) + β φ ( σ ϱ 1 − β ) ≥ φ ( σ 1 − β ϱ − β ( 1 − β ) σ β ϱ β ( 1 − β ) ) = φ ( σ ) = φ ( ν 1 α ν 2 1 − α ) .

We observe that 0 < α ≤ α + β ln ϱ ln ν 2 − ln ν 1 ≤ 1 , 0 ≤ ( 1 − α ) − β ln ϱ ln ν 2 − ln ν 1 ≤ 1 − α < 1 , 0 ≤ α ≤ α − ( 1 − β ) ln ϱ ln ν 2 − ln ν 1 ≤ α ≤ 1 , and 0 < 1 − α ≤ ( 1 − α ) + ( 1 − β ) ln ϱ ln ν 2 − ln ν 1 ≤ 1 . Thus, by Remark 1 φ ∘ exp is convex on [ ln ν 0 , ∞ ) and hence we obtain

(2.4) W ∘ exp ( ln ϱ ) = ( 1 − β ) φ ∘ exp ( ln σ − β ln ϱ ) + β φ ( ln σ + ( 1 − β ) ln ϱ ) = ( 1 − β ) φ ∘ exp α + β ln ϱ ln ν 2 − ln ν 1 ln ν 1 + ( 1 − α ) − β ln ϱ ln ν 2 − ln ν 1 ln ν 2 + β φ α − ( 1 − β ) ln ϱ ln ν 2 − ln ν 1 ln ν 1 + ( 1 − α ) + ( 1 − β ) ln ϱ ln ν 2 − ln ν 1 ln ν 2 ≤ ( 1 − β ) α + β ln ν 2 − ln ν 1 ln ϱ φ ∘ exp ( ln ν 1 ) + ( 1 − β ) ( 1 − α ) − β ln ν 2 − ln ν 1 ln ϱ φ ∘ exp ( ln ν 2 ) + β α − 1 − β ln ν 2 − ln ν 1 ln ϱ φ ∘ exp ( ln ν 1 ) + β ( 1 − α ) + 1 − β ln ν 2 − ln ν 1 ln ϱ φ ∘ exp ( ln ν 2 ) = α φ ( ν 1 ) + ( 1 − α ) φ ( ν 2 ) .

From (2.3) and (2.4), we obtain

(2.5) φ ( ν 1 α ν 2 1 − α ) ≤ W ( ϱ ) ≤ α φ ( ν 1 ) + ( 1 − α ) φ ( ν 2 ) .

Finally, for ϱ 1 , ϱ 2 ∈ ( 0 , ν 0 ] , such that 0 < ln ϱ 1 < ln ϱ 2 ≤ ln ν 0 , since W ∘ exp ( ln ϱ ) is convex, it follows from (2.3) that

W ∘ exp ( ln ϱ 2 ) − W ∘ exp ( ln ϱ 1 ) ln ϱ 2 − ln ϱ 1 = W ( ϱ 2 ) − W ( ϱ 1 ) ln ϱ 2 − ln ϱ 1 ≥ 0 .

This shows that W is increasing on ( 0 , ν 0 ] .

Since ϑ is non-negative, multiplying (2.5) with ϑ ( σ ϱ − β ) ξ , integrating the resulting inequalities over [ 0 , ϱ ] , and using ϑ ( σ ϱ − β ) = ϑ ( σ ϱ 1 − β ) , we have

(2.6) φ ( ν 1 α ν 2 1 − α ) ∫ 0 ϱ ϑ ( σ ϱ − β ) ξ d ξ ≤ ( 1 − β ) ∫ 0 ϱ φ ( σ ϱ − β ) ϑ ( σ ϱ − β ) ξ d ξ + β ∫ 0 ϱ φ ( σ ϱ 1 − β ) ϑ ( σ ϱ 1 − β ) ξ d ξ ≤ [ α φ ( ν 1 ) + ( 1 − α ) φ ( ν 2 ) ] ∫ 0 ϱ ϑ ( σ ϱ − β ) ξ d ξ .

By using Identity (2.2) in (2.6), we obtain (2.1).□

Remark 2

If we choose α = ϑ ϑ + q , β = 1 2 , ϱ = λ 2 in Theorem 8, then

(2.7) φ ν 1 ϑ ϑ + q ν 2 q ϑ + q ∫ λ − 1 σ λ σ ϑ ( ξ ) ξ d ξ ≤ 1 − β β ∫ λ − 1 σ σ φ ( ξ ) ϑ ( ξ ) ξ d ξ + β 1 − β ∫ σ λ σ φ ( ξ ) ϑ ( ξ ) ξ d ξ ≤ [ α φ ( ν 1 ) + ( 1 − α ) φ ( ν 2 ) ] ∫ λ − 1 σ λ σ ϑ ( ξ ) ξ d ξ .

Remark 3

If we choose α = β = 1 2 , ϱ = ν 0 = ν 2 ν 1 − 1 in Theorem 8, then we obtain (1.6).

Remark 4

If we choose α = β = 1 2 , ϱ = ν 0 = ν 2 ν 1 − 1 in Theorem 8, then we obtain (1.5).

Theorem 9

Let φ , σ , and ν 0 be defined as in Theorem  8, 0 < α < 1 , 0 < β < 1 , α + β ≤ 1 , and let X be defined on [ 0 , 1 ] as

(2.8) X ( ϱ ) = 1 − β α ( ln ν 2 − ln ν 1 ) ∫ 0 ν 2 α 1 − β ν 1 − α 1 − β [ ( 1 − β ) φ ( σ ξ − ϱ β ) + β φ ( σ ξ ( 1 − β ) ϱ ) ] d ξ ξ .

Then, X is GA-convex on ( 0 , 1 ] , monotonically increasing on [ 0 , 1 ] and

φ ( ν 1 α ν 2 1 − α ) ≤ X ( ϱ ) ≤ X ( 1 ) = 1 − β α ( ln ν 2 − ln ν 1 ) ∫ 0 ν 2 α 1 − β ν 1 − α 1 − β 1 ξ [ ( 1 − β ) φ ( σ ξ − ϱ β ) + β φ ( σ ξ ( 1 − β ) ϱ ) ] d ξ ≤ α φ ( ν 1 ) + ( 1 − α ) φ ( ν 2 ) .

Proof

Since φ is G A -convex on [ ν 1 , ν 2 ] this prove the G A -convexity of X on ( 0 , ν 0 ] . By using the condition α + β ≤ 1 implies that ν 0 = α 1 − β ν 2 ν 1 − 1 . Since the mapping W : [ 0 , ν 0 ] → R defined by

(2.9) W ( ϱ ) = ( 1 − β ) φ ( σ ϱ − β ) + β φ ( σ ϱ 1 − β )

has been proved to be monotonically increasing on [ 0 , ν 0 ] , thus the mapping X is also monotonically increasing on [ 0 , 1 ] .

Inequality (2.8) follows from inequality (2.5) and the fact that X is monotonically increasing on [ 0 , 1 ] . This accomplished the proof of the theorem.□

The next theorem can be proved similarly.

Theorem 10

Let φ , χ , ν 0 , α , and β be defined as in Theorem  9. Let X 1 be defined on [ 0 , 1 ] as

(2.10) X 1 ( ϱ ) = 1 − β α ( ln ν 2 − ln ν 1 ) ∫ 0 ν 2 α 1 − β ν 1 − α 1 − β ( 1 − β ) φ σ ν 1 α β 1 − β ν 2 − α β 1 − β ξ β ( 1 − ϱ ) + β φ ( σ ν 1 − α ν 2 α ξ ( 1 − β ) ( 1 − ϱ ) ) d ξ ξ .

Then, X 1 is GA-convex monotonically increasing on [ 0 , 1 ] , and

(2.11) ( 1 − β ) 2 α β ( ln ν 2 − ln ν 1 ) ∫ ν 1 α β 1 − β ν 2 1 − α β 1 − β σ φ ( ξ ) ξ d ξ + β α ( ln ν 2 − ln ν 1 ) ∫ σ ν 2 φ ( ξ ) ξ d ξ ≤ X 1 ( ϱ ) ≤ X 1 ( 1 ) = ( 1 − β ) φ ν 1 α 1 − β ν 2 1 − α 1 − β + β φ ( ν 1 ) ≤ α φ ( ν 1 ) + ( 1 − α ) φ ( ν 2 ) .

Remark 5

Taking α = β = 1 2 in inequality (2.8) reduces to

χ ( ϱ ) = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 1 ξ φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) d ξ .

Remark 6

Taking α = β = 1 2 in inequality (2.10) reduces to

(2.12) V ( ϱ ) = 1 2 ( ln ν 2 − ln ν 1 ) ∫ ν 1 ν 2 1 ξ φ ν 2 1 + ϱ 2 ξ 1 − ϱ 2 + φ ν 1 1 + ϱ 2 ξ 1 − ϱ 2 d ξ .

Theorem 11

Let φ , α , β , χ , ν 0 be defined as in Theorem  9 and let ϑ be defined as in Theorem  8. Let Y be a function defined on [ 0 , 1 ] by

(2.13) Y ( ϱ ) = ∫ 0 s [ ( 1 − β ) φ ( σ ϱ ξ − β ϱ ) ϑ ( σ ξ − β ) + β φ ( σ ξ ( 1 − β ) ϱ ) ϑ ( σ ξ 1 − β ) ] d ξ

for some s ∈ [ 0 , ν 0 ] . Then, Y is GA-convex and monotonically increasing on [ 0 , 1 ] and

(2.14) φ ( ν 1 α ν 2 1 − α ) ∫ σ s β σ s 1 − β ϑ ( ξ ) ξ d ξ ≤ Y ( ϱ ) ≤ Y ( 1 ) = 1 − β β ∫ σ s β σ φ ( ξ ) ϑ ( ξ ) ξ d ξ + β 1 − β ∫ σ σ s 1 − β φ ( ξ ) ϑ ( ξ ) ξ d ξ .

Proof

Since φ is G A -convex and ϑ is non-negative, we see that Y is G A -convex on ( 0 , 1 ] . Next for each ξ ∈ [ 0 , s ] , where s ∈ [ 0 , ν 0 ] , it follows from Theorem 8 that h ( ξ ϱ ) = ( 1 − β ) φ ( σ ϱ ξ − β ϱ ) + β φ ( σ ξ ( 1 − β ) ϱ ) is increasing for ϱ ∈ [ 0 , 1 ] . Using the identity ϑ ( σ ϱ β ) = ϑ ( σ ϱ 1 − β ) , we see that Y ( ϱ ) is increasing on [ 0 , 1 ] . Therefore, inequality (2.14) follows immediately.□

Theorem 12

Let φ , α , β , σ , ν 0 be defined as in Theorem  11, and let ϑ be defined as in Theorem  8. Let Y 1 be a function defined on [ 0 , 1 ] by

(2.15) Y 1 ( ϱ ) = ∫ 0 s 1 ξ [ ( 1 − β ) φ ( σ s − β ξ β ( 1 − ϱ ) ) ϑ ( σ s − β ξ β ) + β φ ( σ s 1 − β ξ − ( 1 − β ) ( 1 − ϱ ) ) ϑ ( σ s 1 − β ξ − ( 1 − β ) ) ] d ξ

for some s ∈ [ 0 , ν 0 ] . Then, Y 1 is G A -convex ( 0 , 1 ] and monotonically increasing on [ 0 , 1 ] , and

(2.16) 1 − β β ∫ σ s − β σ φ ( ξ ) ϑ ( ξ ) ξ d ξ + β 1 − β ∫ σ σ s 1 − β φ ( ξ ) ϑ ( ξ ) ξ d ξ ≤ Y ( ϱ ) ≤ Y ( 1 ) = [ ( 1 − β ) φ ( σ s − β ) + ( 1 − β ) φ ( σ s 1 − β ) ] × ∫ σ s − β σ s 1 − β ϑ ( ξ ) ξ d ξ ≤ [ ( 1 − α ) φ ( ν 1 ) + α φ ( ν 2 ) ] ∫ σ s − β σ s 1 − β ϑ ( ξ ) ξ d ξ .

Proof

Since φ is G A -convex and ϑ is non-negative, we see that Y is G A -convex on ( 0 , 1 ] . Next for each ξ ∈ [ 0 , ϱ ] , where ϱ ∈ [ 0 , ν 0 ] , it follows from Theorem 8 that h ( ϱ ) = ( 1 − β ) φ ( σ s − β ) + β φ ( σ s 1 − β ) and k ( ϱ ) = s ξ − ( 1 − ϱ ) are increasing on [ 0 , ν 0 ] and [ 0 , 1 ] , respectively. Hence

h ( k ( ϱ ) ) = ( 1 − β ) φ ( σ s − β ξ β ( 1 − ϱ ) ) ϑ ( σ s − β ξ β ) + β φ ( σ s 1 − β ξ − ( 1 − β ) ( 1 − ϱ ) ) ϑ ( σ s 1 − β ξ − ( 1 − β ) )

is increasing on [ 0 , 1 ] . Using the identity ϑ ( σ ϱ − β ) = ϑ ( σ ϱ 1 − β ) , we see that Y 1 ( ϱ ) is increasing on [ 0 , 1 ] . Therefore, Inequalities (2.16) follows from

φ ( ν 1 α ν 2 1 − α ) ≤ W ( k ( ϱ ) ) ≤ ( 1 − α ) φ ( ν 1 ) + α φ ( ν 2 )

and (2.16).□

Remark 7

Choose α = β = 1 2 , s = ν 0 = ν 1 − 1 ν 2 in Theorems 11 and 12. Then, Inequalities (2.14) and (2.16) reduce to

(2.17) φ ( ν 1 ν 2 ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ Y ( ϱ ) ≤ Y ( 1 ) = ∫ ν 1 ν 2 φ ( ξ ) ϑ ( ξ ) ξ d ξ ≤ Y 1 ( ϱ ) ≤ Y 1 ( 1 ) = φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ,

where

Y ( ϱ ) = 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) ϑ ( ξ ) ξ d ξ

and

(2.18) Y 1 ( ϱ ) = 1 2 ∫ ν 1 ν 2 1 ξ φ ν 1 1 + ϱ 2 ξ 1 − ϱ 2 ϑ ( ν 1 ξ ) + φ ν 2 1 + ϱ 2 ν 1 1 − ϱ 2 ϑ ( ξ ν 2 ) d ξ .

Remark 8

Inequality (2.17) provides weighted generalizations of Theorems 5 and 7.

In the following results, we provide further refinements of Inequalities (1.5) and (1.6) for G A -convex functions by using Lemma 1.

Theorem 13

Let φ , ϑ , χ ϑ be defined as above. Then, the following Fejér-type inequalities hold:

  1. The following inequality holds:

    (2.19) φ ( ν 1 ν 2 ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ 2 ∫ ν 1 3 4 ν 2 1 4 ν 1 1 4 ν 2 3 4 φ ( ξ ) ϑ ξ 2 ν 1 ν 2 d ξ ξ ≤ ∫ 0 1 χ ϑ ( ϱ ) d ϱ ≤ 1 2 φ ( ν 1 ν 2 ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ + ∫ ν 1 ν 2 φ ( ξ ) ϑ ( ξ ) ξ d ξ .

  2. If φ is differentiable on [ ν 1 , ν 2 ] and ϑ is bounded on [ ν 1 , ν 2 ] , then for all ϱ ∈ [ 0 , 1 ] , the inequality holds:

    (2.20) 0 ≤ ∫ ν 1 ν 2 φ ( ξ ) ϑ ( ξ ) ξ d ξ − χ ϑ ( ϱ ) ≤ ( 1 − ϱ ) ( ln ν 2 − ln ν 1 ) φ ( ν 1 ) + φ ( ν 2 ) 2 − ∫ ν 1 ν 2 φ ( ξ ) ξ d ξ ∥ ϑ ∥ ∞ ,

    where ∥ ϑ ∥ ∞ = sup ξ ∈ [ ν 1 , ν 2 ] ϑ ( ξ ) .

  3. If φ is differentiable on [ ν 1 , ν 2 ] , then for all ϱ ∈ [ 0 , 1 ] , we have the inequality

    (2.21) 0 ≤ φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ − χ ϑ ( ϱ ) ≤ ( ln ν 1 − ln ν 2 ) ( ν 2 φ ′ ( ν 2 ) − ν 1 φ ′ ( ν 1 ) ) 4 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ .

Proof

(i) Using techniques of integration and the hypothesis of ϑ , we have the following identities:

(2.22) φ ( ν 1 ν 2 ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ 2 d ξ = 4 ∫ ν 1 ν 1 ν 2 ∫ 0 1 2 φ ( ν 1 ν 2 ) ϑ ( ξ ) ξ d ϱ d ξ ,

(2.23) 2 ∫ ν 1 3 4 ν 2 1 4 ν 1 1 4 ν 2 3 4 φ ( ξ ) ϑ ξ 2 ν 1 ν 2 d ξ ξ = 2 ∫ ν 1 ν 1 ν 2 ∫ 0 1 2 φ ξ 1 2 ν 1 1 4 ν 2 1 4 + φ ξ − 1 2 ν 1 3 4 ν 2 3 4 ϑ ( ξ ) ξ d ϱ d ξ ,

(2.24) ∫ 0 1 χ ϑ ( ϱ ) d ϱ = ∫ ν 1 ν 1 ν 2 ∫ 0 1 2 [ φ ( ξ 1 − ϱ ( ν 1 ν 2 ) ϱ ) + φ ( ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) ) ] ϑ ( ξ ) ξ d ϱ d ξ + ∫ ν 1 ν 1 ν 2 ∫ 0 1 2 [ φ ( ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ ) + φ ( ξ − ( 1 − ϱ ) ( ν 1 ν 2 ) 2 − ϱ ) ] ϑ ( ξ ) ξ d ϱ d ξ ,

and

(2.25) 1 2 φ ( ν 1 ν 2 ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ + ∫ ν 1 ν 2 φ ( ξ ) ϑ ( ξ ) ξ d ξ = ∫ ν 1 ν 1 ν 2 ∫ 0 1 2 [ φ ( ξ ) + φ ( ν 1 ν 2 ) ] ϑ ( ξ ) ξ d ϱ d ξ + ∫ ν 1 ν 1 ν 2 ∫ 0 1 2 [ φ ( ν 1 ν 2 ) + φ ( ν 1 ν 2 ξ − 1 ) ] ϑ ( ξ ) ξ d ϱ d ξ .

By using Lemma 1, we observe that the following inequalities hold for all ϱ ∈ 0 , 1 2 and ξ ∈ [ ν 1 , ν 1 ν 2 ] :

The inequality

(2.26) 4 φ ( ν 1 ν 2 ) ≤ 2 φ ξ 1 2 ν 1 1 4 ν 2 1 4 + φ ξ − 1 2 ν 1 3 4 ν 2 3 4

holds for ξ 1 = ξ 2 = ν 1 ν 2 , λ 1 = ξ 1 2 ν 1 1 4 ν 2 1 4 , λ 2 = ξ − 1 2 ν 1 3 4 ν 2 3 4 .

The inequality

(2.27) 2 φ ξ 1 2 ν 1 1 4 ν 2 1 4 ≤ φ ( ξ 1 − ϱ ( ν 1 ν 2 ) ϱ ) + φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ )

holds for ξ 1 = ξ 2 = ξ 1 2 ν 1 1 4 ν 2 1 4 , λ 1 = ξ 1 − ϱ ( ν 1 ν 2 ) ϱ , λ 2 = ξ ϱ ( ν 1 ν 2 ) 1 − ϱ .

The inequality

(2.28) 2 φ ξ − 1 2 ν 1 3 4 ν 2 3 4 ≤ φ ( ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ ) + φ ( ξ − ( 1 − ϱ ) ( ν 1 ν 2 ) 2 − ϱ )

holds for ξ 1 = ξ 2 = ξ − 1 2 ν 1 3 4 ν 2 3 4 , λ 1 = ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ , λ 2 = ξ − ( 1 − ϱ ) ( ν 1 ν 2 ) 2 − ϱ .

The inequality

(2.29) φ ( ξ 1 − ϱ ( ν 1 ν 2 ) ϱ ) + φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) ≤ φ ( ξ ) + φ ( ν 1 ν 2 )

holds for ξ 1 = ξ 1 − ϱ ( ν 1 ν 2 ) ϱ , ξ 2 = ξ ϱ ( ν 1 ν 2 ) 1 − ϱ , λ 1 = ξ , λ 2 = ν 1 ν 2 .

Finally, the inequality

(2.30) φ ( ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ ) + φ ( ξ − ( 1 − ϱ ) ( ν 1 ν 2 ) 2 − ϱ ) ≤ φ ( ν 1 ν 2 ) + φ ( ν 1 ν 2 ξ − 1 ) .

Multiplying Inequalities (2.26)–(2.30) with ϑ ( ξ ) ξ and integrating them over ϱ on 0 , 1 2 , over ξ on [ ν 1 , ν 1 ν 2 ] , and using identities (2.22)–(2.25), we derive (2.19).

(ii) Since φ : [ ν 1 , ν 2 ] → R is G A -convex on [ ν 1 , ν 2 ] , hence g : [ ln ν 1 , ln ν 2 ] → R defined by g ( ξ ) = φ ∘ exp ( ξ ) is convex on [ ln ν 1 , ln ν 2 ] . Thus, by integration by parts, we obtain the following identity:

(2.31) ∫ ln ν 1 ln ν 1 + ln ν 2 2 ln ν 1 + ln ν 2 2 − ξ [ g ′ ( ln ν 1 + ln ν 2 − ξ ) − g ′ ( ξ ) ] d ξ = ln ν 2 − ln ν 1 2 [ g ( ln ν 1 ) + g ( ln ν 2 ) ] − ∫ ln ν 1 ln ν 1 + ln ν 2 2 [ g ( ln ν 1 + ln ν 2 − ξ ) + g ( ξ ) ] d ξ .

The equality (2.31) is equivalent to the following equality:

(2.32) ∫ ν 1 ν 1 ν 2 1 ξ ln ν 1 + ln ν 2 2 − ln ξ [ ν 1 ν 2 ξ − 1 φ ′ ( ν 1 ν 2 ξ − 1 ) − ξ φ ′ ( ξ ) ] d ξ = ( ln ν 2 − ln ν 1 ) φ ( ν 1 ) + φ ( ν 2 ) 2 − ∫ ν 1 ν 2 φ ( ξ ) ξ d ξ .

Using substitution rules for integration and the hypothesis of ϑ , we have the following identities:

(2.33) ∫ ν 1 ν 2 φ ( ξ ) ϑ ( ξ ) ξ d ξ = ∫ ν 1 ν 1 ν 2 [ φ ( ξ ) + φ ( ν 1 ν 2 ξ − 1 ) ] ϑ ( ξ ) ξ

and

(2.34) χ ϑ ( ϱ ) = ∫ ν 1 ν 1 ν 2 [ φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) + φ ( ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ ) ] ϑ ( ξ ) ξ d ξ .

Now, using the convexity of g ( ξ ) = φ ∘ exp ( ξ ) on [ ln ν 1 , ln ν 2 ] and the hypothesis of ϑ , the following inequality holds for all ϱ ∈ [ 0 , 1 ] and ξ ∈ ln ν 1 , ln ν 1 + ln ν 2 2 :

(2.35) g ( ξ ) − g ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) + g ( ln ν 1 + ln ν 2 − ξ ) − g ϱ ( ln ν 1 + ln ν 2 − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) ≤ ( 1 − ϱ ) ξ − ln ν 1 + ln ν 2 2 g ′ ( ξ ) ϑ ∘ exp ( ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 − ξ g ′ ( ln ν 1 + ln ν 2 − ξ ) ϑ ∘ exp ( ξ ) = ( 1 − ϱ ) ln ν 1 + ln ν 2 2 − ξ [ g ′ ( ln ν 1 + ln ν 2 − ξ ) − g ′ ( ξ ) ] ϑ ∘ exp ( ξ ) ,

which is equivalent to

(2.36) φ ∘ exp ( ξ ) − φ ∘ exp ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) + φ ∘ exp ( ln ν 1 + ln ν 2 − ξ ) − φ ∘ exp ϱ ( ln ν 1 + ln − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) ≤ ( 1 − ϱ ) ln ν 1 + ln ν 2 2 − ξ [ exp ( ξ ) φ ′ ( exp ( ξ ) ) + ν 1 ν 2 exp ( ξ − 1 ) ( 1 − ϱ ) φ ′ ( ν 1 ν 2 exp ( ξ − 1 ) ) ] sup ξ ∈ [ ln ν 1 , ln ν 2 ] ∣ ϑ ∘ exp ( ξ ) ∣ .

Integrating the above inequalities over ξ on ln ν 1 , ln ν 1 + ln ν 2 2 , we obtain

(2.37) ∫ ln ν 1 ln ν 1 + ln ν 2 2 φ ∘ exp ( ξ ) − φ ∘ exp ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) d ξ + ∫ ln ν 1 ln ν 1 + ln ν 2 2 φ ∘ exp ( ln ν 1 + ln ν 2 − ξ ) − φ ∘ exp ϱ ( ln ν 1 + ln − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) d ξ ≤ ( 1 − ϱ ) sup ξ ∈ [ ln ν 1 , ln ν 2 ] ∣ ϑ ∘ exp ( ξ ) ∣ ∫ ln ν 1 ln ν 1 + ln ν 2 2 ln ν 1 + ln ν 2 2 − ξ × [ ν 1 ν 2 exp ( ξ − 1 ) φ ′ ( ν 1 ν 2 exp ( ξ − 1 ) ) − exp ( ξ ) φ ′ ( exp ( ξ ) ) ] d ξ .

After making use of suitable substitution, inequality (2.37) takes the form:

(2.38) ∫ ν 1 ν 1 ν 2 1 ξ [ φ ( ξ ) − φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) ] ϑ ( ξ ) d ξ + ∫ ν 1 ν 1 ν 2 1 ξ φ ( ν 1 ν 2 ξ − 1 ) − φ ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ 2 ϑ ( ξ ) d ξ ≤ ∥ ϑ ∥ ∞ ( 1 − ϱ ) × ∫ ν 1 ν 1 ν 2 ln ν 1 + ln ν 2 2 − ln ξ [ ν 1 ν 2 ξ − 1 φ ′ ( ν 1 ν 2 ξ − 1 ) − ξ φ ′ ( ξ ) ] d ξ ξ .

Inequality (2.20) follows from (2.31), (2.32), (2.33), (2.34), and (2.38).

(iii) We use the fact that φ : [ ν 1 , ν 2 ] → R is G A -convex on [ ν 1 , ν 2 ] , hence g : [ ln ν 1 , ln ν 2 ] → R defined by g ( ξ ) = φ ∘ exp ( ξ ) is convex on [ ln ν 1 , ln ν 2 ] . Thus,

g ( ln ν 1 ) − g ln ν 1 + ln ν 2 2 2 ≤ ln ν 1 − ln ν 2 4 g ′ ( ln ν 1 )

and

g ( ln ν 2 ) − g ln ν 1 + ln ν 2 2 2 ≤ ln ν 2 − ln ν 1 4 g ′ ( ln ν 2 ) .

Adding the above inequalities

(2.39) g ( ln ν 1 ) + g ( ln ν 2 ) 2 − g ln ν 1 + ln ν 2 2 ≤ ( ln ν 2 − ln ν 1 ) ( g ′ ( ln ν 2 ) − g ′ ( ln ν 1 ) ) 4 .

Inequality (2.39) becomes

(2.40) φ ( ν 1 ) + φ ( ν 2 ) 2 − φ ( ν 1 ν 2 ) ≤ ( ln ν 2 − ln ν 1 ) ( ν 2 φ ′ ( ν 2 ) − ν 1 φ ′ ( ν 1 ) ) 4 .

Multiplying both right-hand side and left-hand side of (2.40) with ϑ ( ξ ) ξ 2 and integrating over [ ν 1 , ν 2 ] , we obtain

(2.41) φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ − φ ( ν 1 ν 2 ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ ( ln ν 2 − ln ν 1 ) ( ν 2 φ ′ ( ν 2 ) − ν 1 φ ′ ( ν 1 ) ) 4 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ .

From (2.17) and (2.41) we obtain (2.21).□

Corollary 1

Suppose that the assumption of Theorem  13  is satisfied and ϑ ( ξ ) = 1 ln ν 2 − ln ν 1 , ξ ∈ [ ν 1 , ν 2 ] , then

  1. The following inequalities hold:

    (2.42) φ ( ν 1 ν 2 ) ≤ 2 ∫ ν 1 3 4 ν 2 1 4 ν 1 1 4 ν 2 3 4 φ ( ξ ) ξ ≤ ∫ 0 1 χ ( ϱ ) d ϱ ≤ 1 2 φ ( ν 1 ν 2 ) + 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 φ ( ξ ) ξ d ξ .

  2. The following inequalities hold for all ϱ ∈ [ 0 , 1 ] :

    (2.43) 0 ≤ 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 φ ( ξ ) ξ d ξ − χ ( ϱ ) ≤ ( 1 − ϱ ) φ ( ν 1 ) + φ ( ν 2 ) 2 − 1 ln ν 2 − ln ν 1 ∫ ν 1 ν 2 φ ( ξ ) ξ d ξ .

  3. The following inequalities are valid for all ϱ ∈ [ 0 , 1 ] :

    (2.44) 0 ≤ φ ( ν 1 ) + φ ( ν 2 ) 2 − χ ( ϱ ) ≤ ( ln ν 2 − ln ν 1 ) ( ν 2 φ ′ ( ν 2 ) − ν 1 φ ′ ( ν 1 ) ) 4 .

In the following theorems, we discuss inequalities for the functions χ , χ ϑ , χ 1 , T and T ϑ as considered above:

Theorem 14

Let φ , ϑ , χ 1 , χ ϑ be defined as above. Then, we have the following Fejér-type inequalities:

  1. The following inequality holds for all ϱ ∈ [ 0 , 1 ] :

    (2.45) χ ϑ ( ϱ ) ≤ χ 1 ( ϱ ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ .

  2. The following inequalities hold:

    (2.46) 2 ∫ ν 1 3 4 ν 2 1 4 ν 1 1 4 ν 2 3 4 φ ( ξ ) ϑ ξ 2 ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ 1 2 φ ν 1 1 4 ν 2 3 4 + φ ν 1 3 4 ν 2 1 4 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ ( ln ν 2 − ln ν 1 ) ∫ 0 1 χ 1 ( ϱ ) ϑ ( ν 1 1 − ϱ ν 2 ϱ ) d ϱ ≤ 1 2 φ ( ν 1 ν 2 ) + φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ .

  3. If φ is differentiable on [ ν 1 , ν 2 ] and ϑ is bounded on [ ν 1 , ν 2 ] , then, for all ϱ ∈ [ 0 , 1 ] , we have the following inequality:

    (2.47) 0 ≤ χ ϑ ( ϱ ) − φ ( ν 1 ν 2 ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ ( ln ν 2 − ln ν 1 ) [ χ 1 ( ϱ ) − χ ( ϱ ) ] ∥ ϑ ∥ ∞ ,

    where ∥ ϑ ∥ ∞ = sup ξ ∈ [ ν 1 , ν 2 ] ϑ ( ξ ) .

Proof

(i) Using techniques of integration and the hypothesis of ϑ , we have that the following identity holds on [ 0 , 1 ] :

(2.48) χ 1 ( ϱ ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ = ∫ ν 1 ν 1 ν 2 φ ν 1 1 + ϱ 2 ν 2 1 − ϱ 2 + φ ν 1 1 − ϱ 2 ν 2 1 + ϱ 2 ϑ ( ξ ) ξ d ξ .

By Lemma 1, the following inequality

(2.49) φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) + φ ( ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ ) ≤ φ ν 1 1 + ϱ 2 ν 2 1 − ϱ 2 + φ ν 1 1 − ϱ 2 ν 2 1 + ϱ 2

holds for all ξ ∈ [ ν 1 , ν 1 ν 2 ] with

ξ 1 = ξ ϱ ( ν 1 ν 2 ) 1 − ϱ , ξ 2 = ξ − ϱ ( ν 1 ν 2 ) 1 + ϱ , λ 1 = ν 1 1 + ϱ 2 ν 2 1 − ϱ 2 and λ 2 = ν 1 1 − ϱ 2 ν 2 1 + ϱ 2

Multiplying both right-hand side and left-hand side of (2.49) with ϑ ( ξ ) ξ , integrating over [ ν 1 , ν 1 ν 2 ] , and using (2.34) and (2.49), we obtain (2.45).

(ii) We can observe that

(2.50) 1 2 φ ν 1 1 4 ν 2 3 4 + φ ν 1 3 4 ν 2 1 4 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ = φ ν 1 1 4 ν 2 3 4 + φ ν 1 3 4 ν 2 1 4 ∫ ν 1 ν 1 ν 2 ϑ ( ξ ) ξ d ξ .

By using G A -symmetric assumption on ϑ , we obtain

(2.51) 2 ∫ ν 1 3 4 ν 2 1 4 ν 1 1 4 ν 2 3 4 φ ( ξ ) ϑ ξ 2 ν 1 ν 2 ϑ ( ξ ) ξ d ξ = ∫ ν 1 ν 1 ν 2 φ ν 1 1 4 ν 2 1 4 ξ ξ 2 + φ ν 1 3 4 ν 2 3 4 ξ − ξ 2 ϑ ( ξ ) ξ d ξ .

We can also see that the following identity holds:

(2.52) ( ln ν 2 − ln ν 1 ) ∫ 0 1 χ 1 ( ϱ ) ϑ ( ν 1 1 − ϱ ν 2 ϱ ) d ϱ = ( ln ν 2 − ln ν 1 ) ∫ 1 2 1 φ ν 1 1 + ϱ 2 ν 2 1 − ϱ 2 ϑ ( ν 1 1 − ϱ ν 2 ϱ ) d ϱ + ∫ 0 1 2 φ ν 1 1 + ϱ 2 ν 2 1 − ϱ 2 ϑ ( ν 1 1 − ϱ ν 2 ϱ ) d ϱ + ∫ 0 1 2 φ ν 1 1 − ϱ 2 ν 2 1 + ϱ 2 ϑ ( ν 1 1 − ϱ ν 2 ϱ ) d ϱ + ∫ 1 2 1 φ ν 1 1 − ϱ 2 ν 2 1 + ϱ 2 ϑ ( ν 1 1 − ϱ ν 2 ϱ ) d ϱ = 1 2 ∫ ν 1 ν 1 ν 2 φ ( ν 1 ξ ) + φ ν 1 ν 2 1 2 ξ − 1 2 + φ ( ξ ν 2 ) + φ ν 2 ν 1 1 2 ξ − 1 2 ϑ ( ξ ) ξ d ξ .

Finally, we also have

(2.53) 1 2 φ ( ν 1 ν 2 ) + φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ = φ ( ν 1 ν 2 ) + φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 1 ν 2 ϑ ( ξ ) ξ d ξ .

By Lemma 1, the following inequalities hold for all ξ ∈ [ ν 1 , ν 1 ν 2 ] :

The inequality

(2.54) φ ν 1 1 4 ν 2 1 4 ξ 1 2 + φ ν 1 3 4 ν 2 3 4 ξ − 1 2 ≤ φ ν 1 1 4 ν 2 3 4 + φ ν 1 3 4 ν 2 1 4

holds with the choices of ξ 1 = ν 1 1 4 ν 2 1 4 ξ 1 2 , ξ 2 = ν 1 3 4 ν 2 3 4 ξ − 1 2 , λ 1 = ν 1 3 4 ν 2 1 4 , and λ 2 = ν 1 1 4 ν 2 3 4 .

The inequality

(2.55) φ ν 1 1 4 ν 2 3 4 ≤ 1 2 φ ν 1 ν 2 1 2 ξ − 1 2 + φ ( ν 1 ξ )

holds with the choices of ξ 1 = ξ 2 = ν 1 1 4 ν 2 3 4 , λ 1 = ν 1 ξ , and λ 2 = ν 1 ν 2 1 2 ξ − 1 2 .

The inequality

(2.56) φ ν 1 3 4 ν 2 1 4 ≤ 1 2 φ ν 2 ν 1 1 2 ξ − 1 2 + φ ( ξ ν 2 )

holds with the choices of ξ 1 = ξ 2 = ν 1 3 4 ν 2 1 4 , λ 1 = ξ ν 2 , λ 2 = ν 2 ν 1 1 2 ξ − 1 2 .

The inequality

(2.57) φ ν 1 ν 2 1 2 ξ − 1 2 + φ ( ν 1 ξ ) ≤ φ ( ν 1 ) + φ ( ν 1 ν 2 )

holds with the choices of ξ 1 = ν 1 ξ , ξ 2 = ν 1 ν 2 1 2 ξ − 1 2 , λ 1 = ν 1 , λ 2 = ν 1 ν 2 .

The inequality

(2.58) φ ν 1 1 2 ν 2 ξ − 1 2 + φ 2 ν 2 ξ ν 2 + ξ ≤ φ ( ν 1 ν 2 ) + φ ( ν 2 )

holds with the choices of ξ 1 = ξ ν 2 , ξ 2 = ν 1 1 2 ν 2 ξ − 1 2 , λ 1 = ν 1 ν 2 , λ 2 = ν 2 .

Multiplying (2.54)–(2.58) with ϑ ( ξ ) ξ , integrating them over [ ν 1 , ν 1 ν 2 ] , and using (2.50)–(2.53), we obtain (2.46).

(iii) By integration by parts, we obtain

(2.59) ϱ ∫ ln ν 1 ln ν 1 + ln ν 2 2 ξ − ln ν 1 + ln ν 2 2 g ′ ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 + ln ν 1 + ln ν 2 2 − ξ g ′ ϱ ( ln ν 1 + ln ν 2 − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 d ξ = ϱ ∫ ln ν 1 ln ν 2 ξ − ln ν 1 + ln ν 2 2 g ′ ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 d ξ = ln ν 2 − ln ν 1 2 φ ν 1 1 + ϱ 2 ν 2 1 − ϱ 2 + φ ν 1 1 − ϱ 2 ν 2 1 + ϱ 2 − ∫ ν 1 ν 2 1 ξ φ ( ξ ϱ ( ν 1 ν 2 ) 1 − ϱ ) d ξ = ( ln ν 2 − ln ν 1 ) [ χ 1 ( ϱ ) − χ ( ϱ ) ] .

Using the convexity of g and the hypothesis of ϑ , the following inequality holds for all ϱ ∈ [ 0 , 1 ] and ξ ∈ ln ν 1 , ln ν 1 + ln ν 2 2 :

(2.60) g ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 − g ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) + g ϱ ( ln ν 1 + ln ν 2 − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 − g ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) ≤ ϱ ξ − ln ν 1 + ln ν 2 2 g ′ ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) + ϱ ln ν 1 + ln ν 2 2 − ξ g ′ ϱ ( ln ν 1 + ln ν 2 − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) = ϱ ln ν 1 + ln ν 2 2 − ξ g ′ ϱ ( ln ν 1 + ln ν 2 − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 − g ′ ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ϑ ∘ exp ( ξ ) ≤ ϱ ln ν 1 + ln ν 2 2 − ξ g ′ ϱ ( ln ν 1 + ln ν 2 − ξ ) + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 − g ′ ϱ ξ + ( 1 − ϱ ) ln ν 1 + ln ν 2 2 ∥ ϑ ∥ ∞ .

Integrating (2.60), using (2.59) and (2.17), we obtain (2.47).□

Corollary 2

According to the assumptions of Theorem  14  with ϑ ( ξ ) = 1 ln ν 2 − ln ν 1 , ξ ∈ [ ν 1 , ν 2 ] , the following inequalities hold:

  1. The following inequality holds for all ϱ ∈ [ 0 , 1 ] :

    χ ( ϱ ) ≤ χ 1 ( ϱ ) .

  2. The following inequality holds:

    (2.61) 2 ∫ ν 1 3 4 ν 2 1 4 ν 1 1 4 ν 2 3 4 φ ( ξ ) ξ d ξ ≤ 1 2 φ ν 1 1 4 ν 2 3 4 + φ ν 1 3 4 ν 2 1 4 ≤ ∫ 0 1 χ 1 ( ϱ ) d ϱ ≤ 1 2 φ ( ν 1 ν 2 ) + φ ( ν 1 ) + φ ( ν 2 ) 2 .

  3. The inequality

    (2.62) 0 ≤ χ ( ϱ ) − φ ( ν 1 ν 2 ) ≤ χ 1 ( ϱ ) − χ ( ϱ )

    holds for all ϱ ∈ [ 0 , 1 ] .

Theorem 15

Let φ , ϑ , χ 1 , χ ϑ , T ϑ be defined as above. Then, we have the following results:

  1. T ϑ is GA-convex on ( 0 , 1 ] .

  2. The following inequalities hold for all ϱ ∈ [ 0 , 1 ] :

    (2.63) χ 1 ( ϱ ) ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ T ϑ ( ϱ ) ≤ ( 1 − ϱ ) ∫ ν 1 ν 2 φ ( ξ ) ϑ ( ξ ) ξ d ξ + ϱ ⋅ φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ≤ φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ ,

    (2.64) χ ϑ ( 1 − ϱ ) ≤ T ϑ ( ϱ ) ,

    and

    (2.65) χ ϑ ( ϱ ) + χ ϑ ( 1 − ϱ ) 2 ≤ T ϑ ( ϱ ) .

  3. The following bound is true:

    (2.66) sup ϱ ∈ [ 0 , 1 ] T ϑ ( ϱ ) = φ ( ν 1 ) + φ ( ν 2 ) 2 ∫ ν 1 ν 2 ϑ ( ξ ) ξ d ξ .

Proof

(i) Since φ is G A -convex and ϑ is non-negative, we see that T ϑ is G A -convex on ( 0 , 1 ] .

(ii) We observe that the following identity holds on [ 0 , 1 ] :

(2.67) T ϑ ( ϱ ) = 1 2 ∫ ν 1 ν 1 ν 2 [ φ ( ν 1 ϱ ξ 1 − ϱ ) + φ ( ν 1 ϱ ( ν 1 ν 2 ξ − 1 ) 1 − ϱ ) + φ ( ν 2 ϱ ξ 1 − ϱ ) + φ ( ν 2 ϱ ( ν 1 ν 2 ξ − 1 ) 1 − ϱ ) ] ϑ ( ξ ) ξ d ξ .

By Lemma 1, the following inequalities hold for all ξ ∈ [ ν 1 , ν 1 ν 2 ] :

(2.68) 2 φ ( ν 1 ϱ ( ν 1 ν 2 ) 1 − ϱ ) ≤ φ ( ν 1 ϱ ξ 1 − ϱ ) + φ ( ν 1 ϱ ( ν 1 ν 2 ξ − 1 ) 1 − ϱ )

with

ξ 1 = ξ 2 = ν 1 ϱ ( ν 1 ν 2 ) 1 − ϱ , λ 1 = ν 1 ϱ ξ 1 − ϱ , and λ 2 = ν 1 ϱ ( ν 1 ν 2 ξ − 1 ) 1 − ϱ .

(2.69) 2 φ ( ν 2 ϱ ( ν 1 ν 2 ) 1 − ϱ ) ≤ φ ( ν 2 ϱ ξ 1 − ϱ ) + φ ( ν 2 ϱ ( ν 1 ν 2 ξ − 1 ) 1 − ϱ )

with

ξ 1 = ξ 2 = ν 2 ϱ ( ν 1 ν 2 ) 1 − ϱ , λ 1 = φ ( ν 2 ϱ ξ 1 − ϱ ) and λ 2 = ν 2 ϱ ( ν 1 ν 2 ξ − 1 ) 1 − ϱ .

Multiplying Inequalities (2.68) and (2.69) with ϑ ( ξ ) ξ , integrating them over ξ on [ ν 1 , ν 1 ν 2 ] , and using identities (2.48) and (2.67), we derive the first inequality of (2.63). Using the G A -convexity of φ and inequality (2.17), the last part of (2.63) holds. Using again the G A -convexity of φ , we obtain

(2.70) χ ϑ ( 1 − ϱ ) = ∫ ν 1 ν 2 φ ( ξ 1 − ϱ ( ν 1 ν 2 ) ϱ ) ϑ ( ξ ) ξ d ξ = ∫ ν 1 ν 2 φ ν 1 ϱ 2 ξ 1 − ϱ 2 ν 1 ϱ 2 ξ 1 − ϱ 2 ϑ ( ξ ) ξ d ξ ≤ 1 2 ∫ ν 1 ν 2 [ φ ( ν 1 ϱ ξ 1 − ϱ ) + φ ( ν 2 ϱ ξ 1 − ϱ ) ] ϑ ( ξ ) ξ d ξ = T ϑ ( ϱ ) .

From (2.45), (2.63), and (2.70), we obtain (2.65).

(iii) (2.66) holds due to inequality (2.63). The theorem is thus accomplished.□

3 Conclusion

The topic of mathematical inequalities has become an emerging topic since the past four decades and lot of research has been produced by a number of mathematicians with novel results. This topic has lot of applications in applied mathematics, pure mathematics, and other applied sciences. A number of novel results have been established using convexity and its generalizations with applications in numerical analysis, fixed point theory, differential equations, and optimization. In this study, we have used the GA-convexity as a generalization of convexity to obtain new Fejér-type inequalities with the help of some mappings defined over the interval [ 0 , 1 ] . We have discussed some very interesting properties of those mappings and as a consequence, we obtain refinements of number of results previously obtained in this topic.

Acknowledgments

The author would be very thankful to all the anonymous referees for their very useful and constructive comments in order to improve the article.

  1. Funding information: This work is supported by the Deanship of Scientific Research, King Faisal University under the Ambitious Researcher Track (Research Project Number GrantA141).

  2. Author contributions: I am the only author for conceptualization and writing up this manuscript.

  3. Conflict of interest: The author declares no conflict of interest.

  4. Data availability statement: No data have been used in the manuscript.

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Received: 2023-05-31
Revised: 2023-08-16
Accepted: 2024-02-08
Published Online: 2024-08-02

© 2024 the author(s), 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/dema-2024-0006
Keywords for this article
Hermite-Hadamard inequality; convex function; harmonic GA-convex function; Fejér inequality
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  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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