Abstract
The study of convex functions is one of the most researched of the classical fields. Analysis of the geometric characteristics of these functions is a core area of research in this field; however, a paradigm shift in this research is the application of convexity in optimization theory. The Jensen-Mercer type inequalities are studied extensively in recent years. In the present paper, we extend Jensen-Mercer type inequalities for strong convex function. Some improved inequalities in Hölder sense are also derived. The previously established results are generalized and strengthened by our results.
1. Introduction and Preliminary Results
Convex functions and their consequences are useful in the establishment of different kinds of inequalities; therefore, they are considered the base of theory of inequalities in mathematical analysis. A real valued function is said to be convex on the interval , if holds for all and . The function is said to be concave if reverse of inequality (1) holds.
Convex functions are also very important in the fields of mathematical analysis, mathematical statistics, and optimization theory. These functions motivate towards a nice theory named convex analysis (see [1–3]). Convex functions have been defined in various ways by using different techniques, for example, by support function, by chords joining two points, and Jensen’s inequality. Inequality (1) represents the convex function analytically and provides encouragement to define further general notions.
The study of convex functions [4–14] began with Jensen’s thought-provoking concepts and interesting work over the period from 1905 to 1906. It is used in the analysis as an efficient tool for solving optimization issues. Additionally, inequalities involving convex functions are very stimulating in the development of different sections of mathematics, such as mathematical finance, economics, management sciences, and optimization theory.
If the function is convex, then the inequality is called the Hermite-Hadamard inequality [15, 16].
Definition 1 (Convex set) (see [17]). A set is considered to be convex if the line segment between any two points in lies in ; i.e.,
Authors [18–20] expanded on the idea of a strongly convex function by replacing the nonnegative term with a real-valued nonnegative function and defined it as follows:
Definition 2 (Strongly convex function (see [18])). A function is strongly convex, if holds for all and .
Definition 3 (Riemann-Liouville fractional integral). For a function , the Riemann-Liouville fractional integral operator of order with is defined as Many scholars have recently analyzed a variety of inequalities by using the Riemann-Liouville fractional integrals (see [21–24]).
Definition 4 (Hadamard fractional integral). For a function , the Hadamard fractional integral of order for all is defined as where .
Definition 5 (Conformable fractional integral). Let and . A function is -fractional integrable on if the integral exists and is finite.
This paper is aimed at establishing Hermite-Jensen-Mercer type inequalities and some other inequalities including improved Hölder inequality for strong convex function.
2. New Hermite-Jensen-Mercer Type Inequalities
Theorem 6. Let and be a strong convex function. Then, the inequality holds for all
Proof. To prove that the first inequality holds, take
Since is a strong convex function, so
Suppose and ; then, for and , we have
Multiplying both sides of Equation (11) with , we get
Integrating the above inequality with respect to over the range and then combining the result with the integral operator yield
Now, by altering the variables, we can obtain
So,
Therefore,
As a result, the first inequality of (8) is proved.
We can prove the second inequality of (8) by using strong convexity of for over Adding (17) and (20) leads to
Multiply 9 with , and integrating the obtained inequality w.r.t to over [0,1] gives
This completes the proof.
Remark 7. It is obvious from Theorem 6 that (1)Theorem 2.1 of [25] is obtained if we take , , and in Theorem 6(2)Theorem 2.1 of [26] is obtained if we take , , , and in Theorem 6(3)Theorem 2 of [27] is obtained by taking , , , and in Theorem 6
Theorem 8. Let and be a strong convex function. Then, the inequalities and hold .
Proof. The Jensen-Mercer inequality dictates that .
Taking and for and in (25), we get
Multiplying the above inequality by and integrating the obtained inequality with respect to over [0,1] give that is,
That proves the first inequality of (24).
To prove the second inequality of (24), from the definition of strong convexity of , for , we get
Multiplying the above inequality by and then integrating with respect to over [0,1], we have where implies
Adding on both side of above inequality, which gives (23).
To prove (24), using strong convexity of , we get
Let and ; then, (34) leads
Multiplying the above inequality by and then integrating with respect to over [0,1], we have which can be written as
If follows from the definition of strong convexity of that
Adding the above two inequalities and with the help of Jensen-Mercer inequality, we have
Multiplying the above inequality by and then integrating with respect to over [0,1], we have
that is,
Combining (38) and (43) leads to (24).
Remark 9. Let ; then, Theorem 8 leads to which was proved in Theorem 2.1 of [28].
Lemma 10. Let and be a differentiable mapping such that Then, the inequality holds .
Proof. Suppose where Using integration by parts, we get Similarly, using integration by parts for , we get Therefore, inequality (43) follows from (47)–(50).
Remark 11. (1)If we take , , and in Lemma 10, then we can get Lemma 3.1 of [26](2)If we take , , and , then Lemma 10 reduces to Lemma 1.1 of [29]
Lemma 12. Let and be a differentiable mapping such that . Then, the identity holds .
Proof. Suppose Then, we clearly see that and Therefore, identity (51) follows from (52)–(54).
Corollary 13. If we take , then Lemma 12 leads to the equality
Remark 14. If we take and in Corollary 13, then (55) becomes which was proved in Lemma 2.1 of [30].
Theorem 15. Let and be a differentiable mapping such that and is a convex mapping on . Then, the inequality holds for all
Proof. It follows from Lemma 10, Jensen-Mercer inequality, power mean inequality, and the convexity of function that Using the definition of strong convexity Therefore, inequality (57) can be derived after some simple calculation.
Remark 16. From Theorem 15, we clearly see that (1)If we take , , and in Theorem 15, then we get Theorem 3.1 of [25](2)If we take , , , and in Theorem 15, then we get Theorem 3.1 of [26](3)If we take , , , and in Theorem 15, then we get Theorem 5 of [29] in the case of
Theorem 17. Let and be a differentiable mapping such that and is a convex mapping on . Then, the inequality holds for all .
Proof. It follows from Lemma 10, Jensen-Mercer inequality, power-mean inequality, and the convexity of function that Using definition of strong convexity, we have Making simple simplification, we get (60) from (61).
Remark 18. Theorem 17 leads to (1)If we take , , and in Theorem 17, then we get Theorem 2.12 of [25](2)If we take , , , and in Theorem 17, then we get Theorem 3.1 of [26](3)If we take , , , and in Theorem 17, then we get Theorem 5 of [29] in the case of
Theorem 19. Let and be a differentiable mapping such that and is a convex mapping on . Then, the inequality
Proof. It follows from Lemma 10, Jensen-Mercer inequality, power mean inequality, and the convexity of function that It follows from the strong convexity of which completes the proof.
Corollary 20. Let and . Then, Theorem 19 leads to
Theorem 21. Let with and be a differentiable mapping such that and is a convex mapping on . Then, the inequality holds .
Proof. It follows from Lemma 10, Jensen-Mercer inequality, strong convexity of , and Holder integral inequality that By making necessary changes, we get (67).
Theorem 22. Let and be a differentiable mapping such that and is a convex mapping on. Then, one has
Proof. By using Lemma 12 and similar arguments as in the proofs of previous theorem, we have which completes the proof.
3. New Inequalities by Improved Hölder Inequality
Theorem 23. Let with and be a differentiable mapping such that and is a strong convex mapping on . Then, one has
Proof. It follows from Lemma 10, Jensen-Mercer inequality, the convexity of , and Holder-Iscan integral inequality given in Theorem 1.4 of [31] that Applying definition of strong convexity, By some computations, one can get the required result.
Theorem 24. Let with and be a differentiable mapping such that and is a strong convex mapping on . Then, one has
Proof. It follows from Lemma 10, Jensen-Mercer inequality, the convexity of , and the improved power-mean integral inequality given in Theorem 1.5 of [31] that Using definition of strong convexity,
4. Applications to Special Means
Means are important in applied and pure mathematics, especially they are used frequently in numerical approximation. In literature, they are order in the following way:
The arithmetic mean of two numbers , such that a is defined as
The generalized logarithmic mean is defined as follows:
Proposition 25. Assume and ; then, holds for where
Proof. From Theorem 23, we have with holds. Setting and in the above theorem, we obtain Now use the following: For , we have This implies that By using these means, we get By using the results of Section 3, we get some application to special means.
Proposition 26. Let ; then,
Proof. We obtain the result immediately from Theorem 23.
Proposition 27. Let , then
Proof. We obtain the result immediately from Theorem 24.
Data Availability
All data required for this research is included within the paper.
Conflicts of Interest
Authors of this paper declare that they have no competing interests.
Authors’ Contributions
X.W. analyzed and approved the results, wrote the final version of the paper, and arranged the funding. A.H. proved the main results. M.S.S. proposed the problem and supervised this work. S.U.Z. wrote the first version of the paper.
Acknowledgments
This paper is supported by the Department of Mathematics, University of Okara, Okara, Pakistan.