1. Introduction
The concept of convexity and its generalization is a rapidly growing area of mathematics, with numerous applications in machine (deep) learning, optimization, engineering, management, control theory, economics, and other disciplines. Nowadays, numerous mathematicians, economists, and physicists use this novel concept for their theoretical aspects as well. Mathematicians incorporate it to provide solution to problems that arise in different branches of sciences. This hypothesis gives us fascinating and amazing mathematical strategies to tackle and to take care of numerous issues that emerge in pure and applied sciences. During the last few decades, numerous scientists especially mathematicians have added to the advancement of this beautiful concept in different directions. It is especially important in the study of optimization problems, where it is distinguished by many convenient properties (for example, any minimum of a convex function is a global minimum, or the maximum is attained at a boundary point). This explains why there is a very rich theory of convex functions and convex sets. Optimization of convex functions has many practical applications (circuit design, controller design, modeling, etc.). Due to a lot of importance, it has gathered a lot of attention and has been the focus of several researchers. For the attention of the readers, we encourage the references [
1,
2,
3,
4,
5,
6].
It has a strong connection and also plays a significant role in the development of the theory of inequalities. Both convexity and inequality are closely related to each other. The concept of inequality is one of the main instruments in many parts of designing and science, for example, optimization engineering, probability theory, numerical analysis, finite element method, modeling, etc. At last, the hypothesis of inequalities might be viewed as an autonomous branch of mathematics. During the last twenty years, numerous mathematicians and researchers thought of their incredible commitments and considerations to investigate the Hermite–Hadamard inequality. In this manner, a significant amount of literature can be found for the researchers. For the attention of the readers, we encourage exploring references [
7,
8,
9,
10,
11,
12].
In the year 1929, Bernstein, introduced the term exponential convexity to the literature. After Bernstein, Widder [
13] explored these functions as a subclass of convex functions in a given interval
. The sizeable and worthwhile research on big data analysis and extensive learning has recently increased the attentiveness in information theory involving exponentially convex functions. So especially in the last few decades, different mathematicians worked on the idea of exponential type convexity in different directions and contributed to the field of analysis. Due to the earlier works, these functions have proceeded as a remarkable and new class of convex functions, which have worthy benefits in technology, data science, information sciences, data mining, statistics, stochastic optimization, statistical learning, and sequential prediction. The reader may refer to [
14,
15,
16,
17,
18,
19] for the background of exponential convexity.
Motivated by the ongoing research activities, this paper aims to introduce a new class of exponential type convex function, called generalized exponential type -convex function. We explore some algebraic properties and examples in the manner of newly introduced definitions. A new version of Hermite–Hadamard inequality and its refinements are investigated employing this new convexity.
2. Preliminaries
In this section we recall some known concepts.
In the year 1905, Jensen, presented the meaning of convex function, which reads as follows:
Definition 1 ([
1,
20]).
Let be a convex subset of a real vector space and let be a function. Then a function is said to be convex, if holds for all and Any paper on Hermite inequalities seems to be incomplete without mentioning the well-known Hermite–Hadamard inequality. This inequality states that, if
is convex in
for
and
, then
The Hermite–Hadamard inequality plays an amazing and magnificent role in the literature. Several mathematicians have collaborated variant concepts in the subject of inequalities and its applications. Interested readers can refer to [
21,
22,
23,
24].
The family of
-convex functions was first time explored and introduced by G. Toader in [
25].
Definition 2 ([
25]).
A function is said to be -convex, where if holds ∀
and If the above inequality (3) holds in the reversed sense, then is said to be -concave. Definition 3 ([
26]).
Let be a nonnegative function. Then is said exponential type convex, if holds ∀ and Definition 4 ([
27]).
A nonnegative real-valued function is known as an n-polynomial convex function, if holds for every , , and . Inspired by the above results and literature of inequality theory, we organize the paper as follow: In
Section 3, we elaborate the concept and fundamental properties of the newly introduced definition, namely the generalized exponential type
-convex function. In
Section 4, we deduce new generalization of Hermite–Hadamard type inequality in the manner of a newly introduced concept. Next, in
Section 5, we establish some modifications and estimations of the Hermite–Hadamard type inequality in mode of newly introduced idea. Finally, in
Section 6, we give a brief conclusion.
3. Algebraic Properties of Generalized Exponential Type –Convex Functions
The principal focus of this section, we present our main definition of generalized exponential type -convex function and its associated properties.
Definition 5. Let be a nonnegative function, then is said to be a generalized exponential type -convex, ifholds ∀, and . Remark 1. For we attain exponential type convexity, explored by Kadakal and İşcan in [26]. Remark 2. The range of the exponential type -convex functions for is
Proof. Let
be arbitrary. Using the definition of exponential type
-convex function, for
, we have
□
We explore some relations between the class generalized exponential type -convex functions and other class of generalized convex functions.
Lemma 1. The following inequalities and hold, .
Proof. The proof is clearly seen and hence omitted. □
Proposition 1. If , then every nonnegative -convex function is generalized exponential type -convex function.
Proof. Since
, and by using Lemma 1, we have
□
Theorem 1. Let If and are generalized exponential type -convex functions for then is generalized exponential type -convex function.
Proof. Let
and
be generalized exponential type
-convex functions, then
□
Theorem 2. Let If is generalized exponential type -convex functions for then nonnegative real number is generalized exponential type -convex function.
Proof. Let
be generalized exponential type
-convex, then
□
Theorem 3. Let be -convex function for and and is nondecreasing and generalized exponential type -convex function. Then for the same fixed numbers the function is generalized exponential type -convex.
Proof. ,
, and
we have
□
Theorem 4. Let be a class of generalized exponential type -convex functions for and let If then E is an interval and is generalized exponential type -convex function on
Proof. For all
,
and
we have
□
Theorem 5. If the function is generalized exponential type -convex for then is bounded on
Proof. Suppose
be a point and
and
and then ∃
such that
Thus, since
and
, we have
□
4. New Generalization of H–H Type Inequality Using Generalized Exponential Type -Convex Function
The subject of this section is to deduce new generalizations of H–H type integral inequality involving generalized exponential type -convex function .
Theorem 6. Let be generalized exponential type -convex function for and If then Proof. Let us denote,
Using the definition of exponential type
-convexity of
, we have
Now, integrating on both sides in the last inequality with respect to
over
we obtain
This completes the left-side inequality. For the right-side inequality, using exponential type
-convexity of
, we obtain
The proof is completed. □
Corollary 1. If in Theorem 6, we obtain (Theorem 3.1, [26]). 5. Refinements of H–H Type Inequality via Generalized Exponential Type -Convex Function
Let us establish some refinements of the H–H inequality for functions whose first derivative in absolute value at certain power is generalized exponential type -convex. First we need some new useful lemmas.
Lemma 2. Let and a mapping is differentiable on with and . If then Proof. Using the integrating by parts, we have
This completes the proof. □
Lemma 3. Let and is differentiable on with and . If then Proof. Using the integrating by parts, we have
which completes the proof. □
Lemma 4. Let and a function is differentiable on with and . If then Proof. Applying the by parts integration, we have
□
Lemma 5. Let and a function is differentiable on with and . If then Proof. Applying the by parts integration, we have
□
Lemma 6. Let and a function is differentiable on with and . If then Proof. Applying the by parts integration, we have
□
Theorem 7. Let be differentiable on with and . If is generalized exponential type -convex function on for and then for the following inequality holds: Proof. From Lemma 2, Hölder’s inequality and generalized exponential type
-convexity of
we have
This completes the proof. □
Remark 3. If in Theorem 7, we have Theorem 8. Let be differentiable on with and . If is generalized exponential type -convex function on for and we have Proof. From Lemma 2, power mean inequality and generalized exponential type
-convexity of
we have
This completes the proof. □
Remark 4. If in Theorem 8, we obtain Theorem 9. Let is differentiable on with and . If is generalized exponential type -convex function on for and then for , we have: Proof. From Lemma 3, Hölder’s inequality and generalized exponential type
-convexity of
we have
This completes the proof. □
Remark 5. If in Theorem 9, we obtain Theorem 10. Let is differentiable on with and . If is generalized exponential type -convex function on for and then we have: Proof. From Lemma 3, power mean inequality and generalized exponential type
-convexity of
we have
This completes the proof. □
Remark 6. If in Theorem 10, we have Theorem 11. Let is differentiable on with and . If is generalized exponential type -convex function on for and then for , we have Proof. Using Lemma 4, property of modulus, Hölder’s inequality and generalized exponential type
-convex function
we have
Consequently, the proof is completed. □
Remark 7. If we put in above theorem, then Theorem 12. Let is differentiable on with and . If is generalized type -convex function on for then for , we have Proof. Using Lemma 4, property of modulus, power mean inequality and generalized exponential type
-convex function
we have
Consequently, the proof is completed. □
Remark 8. If we put in above theorem, then Theorem 13. Let is differentiable on with and . If is generalized exponential type -convex function on for and then for , we have Proof. Using Lemma 5, property of modulus, Hölder’s inequality and generalized exponential type
-convex function
we have
Consequently, the proof is completed. □
Remark 9. If we put in above theorem, then Theorem 14. Let is differentiable on with and . If is generalized exponential type -convex function for then for , we have Proof. Using Lemma 5, property of modulus, power mean inequality and generalized exponential type
-convex function
we have
Consequently, the proof is completed. □
Remark 10. If we put in above theorem, then Theorem 15. Let is differentiable on with and . If is generalized exponential type -convex function on for and then for and we have Proof. Using Lemma 6, property of modulus, Hölder’s inequality and generalized exponential type
-convex function
we have
Consequently, the proof is completed. □
Remark 11. If we put in above theorem, then Theorem 16. Let is differentiable on with and . If is generalized exponential type -convex function on for then for , we have Proof. Using Lemma 6, property of modulus, power mean inequality and generalized exponential type
-convex function
we have
Consequently, the proof is completed. □
Remark 12. If we put in above theorem, then 6. Conclusions
In this paper, the authors have introduced a new concept namely generalized exponential type -convex function. A new evaluation of H–H type inequality for the new class of convexity is introduced. Further, the authors have introduced five new integral identities concerning -convexity, then employing these identities some H–H type integral inequalities for generalized exponential type -convex functions are presented. Special cases of our established results are also presented for some particular values of parameters of and . Since convex functions have enormous applications in numerous mathematical fields, we trust that our results and techniques can be applied in convex analysis, deep learning, quantum calculus, fractional calculus, economics, etc.
Author Contributions
Conceptualization, M.T., H.A. and C.C.; methodology, M.T., H.A. and C.C.; software, M.T. and H.A.; validation, C.C., H.A.-Z., A.E.A. and S.A.; formal analysis, H.A. and C.C.; investigation, M.T. and H.A.; data curation, M.T., H.A., C.C., H.A.-Z., A.E.A. and S.A.; writing—original draft preparation, M.T., H.A. and C.C.; writing—review and editing, M.T. and H.A.; supervision, H.A.-Z., A.E.A. and S.A.; project administration, H.A. and C.C.; funding acquisition, H.A.-Z., A.E.A. and S.A. All authors have read and agreed to the published version of the manuscript.
Funding
Research Supporting Project number (RSP-2021/167), King Saud University, Riyadh, Saudi Arabia.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
Research Supporting Project number (RSP-2021/167), King Saud University, Riyadh, Saudi Arabia.
Conflicts of Interest
The authors declare no conflict of interest.
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