ABSTRACT
In this paper, we investigate a novel class of Hermite-Hadamard inequalities applicable to functions with h-convex absolute derivatives. Graphical representations are provided to bolster the validity of our key findings. Some limiting results of our main findings are discussed as corollaries. Furthermore, we establish error estimates in terms of trapezoid formulae for differences between generalized means.
1. Introduction
Fractional calculus is a mathematical framework that generalizes the traditional operations of differentiation and integration to non-integer orders. This mathematical concept finds application in various fields, including science and engineering (Sun et al. Citation2018), biomedical engineering (Magin Citation2004, Citation2012) and electrochemistry (Oldham Citation2010). Fractional calculus plays a significant role in inequalities and the analysis of convex functions. Here’s how fractional calculus is related to these concepts: Fractional differential inequalities are inequalities that involve fractional derivatives of functions. These inequalities are essential in analysing the behaviour of solutions to fractional differential equations.
Convex functions are mathematical functions with specific properties, such as non decreasing slopes and being bowl-shaped (Luisa Citation1990). Fractional calculus can be used to analyse and establish properties of convex functions. In particular, fractional derivatives can be employed to determine the convexity and concavity of functions and provide conditions under which functions are convex.
Inequalities play significant role in establishing these conditions. Inequalities in convex analysis, which is a branch of mathematics that deals with convex sets and functions. Convex analysis has numerous applications across various fields, including mathematics, optimization, economics, engineering and computer science, as discussed in (Niculescu et al. Citation2006; Nguyen and Tran Citation2021). Convex analysis and convexity are intimately related concepts in mathematics. Convexity is at the core of convex analysis, providing the foundation for this field. Convex analysis studies and characterizes convexity and associated properties by using several kinds of inequalities. Additionally, Jensen’s inequality plays a pivotal role in probability theory and statistics by demonstrating how the expected value of a convex function compares to the convex function (Dwivedi and Sharma Citation2022; Fahad et al. Citation2023). Hölder’s inequality, Young’s inequality and Minkowski’s inequality are also integral to convex analysis, providing tools for handling norms, inner products and integral inequalities (McShane Citation1937; Gardner Citation2002; Aldaz Citation2008; Maligranda Citation2018; Butt et al. Citation2022, Citation2023). The Hermite-Hadamard inequality is a fundamental result in mathematical analysis closely related to convex functions. The mathematician showcased modifications, extensions and refinements of classical inequalities, including the Hermite-Hadamard inequalities (Mitrinovic et al. Citation1993; Dragomir Citation2011; Kalsoom et al. Citation2022; Khan et al. Citation2022). It establishes an important connection between convexity and integral inequalities and is named after mathematicians Charles Hermite and Jacques Hadamard, as discussed in (Wu et al. Citation2021; Kashuri et al. Citation2023). Raissouli et al. explored an algorithm, coupled and recursively refining the Hermite-Hadamard inequality on a simplex to represent the mean value of the integral (Raissouli and Dragomir Citation2015). Generalized inequalities of the Hermite-Hadamard type for s-convex, -concave and -convex function are derived using classical and Riemann-Liouville fractional integrals (Hudzik and Maligranda Citation1994; Özdemir et al. Citation2013, Citation2016; Feckan et al. Citation2013; Khan et al. Citation2017). The brief disussion of several convex functions is explored in (Özdemir et al. Citation2010, Citation2011; Zhang et al. Citation2010; Bai et al. Citation2012; Wang et al. Citation2012, Citation2022; Deng and Wang Citation2013). The left Riemann-Liouville fractional Hermite-Hadamard inequalities using Green’s function and Jensen’s inequality (Adil Khan et al. Citation2018) and Schur convexity with Hadamard’s inequality (Chu et al. Citation2010). Let’s recall the fundamentals used to obtain main results.
In (Varošanec Citation2007), the -convex function defined as follow:
Definition 1.
A non negative function is called -convex. If , and , we have
In (Hermite Citation1883; Hadamard Citation1893) C. Hermite and J. Hadamard state the following fundamental result which is known as Hermite-Hadamard inequality.
Theorem 1.
Let is a convex function and . Then, the following inequalities hold:
The following sections of the paper are organized as follows: In Section 2, we explore the concept of -convexity and examine its fundamental characteristics. In Section 3, we enhance the well-known Hermite-Hadamard inequalities by incorporating the idea of -convexity. We then validate these improved inequalities through graphical representations in both two and three-dimensional spaces. In Section 4, our focus shifts to the exploration of error estimation in generalized means. This exploration effectively demonstrates how the results established in the preceding sections can be practically applied and adapted for various purposes. In the last part, we provide a brief summary of the key discoveries and emphasize the broader significance of our research.
2. Some algebraic properties to -convex function
Algebraic properties hold immense significance in mathematics and various fields beyond. These properties find practical applications in science, engineering, computer science and economics, providing a universal language for problem-solving and modelling real-world phenomena. In particular, the boundedness of convex functions that motivates many desirable properties e.g. optimization, analysis and interpretation in various mathematical and practical contexts. In this section, we will discuss some algebraic properties associated with -convex functions, similar to the algebraic properties of exponential trigonometric convex functions discussed in (Kadakal et al. Citation2021).
Theorem 2.
Consider an arbitrary family of -convex functions and let . If the set is not empty, then forms an interval and the function ℶ is -convex on this interval .
Proof.
Consider and let be arbitrary. Then
Since the interval between any two of its points contains every point and ℶ is an -convex function, this completes the proof.□
Theorem 3.
Let ℶ be -convex function, then the function ℶ on the interval is bounded.
Proof.
Let and be an arbitrary point. Then there exist such that . Thus, since and , we have
Since
Furthermore, there exists a , for any such that and . So without loss of generality we can assume, . Thus, we obtain the following:
By considering to be an upper bound, we get
Therefore, the proof is done.□
Theorem 4.
If the sequence ℶ consists of convex functions that converge to a finite limit function ℶ on the interval , then ℶ is also convex.
Proof.
Let and ,
So, from above it follows that ℶ is convex.
3. Hermite-Hadamard inequality for -convex function
In this section, we establish our primary findings related to the Hermite-Hadamard inequality for differentiable convex functions. To derive our results, we require the following lemma presented in (Barsam et al. Citation2021).
Lemma 1.
Let ℶ be a differentiable function on , such that . If ℶ for all . Then, we have the following equality.
Theorem 5.
Let ℶ is a differentiable function on , where such that . If is -convex function on for all . Then, we have the following inequality
where is bounded by .
Proof.
By using Lemma 1, we have
By applying -convexity of , we get
Hence, the proof is done.
Example 1.
To validate the inequality presented in Theorem 5, one may inspect the graph of inequality (1). For this analysis, we substitute ℶ and obtain the following result.
Case (i) By setting the parameters and with , and denoting the left, middle and right components and considering the ranges of and , the graph of the double inequality (2) can be presented as follows.
Case (ii) By setting the parameters as , and and specifying the functions , and as the left, middle and right components and while examining the range of , the graph of the double inequality (2) can be presented as follows.
The visually confirm the results obtained in 3D and 2D graphs of EquationEquation (2)(2) (2) .
Corollary 1.
With the assumptions that ℶ is a differentiable function on and such that . If is -convex function on . If we choose , we obtain the following limiting result.
Corollary 2.
If we choose in Theorem 5, then we have the following limiting relation
Theorem 6.
Let ℶ is a differentiable function on , where such that . If is -convex function on and . Then, we have the following inequality
where is bounded by .
Proof.
By using Lemma 1, we have
By applying -convexity of , we get
Hence, the proof is done.
Example 2.
To validate the inequality presented in Theorem 6, one may inspect the graph of inequality (3). For this analysis, we substitute and obtain the following result.
Case (i) By setting the parameters and with , and denoting the left, middle and right components and considering the ranges of and , the graph of the double inequality (4) can be presented as follows.
Case (ii) By setting the parameters as , and and specifying the functions , and as the left, middle and right components and while examining the range of , the graph of the double inequality (4) can be presented as follows.
The visually confirm the results obtained in 3D and 2D graphs of EquationEquation (3)(3) (3) .
Corollary 3.
With the assumption that is a differentiable function on and such that . If is -convex function on and . If we choose , we get
Corollary 4.
If we put in Theorem 6, then we have
Theorem 7.
Let is a differentiable function on with , where such that . If is -convex function on and with . Then, we have the following inequality
where is bounded by .
Proof.
By using Lemma 1, we have
on the other hand
Substituting
Also
and
So
This completes the proof of the result.
Example 3.
To validate the inequality presented in Theorem 7, one may inspect the graph of inequality (5). For this analysis, we substitute and obtain the following result.
Case (i) By setting the parameters and with , and denoting the left, middle and right components and considering the ranges of and , the graph of the double inequality (6) can be presented as follows.
Case (ii) By setting the parameters as , and and specifying the functions , and as the left, middle and right components and while examining the range of , the graph of the double inequality (6) can be presented as follows.
The visually confirm the results obtained in 3D and 2D graphs of EquationEquation (6)(6) (6) .
Corollary 5.
Suppose that is a differentiable function on with , where such that . And with the assumption that is -convex function on and with . If we choose as or . Then, we get the following inequality
Corollary 6.
If we choose in Theorem 7, then we have
Theorem 8.
Let is a differentiable function on with , where such that . If is -convex function on and with . Then, we have the following inequality
where is bounded by .
Proof.
By using Lemma 1, we have
By applying -convexity of , we get
By substituting , we obtain the required result.□
Example 4.
To validate the inequality presented in Theorem 8, one may inspect the graph of inequality (7). For this analysis, we substitute and obtain the following result.
Case (i) By setting the parameters and with , and denoting the left, middle and right components and considering the ranges of and , the graph of the double inequality (8) can be presented as follows.
Case (ii) By setting the parameters as , and and specifying the functions , and as the left, middle and right components and while examining the range of , the graph of the double inequality (8) can be presented as follows.
The visually confirm the results obtained in 3D and 2D graphs of EquationEquation (8)(8) (8) .
Theorem 9.
Let is a differentiable function on with , where such that . If is -convex function on and with . Then, we have the following inequality
where is bounded by .
Proof.
By using Lemma 1, we have
By applying -convexity of , we get
Hence, the proof is done.
Example 5.
To validate the inequality presented in Theorem 9, one may inspect the graph of inequality (9). For this analysis, we substitute and obtain the following result.
Case (i) By setting the parameters and with , and denoting the left, middle and right components and considering the ranges of and , the graph of the double inequality (10) can be presented as follows.
Case (ii) By setting the parameters as , and and specifying the functions , and as the left, middle and right components and while examining the range of , the graph of the double inequality (10) can be presented as follows.
The visually confirm the results obtained in 3D and 2D graphs of EquationEquation (8)(8) (8) .
4. Some application to main results in term of means
The concept of the ‘mean’ is widely used in various fields and contexts to summarize and analyse data. The mean is a measure of central tendency that represents the average or typical value in a data set. The purpose of this section is to illustrate practical uses of key results related to means.
The means are expressed as follows:
(i) The arithmetic means
(ii) The logarithmic mean
Proposition 1.
Let , , then we have the following inequalities.
Proof.
By using Theorem 5 and substituting in (1), we have
By substituting and in (11), we can write as
Proposition 2
Let , , then we have the following inequalities.
Proof.
By using Theorem 6 and substituting in (3), we have
By substituting , and in (12), we can write as
Proposition 3.
Let , , then we have the following inequalities.
Proof.
By using Theorem 7 and substituting in (5), we have
By substituting , and in (13), we can write as
5. Conclusions
Hermite-Hadamard inequalities are a notable set of mathematical results that establish connections between the integral and average value of a real-valued function over a closed interval. These inequalities provide valuable insights into the behaviour of functions and play a crucial role in various branches of mathematics, including calculus, real analysis and convex functions. They have practical applications in diverse areas such as physics, economics, optimization and engineering. In this paper, we have established some novel fractional integral inequalities of Hermite-Hadamard-type for a class of differentiable trigonometrically-convex functions. Several novel estimates of Hermite-Hadamard inequality are produced. In addition, some limiting cases are given as corollaries. Further, new inequalities involving special means, such as arithmetic, geometric, logarithmic and some other well-known means, are generated as a result of some of our main findings. This article is supposed to constructively contribute and applicable to the current literature.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgement
The author Thabet Abdeljawad would like to thank Prince Sultan University through the TAS research lab.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Funding
References
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