# New Fractional Estimates of Simpson-Mercer Type for Twice Differentiable Mappings Pertaining to Mittag-Leffler Kernel

Authors: Saad Ihsan Butt; Saima Rashid; Iram Javed; Khuram Ali Khan; Rostin Matendo Mabela
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main motivation of this study is to introduce a novel auxiliary result of Simpson’s formula by employing the Mercer scheme for twice differentiable functions involving the Atangana-Baleanu (AB) fractional integral operator concerned with the Mittag-Leffler as a nonsingular or nonlocal kernel. Thus, by employing Mercer’s convexity on twice differentiable mappings along with Hölder’s and power-mean inequalities, one can develop a variety of new Simpson’s error estimates. Lastly, some applications toq-digamma function and modified Bessel functions are presented. Furthermore, the graphical illustrations described the efficiency and applicability of the proposed technique with success. We make links between our findings and a number of well-known discoveries in the literature. It is hoped that the proposed methodology will provide a new venue in the numerical techniques for calculating the quadrature formulae.


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## Body

## 1. Introduction

Convex functions have gained a lot of popularity in recent years. Convex functions are frequently used in various areas of current analysis in a variety of mathematical disciplines. They are magical, especially in optimization theory, because they have so many useful properties. Inequality theory and convex functions have a strong relationship. Convex functions can be used to obtain a variety of important and useful inequalities. Due to wide range of implementations, it is among the most advanced branches of mathematical modeling. Convex functions are the topic of research in a number of disciplines due to their applicability in inequality theory and defined as:(1)ϕκϰ+1−κϰ1≤κϕϰ+1−κϕϰ1,where ϕ:ζ1,ζ2⊆R⟶R is a convex function which holds for all ϰ,ϰ1∈ζ1,ζ2 and κ∈0,1.Additional information of different types of convexity and their contribution to inequalities can be found here, see [1, 2]. The improvement and exploration of the integral inequalities referring to convex functions is primarily motivated based on the research and findings presented in these books. Because of their wide range of implementations such as probability theory, information theory, computational problems, and optimization, the Jensen and related inequalities are essential and well-known inequalities for convex functions. See [3, 4] and references there in.One of the most significant inequality that we may say is the natural extension of convex function is Jensen-Mercer inequality [5] given as:
(2)ϕζ1+ζ2−∑j=1nΘjϰj≤ϕζ1+ϕζ2−∑j=1nΘjϕϰj,where ϕ on ζ1,ζ2 is a convex function ∀ϰj∈ζ1,ζ2 and all Θj∈0,1 where
(3)∑j=1nΘj=1.It is the most effective inequality in predicting the estimations of bounds of distance functions in information theory [6].Many researchers put an effort in this direction to build new results such as fractional variants of Hermite Jensen-Mercer type inequalities along with applications [7], Moradi and Furuichi worked for the improvement and generalization of Jensen-Mercer-type inequalities [8], Kian and Moslehian worked for the improvement of operator Jensen-Mercer inequality [9], Niezgoda worked on generalization of Mercer’s result on convex function [10], and Harovath gave some notes on Jensen-Mercer inequality [11]. A lot of work has been done on Jensen-Mercer-type inequality in Yang’s Calculus, one can see [12, 13].An inequality which is notable as Simpson’s inequality is as follows.Theorem 1 (see [14]).
Suppose thatϕ:ζ1,ζ2⟶ℝ is a four-time continuously differentiable mapping on ζ1,ζ2, and let ϕ4∞=supϰ∈ζ1,ζ2ϕ4ϰ<∞, then the following inequality holds:
(4)13ϕζ1+ϕζ22+2ϕζ1+ζ22−1ζ2−ζ1∫ζ1ζ2ϕϰdϰ≤12880ϕ4∞ζ2−ζ14.The results of Simpson type inequalities for convex mappings have been looked by various writers because convex theory is an excellent technique to deal with a sizable number of issues from various mathematical disciplines. Specifically, differentiable functions are utilized to demonstrate some Simpson’s type inequalities fors-convex functions [15], and then, the inequality extended to Riemann-Liouville fractional integrals [16]. A lot of work has been done utilizing this inequality for first derivative, one can see [17, 18].Sarikaya et al. [19] explored numerous Simpson type inequalities for functions whose second derivatives are convex. The first and second outcomes on fractional Simpson inequality for twice differentiable functions were established in [14, 20]. With the help of these articles, the aim of this paper is to extend the results given in [19] for twice differentiable functions to generalized fractional integrals. Nowadays, twice differentiable functions are topic of interest for most of the researchers. We will use Mercer convexity along with twice differentiability for Simpson type inequalities to improve our outcomes and give new bounds.Here, a lemma that is given in [14] stated as follows.Lemma 2.
If there is a mappingϕ:ζ1,ζ2⟶ℝ that is absolutely continuous on ζ1,ζ2 considering ϕ′′∈L1ζ1,ζ2, then the following equality holds:
(5)16ϕζ1+4ϕζ1+ζ22+ϕζ2−2ϖ−1Γϖ+1ζ2−ζ1ϖJζ2−ϖϕζ1+ζ22+Jζ1+ϖϕζ1+ζ22=ζ2−ζ128ϖ+1∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′1+κ2ζ2+1−κ2ζ1+ϕ′′1+κ2ζ1+1−κ2ζ2dκ.We will extend this result for a new fractional integral operator along with mercer convexity. It is not easy to deal with double derivative; we have worked on double derivative for a new operator along with Mercer convexity. It is completely new idea, and in this way, the generalized outcomes will increase its worth and captures interest of many scholars toward this field.The generalization of classical calculus, fractional calculus, is widely used in sciences, particularly engineering. The classical calculus offers an excellent method for modelling and explaining numerous essential dynamic processes in most sections of applied sciences. The hypothesis of fractional calculus was developed to merge and generalisen-fold integration and integer-order differentiation. The field of applied sciences includes fractional analysis. Many results based on fractional models have been published in various fields of science [21].The fractional operators of integral and derivative helps in improving the relationships between mathematics and other specialisations by providing solutions that are more closely related to real-world problems. Fractional integral and derivative operators have evolved over time [22, 23]. Some fractional numerical simulations can be seen in [24, 25]. In their review article “Fractional calculus in the sky,” [26] D. Baleanu and R. P. Agrwal, two esteemed professors, provide the most recent compact review of fractional calculus.Definition 3 (see [27]).
Letζ2>ζ1, ϖ∈0,1 and ϕ∈H1ζ1,ζ2, then the new fractional derivative is:
(6)ζ1ABCDκϖϕκ=Bϖ1−ϖ∫ζ1κϕ′ϰEϖ−ϖκ−ϰϖ1−ϖdϰ.Definition 4 (see [27]).
Letϕ∈H1ζ1,ζ2, ζ1>ζ2, ϖ∈0,1, then we have:
(7)ζ1ABRDκϖϕκ=Bϖ1−ϖddκ∫ζ1κϕϰEϖ−ϖκ−ϰϖ1−ϖdϰ.Definition 5 (see [27]).
Letϕ∈H1ζ1,ζ2, then with nonlocal kernel, the fractional integral operator is defined as:
(8)ζ1ABIκϖϕκ=1−ϖBϖϕκ+ϖBϖΓϖ∫ζ1κϕϰ1κ−ϰ1ϖ−1dϰ1,where ζ2>ζ1,ϖ∈0,1.In [28], AB-fractional integral operator’s right hand side is:
(9)ζ2ABIκϖϕκ=1−ϖBϖϕκ+ϖBϖΓϖ∫κζ2ϕϰ1ϰ1−κϖ−1dϰ1.Because the normalization functionBϖ is positive, any positive function has a positive fractional AB-integral. It is worth noting that the classical integral is obtained when the order is ϖ⟶1, while the initial function is obtained when the order is ϖ⟶0. In the theory of integral inequalities involving AB operators, there has been some recent progress. One can see in [29, 30].The purpose of this analysis is to utilize the AB integral operator to propose novel Mercer type inequalities for convex functions. The utmost objective is to acquire outcomes that create particular Mercer type inequalities by employing AB operators and elaborate the facts more appropriately in terms of the operator’s qualities and kernel structure. Inequalities of the classical Mercer type and their different versions are created in the case when we haveϖ=1 in the obtained results. This significant achievement is attributed to the AB fractional integral operator exhibiting the heredity characteristic. The exponential and power law functions are not as good as the generalized Mittag-Leffler function with robust memory entangled in the AB fractional formulation. Furthermore, the Atangana-Baleanu fractional-order derivative is at the same time Liouville-Caputo and Caputo-Fabrizio thus possesses Markovian and non-Markovian properties. Meanwhile, the graphical construction represents the comparison between the error and error estimates clearly with the aid of MATLAB 2021 package. Finally, the obtained outcomes were backed up by diminished outcomes and implementations.


## 2. Novel Simpson’s Atangana-Baleanu Inequalities

Here, we present Mercer type Simpson’s inequalities for Atangana-Baleanu integral operator for differentiable functions onν1,ν2. For this, we give a new Atangana-Baleanu integral operator auxiliary identity that will serve to produce subsequent results for improvements.Lemma 6.
If there is a mappingϕ:ν1,ν2⟶ℝ that is absolutely continuous on ν1,ν2 considering ϕ′′∈L1ν1,ν2, where ζ1,ζ2∈ν1,ν2, then the following equality holds:
(10)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22=ζ2−ζ128ϖ+1∫011BϖΓϖ131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1+ν2−1+κ2ζ2+1−κ2ζ1+ϕ′′ν1+ν2−1+κ2ζ1+1−κ2ζ2dκ.Proof.
We note that(11)I1=1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1+ν2−1+κ2ζ2+1−κ2ζ1dκ,using integration by parts; we obtain
(12)=1BϖΓϖ131−2ϖ1−κ+2−3κϖκϕ′ν1+ν2−1+κ/2ζ2+1−κ/2ζ1ζ1−ζ2/201−1BϖΓϖ∫012ϖ+13−ϖ+1κϖϕ′ν1+ν2−1+κ/2ζ2+1−κ/2ζ1ζ1−ζ2/2dκ=−21−2ϖ3ζ1−ζ2BϖΓϖϕ′ν1+ν2−ζ1+ζ22−2ζ1−ζ2BϖΓϖ×∫012ϖ+13−ϖ+1κϖϕ′ν1+ν2−1+κ2ζ2+1−κ2ζ1dκ=21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ2+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22−4ϖϖ+1ζ2−ζ12BϖΓϖ∫01κϖ−1ϕν1+ν2−1+κ2ζ2+1−κ2ζ1dκ.
By the change of the variableϰ=ν1+ν2−1+κ/2ζ2+1−κ/2ζ1 for κ∈0,1, we have
(13)=21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ2+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22−8ϖϖ+12ϖ−1BϖΓϖζ2−ζ1ϖ+2∫ν1+ν2−ζ2ν1+ν2−ζ1+ζ2/2ϕϰν1+ν2−ζ1+ζ22−ϰϖ−1dϰ.
Add and subtract1−ϖ/Bϖ8ϖ+12ϖ−1/ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ2/2 in Equation (13); we have
(14)=21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ2+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22−1−ϖBϖ8ϖ+12ϖ−1ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ22+8ϖϖ+12ϖ−1BϖΓϖζ2−ζ1ϖ+2∫ν1+ν2−ζ2ν1+ν2−ζ1+ζ2/2ϕϰν1+ν2−ζ1+ζ22−ùϖ−1dϰ+1−ϖBϖ8ϖ+12ϖ−1ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ22,(15)=21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ2+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22−8ϖ+12ϖ−1ζ2−ζ1ϖ+2ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+1−ϖBϖ8ϖ+12ϖ−1ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ22.
Similarly, we have(16)I2=1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1+ν2−1+κ2ζ1+1−κ2ζ2dκ,using integration by parts; we obtain
(17)=−21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22−4ϖϖ+1ζ2−ζ12BϖΓϖ∫01κϖ−1ϕν1+ν2−1+κ2ζ1+1−κ2ζ2dκ.
By the change of the variableϰ=ν1+ν2−1+κ/2ζ1+1−κ/2ζ2 for κ∈0,1, we have
(18)=−21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22,(19)−8ϖϖ+12ϖ−1BϖΓϖζ2−ζ1ϖ+2∫ν1+ν2−ζ1+ζ2/2ν1+ν2−ζ1ϕϰϰ−ν1+ν2−ζ1+ζ22ϖ−1dϰ.
Add and subtract1−ϖ/Bϖ8ϖ+12ϖ−1/ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ2/2 in Equation (19); we have
(20)=−21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22−1−ϖBϖ8ϖ+12ϖ−1ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ22+8ϖϖ+12ϖ−1BϖΓϖζ2−ζ1ϖ+2∫ν1+ν2−ζ1+ζ2/2ν1+ν2−ζ1ϕϰν1+ν2−ζ1+ζ22−ϰϖ−1dϰ+1−ϖBϖ8ϖ+12ϖ−1ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ22,(21)=−21−2ϖ3ζ2−ζ1BϖΓϖϕ′ν1+ν2−ζ1+ζ22+4ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+8ϖ+13ζ2−ζ12BϖΓϖϕν1+ν2−ζ1+ζ22−8ϖ+12ϖ−1ζ2−ζ1ϖ+2ν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+1−ϖBϖ8ϖ+12ϖ−1ζ2−ζ1ϖ+2ϕν1+ν2−ζ1+ζ22.
Add Equations (15) and (21), also multiplying both sides by ζ2−ζ12/8ϖ+1, then we get
(22)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22.
This concludes Lemma’s proof.Remark 7.
If we chooseϖ=1,ν1=ζ1, and ν2=ζ2 in Lemma 6, then we have
(23)16ϕζ1+4ϕζ1+ζ22+ϕζ2−1ζ2−ζ1∫ζ1ζ2ϕϰdϰ=ζ2−ζ1248∫014κ−3κ2−1ϕ″1+κ2ζ1+1−κ2ζ2+ϕ″1+κ2ζ2+1−κ2ζ1dκ,which is a new equality in literature.Remark 8.
If we chooseν1=ζ1 and ν2=ζ2 in Lemma 6, then we have
(24)16ϕζ1+4ϕζ1+ζ22+ϕζ2−2ϖ−1ζ2−ζ1ϖζ2ABIζ1+ζ2/2ϖϕζ1+ζ22+ζ1ABIζ1+ζ2/2ϖϕζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕζ1+ζ22=ζ2−ζ128ϖ+1∫011BϖΓϖ131−2ϖ1−κ+2−3κϖκ×ϕ′′1+κ2ζ1+1−κ2ζ2+ϕ′′1+κ2ζ2+1−κ2ζ1dκ,which is a new equality via the Atangana-Baleanu in fractional calculus.Theorem 9.
Letϕ be defined as in Lemma 6, and if ϕ′′ is convex on ν1,ν2, then we have the following inequality:
(25)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+1Y1ϖ2ϕ′′ν1+2ϕ′′ν2−ϕ′′ζ1+ϕ′′ζ2,where Y1ϖ is defined by
(26)Y1ϖ=1BϖΓϖ1−ϖ23ϖ+2,if0<ϖ≤12,1BϖΓϖ2κϖϖ+2ϖ+2−1−2ϖκϖ+ϖ+1κϖ23+1−ϖ23ϖ+2,ifϖ>12.Proof.
We will start the proof by looking at modulus in Lemma6,
(27)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+11BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1+ν2−1+κ2ζ2+1−κ2ζ1+ϕ′′ν1+ν2−1+κ2ζ1+1−κ2ζ2dκ.
By using the convexity ofϕ′′ with Jensen-Mercer Inequality, we have
(28)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+11BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1+ϕ′′ν2−1+κ2ϕ′′ζ2+1−κ2ϕ′′ζ1+ϕ′′ν1+ϕ′′ν1−1+κ2ϕ′′ζ1+1−κ2ϕ′′ζ2dκ.
⇒ To evaluate the above integral, assume the mapping ς:0,1⟶ℝ where
(29)ςκ=1BϖΓϖ131−2ϖ1−κ+2−3κϖκwithϖ>0.(1)
If0<ϖ≤1/2, then we have
(30)∫01ςκdκ=1BϖΓϖ1−ϖ23ϖ+2.(2)
Ifϖ>1/2, then there exists a real number κϖ such that 0<ϖ<1, and we have
(31)∫01ςκdκ=1BϖΓϖ2κϖϖ+2ϖ+2−1−2ϖκϖ+ϖ+1κϖ23+1−ϖ23ϖ+2.
Hence, we have(32)⇒16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+1Y1ϖ2ϕ′′ν1+2ϕ′′ν2−ϕ′′ζ1+ϕ′′ζ2.
This concludes theorem’s proof.Remark 10.
If we chooseϖ=1 and κϖ=1/3, in Theorem 9, then we have the inequality
(33)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−1ζ2−ζ1∫ν1+ν2−ζ2ν1+ν2−ζ1ϕϰdϰ≤ζ2−ζ121622ϕ′′ν1+2ϕ′′ν2−ϕ′′ζ1+ϕ′′ζ2,which is Mercer variant of an identity proved by Sarikaya et al. in [19].Remark 11.
If we chooseϖ=1,ν1=ζ1, and ν2=ζ2 in Theorem 9, then κϖ=1/3, and we have the inequality
(34)16ϕζ1+4ϕζ1+ζ22+ϕζ2−1ζ2−ζ1∫ζ1ζ2ϕϰdϰ≤ζ2−ζ12162ϕ′′ζ1+ϕ′′ζ2,which is proved by Sarikaya et al. in [19].Remark 12.
If we chooseν1=ζ1 and ν2=ζ2 in Theorem 9, then we have the inequality
(35)16ϕζ1+4ϕζ1+ζ22+ϕζ2−2ϖ−1Γ1+ϖζ2−ζ1ϖJζ2−ϖϕζ1+ζ22+Jζ1+ϖϕζ1+ζ22≤ζ2−ζ128ϖ+1Ω1ϖϕ′′ζ1+ϕ′′ζ2,which is proved by Hezenci et al. in [14].Remark 13.
If we chooseϖ=1 and κϖ=1/3 in Remark 12 then we have the inequality
(36)16ϕζ1+4ϕζ1+ζ22+ϕζ2−1ζ2−ζ1∫ζ1ζ2ϕϰdϰ≤ζ2−ζ12162ϕ′′ζ1+ϕ′′ζ2,which is proved by Sarikaya et al. in [19].Theorem 14.
Letϕ be defined as in Lemma 6, then for q>1, there is a mapping ϕ′′q that is convex on ν1,ν2, then we have identity:
(37)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+1Y1ϖ,p1BϖΓϖ2ϕ′′ν1q+2ϕ′′ν2q−ϕ′′ζ2q+ϕ′′ζ1q1/q,where p,q>1 are conjugate exponents and Y is defined by
(38)Y1ϖ,p=1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκpdκ1/p.Proof.
Using Hölder’s inequality in Lemma6, we obtain
(39)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+1×1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκpdκ1/p1BϖΓϖ∫01ϕ′′ν1+ν2−1+κ2ζ2+1−κ2ζ1qdκ1/q+1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκpdκ1/p1BϖΓϖ∫01ϕ′′ν1+ν2−1+κ2ζ1+1−κ2ζ2qdκ1/q.
By using the convexity ofϕ′′q with Jensen-Mercer Inequality, we obtain
(40)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+11BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκpdκ1/p×1BϖΓϖ∫01ϕ′′ν1q+ϕ′′ν2q−1+κ2ϕ′′ζ2q+1−κ2ϕ′′ζ1qdκ1/q+1BϖΓϖ∫01ϕ′′ν1q+ϕ′′ν2q−1+κ2ϕ′′ζ2q+1−κ2ϕ′′ζ1qdκ1/q.(41)⇒16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+11BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκpdκ1/p×1BϖΓϖϕ′′ν1q+ϕ′′ν2q−3ϕ′′ζ2q+f′′ζ1q41/q+1BϖΓϖϕ′′ν1q+ϕ′′ν2q−ϕ′′ζ2q+3ϕ′′ζ1q41/q.
This concludes theorem’s proof.Corollary 15.
If we chooseϖ=1,ν1=ζ1, and ν2=ζ2 in Theorem 14, then tϖ=1/3, and we have the inequality
(42)16ϕζ1+4ϕζ1+ζ22+ϕζ2−1ζ2−ζ1∫ζ1ζ2ϕϰdϰ≤ζ2−ζ12162Y1,pϕ′′ζ1q+ϕ′′ζ2q1/q,which is given by Hezenci et al. in [14].Theorem 16.
Letϕ be defined as in Lemma 6, then for q>1, there is a mapping ϕ′′q that is convex on ν1,ν2, then we have identity:
(43)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+1Y1ϖ1−1/qY1ϖϕ′′ν1q+ϕ′′ν2q−Y1ϖ+Y2ϖϕ′′ζ2q+Y1ϖ−Y2ϖϕ′′ζ1q21/q+Y1ϖϕ′′ν1q+ϕ′′ν2q−Y1ϖ+Y2ϖϕ′′ζ1q+Y1ϖ−Y2ϖϕ′′ζ2q21/q,where Y1ϖ is defined in Theorem 9 and Y2ϖ is defined by
(44)Y2ϖ=1BϖΓϖ3−ϖ−2ϖ218ϖ+3,if0<ϖ≤12,1BϖΓϖ2κϖϖ+3ϖ+3−31−2ϖκϖ2+4ϖ+1κϖ318+1−ϖ23ϖ+2,ifϖ>12.Proof.
By applying the power-mean inequality in Lemma6, we get
(45)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+1×1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκdκ1−1/q×1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1+ν2−1+κ2ζ2+1−κ2ζ1qdκ1/q+1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκdκ1−1/q×1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1+ν2−1+κ2ζ1+1−κ2ζ2qdκ1/q.
⇒ To evaluate the above integral, assume the mapping ς:0,1⟶ℝ, where
(46)ςκ=1BϖΓϖ131−2ϖ1−κ+2−3κϖκwithϖ>0.(1)
Let us consider0<ϖ≤1/2, then we have(47)∫01ςκκdκ=1BϖΓϖ3−ϖ−2ϖ218ϖ+3.(2)
Ifϖ>1/2, then there exists a real number κϖ such that 0<ϖ<1, and we have(48)∫01ςκκdκ=1BϖΓϖ2κϖϖ+3ϖ+3−31−2ϖκϖ2+4ϖ+1κϖ318+3+ϖ−2ϖ218ϖ+3.
Sinceϕ′′q is convex and taking into account Jensen-Mercer Inequality, we obtain
(49)1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκϕ′′ν1+ν2−1+κ2ζ2+1−κ2ζ1qdκ≤1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1q+ϕ′′ν2q−1+κ2ϕ′′ζ2q+1−κ2ϕ′′ζ1qdκ=Y1ϖϕ′′ν1q+ϕ′′ν2q−Y1ϖ+Y2ϖϕ′′ζ2q+Y1ϖ−Y2ϖϕ′′ζ1q2.
And similarly,(50)1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκϕ′′ν1+ν2−1+κ2ζ1+1−κ2ζ2qdκ≤1BϖΓϖ∫01131−2ϖ1−κ+2−3κϖκ×ϕ′′ν1q+ϕ′′ν2q−1+κ2ϕ′′ζ1q+1−κ2ϕ′′ζ2qdκ=Y1ϖϕ′′ν1q+ϕ′′ν2q−Y1ϖ+Y2ϖϕ′′ζ1q+Y1ϖ−Y2ϖϕ′′ζ2q2.
Finally, we obtain(51)⇒16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−2ϖ−1ζ2−ζ1ϖν1+ν2−ζ1ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+ν1+ν2−ζ2ABIν1+ν2−ζ1+ζ2/2ϖϕν1+ν2−ζ1+ζ22+21−ϖ2ϖ−1Bϖζ2−ζ1ϖϕν1+ν2−ζ1+ζ22≤ζ2−ζ128ϖ+1Y1ϖ1−1/qY1ϖϕ′′ν1q+ϕ′′ν2q−Y1ϖ+Y2ϖϕ′′ζ2q+Y1ϖ−Y2ϖϕ′′ζ1q21/q+Y1ϖϕ′′ν1q+ϕ′′ν2q−Y1ϖ+Y2ϖϕ′′ζ1q+Y1ϖ−Y2ϖϕ′′ζ2q21/q.
This completes the proof.Remark 17.
If we chooseϖ=1 and κϖ=1/3, in Theorem 16, then we have the inequality
(52)16ϕν1+ν2−ζ1+4ϕν1+ν2−ζ1+ζ22+ϕν1+ν2−ζ2−1ζ2−ζ1∫ν1+ν2−ζ2ν1+ν2−ζ1ϕϰdϰ≤ζ2−ζ1211621−1/q1162ϕ′′ν1q+1162ϕ′′ν2q−593527ϕ′′ζ1q+1333527ϕ′′ζ2q1/q+1162ϕ′′ν1q+1162ϕ′′ν2q−593527ϕ′′ζ2q+1333527ϕ′′ζ1q1/q,which is Mercer variant of an identity proved by Sarikaya et al. in [19].Remark 18.
If we takeϖ=1,ν1=ζ1, and ν2=ζ2 in Theorem 16, then Theorem 16 reduce to
(53)16ϕζ1+4ϕζ1+ζ22+ϕζ2−1ζ2−ζ1∫ζ1ζ2ϕϰdϰ≤ζ2−ζ1211621−1/q593527ϕ′′ζ1q+1333527ϕ′′ζ2q1/q+593527ϕ′′ζ2q+1333527ϕ′′ζ1q1/q,which is proved by Sarikaya et al. in [19].Remark 19.
If we takeν1=ζ1 and ν2=ζ2 in Theorem 16, then Theorem 16 reduce to
(54)16ϕζ1+4ϕζ1+ζ22+ϕζ2−2ϖ−1Γ1+ϖζ2−ζ1ϖJζ2−ϖϕζ1+ζ22+Jζ1+ϖϕζ1+ζ22≤ζ2−ζ128ϖ+1Ω1ϖ1−1/qΩ1ϖ+Ω2ϖϕ′′ζ2q+Ω1ϖ−Ω2ϖϕ′′ζ1q21/q+Ω1ϖ+Ω2ϖϕ′′ζ1q+Ω1ϖ−Ω2ϖϕ′′ζ2q21/q,which is proved by Hezenci et al. in [14].Remark 20.
If we takeϖ=1 in Remark 19, then we have
(55)16ϕζ1+4ϕζ1+ζ22+ϕζ2−1ζ2−ζ1∫ζ1ζ2ϕϰdϰ≤ζ2−ζ1211621−1/q593527ϕ′′ζ1q+1333527ϕ′′ζ2q1/q+593527ϕ′′ζ2q+1333527ϕ′′ζ1q1/q,which is proved by Sarikaya et al. in [19].


## 3. Applications

### 3.1.Q-Digamma Function

Theφq-digamma function, which is described as the logarithmic derivative of the q-gamma function, is an essential function related to the q-gamma function. A few papers had also additionally been utilized that explore the monotonicity and complete monotonicity characteristics for functions linked with the q-gamma and q-digamma functions, which tends to result in remarkable inequalities. One can see in [31, 32].Assume theq-analogue of the digamma function φ for 0<q<1 is the q-digamma function φq and is (see [33, 34]) given as:
(56)φq=−ln1−q+lnq∑k=0∞qk+ϰ1−qk+ϰ=−ln1−q+lnq∑k=0∞qkϰ1−qkϰ.Forq≥1 and ϰ>0,q-digamma function φq can be given as:
(57)φq=−lnq−1+lnqϰ−12−∑k=0∞q−k+ϰ1−q−k+ϰ=−lnq−1+lnqϰ−12−∑k=0∞q−kϰ1−q−kϰ.Proposition 21.
Assume thatζ1,ζ2,ν1,ν2∈ℝ such that 0<ν1<ν2,q≥1,0<q<1, and q−1=1−p−1. Then, the following inequality is valid:
(58)16φq′ν1+ν2−ζ1+4φq′ν1+ν2−ζ1+ζ22+φq′ν1+ν2−ζ2−φqν1+ν2−ζ1−φqν1+ν2−ζ2ζ2−ζ1≤ζ2−ζ1211621−1/q1162φq3ν1q+1162φq3ν2q−593527φq3ζ1q+1333527φq3ζ2q1/q+1162φq3ν1q+1162φq3ν2q−593527φq3ζ2q+1333527φq3ζ1q1/q.Proof.
The assertion can be obtained immediately by using Remark17 with the ϕϰ⟶φq′ϰ for all q>0, and consequently, ϕ′′ϰ≔φq3ϰ is convex on the same interval 0,∞.Proposition 22.
Assume thatζ1,ζ2 are the real numbers such that 0<ζ1<ζ2,q≥1,0<q<1, and q−1=1−p−1. Then, the following inequality is valid:
(59)16φq′ζ1+4φq′ζ1+ζ22+φq′ζ2−φqζ2−φqζ1ζ2−ζ1≤ζ2−ζ1211621−1/q593527φq3ζ1q+1333527φq3ζ2q1/q+593527φq3ζ2q+1333527φq3ζ1q1/q.Proof.
The assertion can be obtained immediately by using Remark18 with the ϕϰ⟶φq′ϰ for all q>0, and consequently, ϕ′′ϰ≔φq3ϰ is convex on the same interval 0,∞.


### 3.2. Modified Bessel Function

Bessel functions were named after Friedrich Wilhelm Bessel (1784-1846); however, Daniel Bernoulli is generally credited with being the first to introduce the concept of Bessels functions in 1732. Numerous results about Bessel functions have been established by utilizing its generating function (see [35]).We know the first type of modified Bessel functionBτ1, which has the series interpretation (see [33], p.77).
(60)Bτ1ϰ=Σn≥0ϰ/2τ1+2nn!Γτ1+n+1.where ϰ∈R and τ1>−1, while the second type modified Bessel function ϕτ1 (see [33], p.78) is usually defined as
(61)ϕτ1ϰ=π2B−τ1ϰ−Bτ1ϰsinτ1π.Consider the functionΨτ1ϰ:R⟶1,∞ defined by
(62)Ψτ1ϰ=2τ1Γτ1+1ϰ−τ1ϕτ1ϰ,Here, first, second- and third-order derivative formula ofΨτ1ϰ is given as in [33]:
(63)Ψτ1′ϰ=ϰ2τ1+1Ψτ1+1ϰ,(64)Ψτ1′′ϰ=ϰ24τ1+1τ1+2Ψτ1+2ϰ+12τ1+1Ψτ1+1ϰ,(65)Ψτ1′′′ϰ=ϰ34τ1+1τ1+2τ1+3Ψτ1+3ϰ+3ϰ4τ1+1τ1+2Ψτ1+2ϰ.Proposition 23.
Suppose thatζ1,ζ2,ν1,ν2∈ℝ such that 0<ν1<ν2, and τ1>−1. Then, we have
(66)16ν1+ν2−ζ12τ1+1Ψτ1+1ν1+ν2−ζ1+4ν1+ν2−ζ1+ζ2/22τ1+1Ψτ1+1ν1+ν2−ζ1+ζ22+ν1+ν2−ζ22τ1+1Ψτ1+1ν1+ν2−ζ2−Ψmν1+ν2−ζ1−Ψmν1+ν2−ζ2ζ2−ζ1≤ζ2−ζ1211621−1/q×1162ν134τ1+1τ1+2τ1+3Ψτ1+3ν1+3ν14τ1+1τ1+2Ψτ1+2ν1q+1162ν234τ1+1τ1+2τ1+3Ψτ1+3ν2+3ν24τ1+1τ1+2Ψτ1+2ν2q−593527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q+1333527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q1/q+1162ν134τ1+1τ1+2τ1+3Ψτ1+3ν1+3ν14τ1+1τ1+2Ψτ1+2ν1q+1162ν234τ1+1τ1+2τ1+3Ψτ1+3ν2+3ν24τ1+1τ1+2Ψτ1+2ν2q−593527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q+1333527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q1/q.Proof.
The required result follows immediately from Remark17 utilizing ϕϰ=Ψτ1′ϰ, ϰ>0, and the identities (66) and (63).Proposition 24.
Suppose thatτ1>−1 and 0<ζ1<ζ2. Then, we have
(67)16ζ12τ1+1Ψτ1+1ζ1+4ζ1+ζ24τ1+1Ψτ1+1ζ1+ζ22+ζ22τ1+1Ψτ1+1ζ2−Ψmζ2−Ψmζ1ζ2−ζ1≤ζ2−ζ1211621−1/q×593527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q+1333527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q1/q+593527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q+1333527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q1/q.Proof.
The required result follows immediately from Remark18 utilizing ϕϰ=Ψτ1′ϰ,ϰ>0, and the identities (66) and (63).


## 3.1.Q-Digamma Function

Theφq-digamma function, which is described as the logarithmic derivative of the q-gamma function, is an essential function related to the q-gamma function. A few papers had also additionally been utilized that explore the monotonicity and complete monotonicity characteristics for functions linked with the q-gamma and q-digamma functions, which tends to result in remarkable inequalities. One can see in [31, 32].Assume theq-analogue of the digamma function φ for 0<q<1 is the q-digamma function φq and is (see [33, 34]) given as:
(56)φq=−ln1−q+lnq∑k=0∞qk+ϰ1−qk+ϰ=−ln1−q+lnq∑k=0∞qkϰ1−qkϰ.Forq≥1 and ϰ>0,q-digamma function φq can be given as:
(57)φq=−lnq−1+lnqϰ−12−∑k=0∞q−k+ϰ1−q−k+ϰ=−lnq−1+lnqϰ−12−∑k=0∞q−kϰ1−q−kϰ.Proposition 21.
Assume thatζ1,ζ2,ν1,ν2∈ℝ such that 0<ν1<ν2,q≥1,0<q<1, and q−1=1−p−1. Then, the following inequality is valid:
(58)16φq′ν1+ν2−ζ1+4φq′ν1+ν2−ζ1+ζ22+φq′ν1+ν2−ζ2−φqν1+ν2−ζ1−φqν1+ν2−ζ2ζ2−ζ1≤ζ2−ζ1211621−1/q1162φq3ν1q+1162φq3ν2q−593527φq3ζ1q+1333527φq3ζ2q1/q+1162φq3ν1q+1162φq3ν2q−593527φq3ζ2q+1333527φq3ζ1q1/q.Proof.
The assertion can be obtained immediately by using Remark17 with the ϕϰ⟶φq′ϰ for all q>0, and consequently, ϕ′′ϰ≔φq3ϰ is convex on the same interval 0,∞.Proposition 22.
Assume thatζ1,ζ2 are the real numbers such that 0<ζ1<ζ2,q≥1,0<q<1, and q−1=1−p−1. Then, the following inequality is valid:
(59)16φq′ζ1+4φq′ζ1+ζ22+φq′ζ2−φqζ2−φqζ1ζ2−ζ1≤ζ2−ζ1211621−1/q593527φq3ζ1q+1333527φq3ζ2q1/q+593527φq3ζ2q+1333527φq3ζ1q1/q.Proof.
The assertion can be obtained immediately by using Remark18 with the ϕϰ⟶φq′ϰ for all q>0, and consequently, ϕ′′ϰ≔φq3ϰ is convex on the same interval 0,∞.


## 3.2. Modified Bessel Function

Bessel functions were named after Friedrich Wilhelm Bessel (1784-1846); however, Daniel Bernoulli is generally credited with being the first to introduce the concept of Bessels functions in 1732. Numerous results about Bessel functions have been established by utilizing its generating function (see [35]).We know the first type of modified Bessel functionBτ1, which has the series interpretation (see [33], p.77).
(60)Bτ1ϰ=Σn≥0ϰ/2τ1+2nn!Γτ1+n+1.where ϰ∈R and τ1>−1, while the second type modified Bessel function ϕτ1 (see [33], p.78) is usually defined as
(61)ϕτ1ϰ=π2B−τ1ϰ−Bτ1ϰsinτ1π.Consider the functionΨτ1ϰ:R⟶1,∞ defined by
(62)Ψτ1ϰ=2τ1Γτ1+1ϰ−τ1ϕτ1ϰ,Here, first, second- and third-order derivative formula ofΨτ1ϰ is given as in [33]:
(63)Ψτ1′ϰ=ϰ2τ1+1Ψτ1+1ϰ,(64)Ψτ1′′ϰ=ϰ24τ1+1τ1+2Ψτ1+2ϰ+12τ1+1Ψτ1+1ϰ,(65)Ψτ1′′′ϰ=ϰ34τ1+1τ1+2τ1+3Ψτ1+3ϰ+3ϰ4τ1+1τ1+2Ψτ1+2ϰ.Proposition 23.
Suppose thatζ1,ζ2,ν1,ν2∈ℝ such that 0<ν1<ν2, and τ1>−1. Then, we have
(66)16ν1+ν2−ζ12τ1+1Ψτ1+1ν1+ν2−ζ1+4ν1+ν2−ζ1+ζ2/22τ1+1Ψτ1+1ν1+ν2−ζ1+ζ22+ν1+ν2−ζ22τ1+1Ψτ1+1ν1+ν2−ζ2−Ψmν1+ν2−ζ1−Ψmν1+ν2−ζ2ζ2−ζ1≤ζ2−ζ1211621−1/q×1162ν134τ1+1τ1+2τ1+3Ψτ1+3ν1+3ν14τ1+1τ1+2Ψτ1+2ν1q+1162ν234τ1+1τ1+2τ1+3Ψτ1+3ν2+3ν24τ1+1τ1+2Ψτ1+2ν2q−593527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q+1333527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q1/q+1162ν134τ1+1τ1+2τ1+3Ψτ1+3ν1+3ν14τ1+1τ1+2Ψτ1+2ν1q+1162ν234τ1+1τ1+2τ1+3Ψτ1+3ν2+3ν24τ1+1τ1+2Ψτ1+2ν2q−593527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q+1333527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q1/q.Proof.
The required result follows immediately from Remark17 utilizing ϕϰ=Ψτ1′ϰ, ϰ>0, and the identities (66) and (63).Proposition 24.
Suppose thatτ1>−1 and 0<ζ1<ζ2. Then, we have
(67)16ζ12τ1+1Ψτ1+1ζ1+4ζ1+ζ24τ1+1Ψτ1+1ζ1+ζ22+ζ22τ1+1Ψτ1+1ζ2−Ψmζ2−Ψmζ1ζ2−ζ1≤ζ2−ζ1211621−1/q×593527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q+1333527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q1/q+593527ζ234τ1+1τ1+2τ1+3Ψτ1+3ζ2+3ζ24τ1+1τ1+2Ψτ1+2ζ2q+1333527ζ134τ1+1τ1+2τ1+3Ψτ1+3ζ1+3ζ14τ1+1τ1+2Ψτ1+2ζ1q1/q.Proof.
The required result follows immediately from Remark18 utilizing ϕϰ=Ψτ1′ϰ,ϰ>0, and the identities (66) and (63).


## 4. Conclusion

The study deal with the investigation of Simpson-Mercer type inequalities for twice differentiable functions. Adopting the novel approach, we extended the study of Simpson-Mercer type integral inequalities using Hölder’s and power-mean integral inequalities via Atangana-Baleanu (AB) fractional integral operators. It is interesting to extend such findings to the Atangana-Baleanu (AB) operator and also to other convexities. Various representations were used to explain the findings to clarify the important aspects of the fractional inequalities under consideration, such as Figures1–3, which illustrate the error bounds for the dominant findings. We believe that our newly introduced idea and concept will have very deep research in the captivating field of numerical analysis and inequalities. The strategy’s effective and productive execution is investigated and confirmed in order to show that it may be applied to coordinated convex functions that emerge in fractional calculus, especially to fractal and fractional integral operators.Figure 1
Three-dimensional illustration of the error and error bounds for (43).Figure 2
Three-dimensional illustration of the error and error bounds for (53).Figure 3
Three-dimensional illustration of the error and error bounds for (55).



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