Journal of Mathematics
Volume 2020 (2020), Article ID 4957141, 9 pages
https://doi.org/10.1155/2020/4957141

Research Article

Strongly Reciprocally -Convex Functions and Some Inequalities

Hao Li,¹ Muhammad Shoaib Saleem,² Ijaz Hussain,² and Muhammad Imran²

¹School of Science, Department Mathematics and Information, University of Changan, Xi’an, China
²Department of Mathematics, University of Okara, Okara, Pakistan

Correspondence should be addressed to Muhammad Shoaib Saleem; shaby455@yahoo.com

Received 22 April 2020; Accepted 28 July 2020; Published 4 September 2020

Academic Editor: Nan-Jing Huang

Copyright © 2020 Hao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we generalize the concept of strong and reciprocal convexity. Some basic properties and results will be presented for the new class of strongly reciprocally -convex functions. Furthermore, we will discuss the Hermite–Hadamard-type, Jensen-type, and Fejér-type inequalities for the strongly reciprocally -convex functions.

1. Introduction

The importance of convex functions and convex sets cannot be ignored, especially in nonlinear programing [1–5] and optimization theory [6], see, for instance, [7–14]. Generalization in the convexity is always appreciable. Also, many generalizations and extensions have been made in the theory of inequalities as well as in convexity. Several inequalities have been studied and established for the convexity of functions, and many generalizations, applications, and refinements take place, see [7, 9, 13, 15–18], for further study.

In the theory of inequalities, the famous inequality, Hermite–Hadamard inequality was established by Jaques Hadamard [19]. If is a convex function, then

holds for all with .

In [10], Lipot Fejér established the weighted version of the Hermite–Hadamard inequality.

If is a convex function, then the inequality

holds for all with and is integrable, nonnegative, and symmetric about .

For more details on the Fejér inequality, see [8, 9, 11, 20–22]. The main motivation of this article is based on [18].

Mathematically, Jensen-type inequality is stated as if is a convex function defined on , then

holds for all , and with .

This inequality has applications in probability and statistics.

The article is organized as follows: Section 2 is devoted to preliminaries and basic results, whereas in the last section, we will develop the main results for strongly reciprocally -convex functions.

2. Preliminaries

This section concerns preliminaries and basic results for the strongly reciprocally -convex functions.

Definition 1 (-convex set; see [23]). An interval is called the -convex set if for all and , where or , , , and .

Definition 2 (-convex function; see [24]). A function is called -convex function if

for all and , where is the -convex set.

Definition 3 (strongly convex function; see [14]). Let be a positive number. A function is called a strongly convex function if

for all and .

Definition 4 (strongly -convex function; see [25]). Let be a positive number. A function is called strongly -convex function if

for all and .

Definition 5 (harmonic convex function; see [22]). Let be an interval. A function is harmonic convex if

for all and .

Definition 6 (harmonic -convex function; see [26]). A function is called a harmonic -convex function if

for all and .

Definition 7. (strongly reciprocally convex function; see [18]). Let and . A function is said to be strongly reciprocally convex with modulus on if the inequality

holds for all and .
Now, we are ready to introduce a new class of convexity named as strongly reciprocally -convex function.

Definition 8 (strongly reciprocally -convex function). A function is called strongly reciprocally -convex with modulus on if the inequality

holds, for all and .

Remark 1.

(1)If we insert in inequality (10), then we retrace the strong and reciprocal convexity [18](2)If we insert in inequality (10), then we retrace the harmonic -convexity [26](3)If we insert and in inequality (10), then we retrace the harmonic convexity [22]

The following proposition expresses the algebraic property of strongly reciprocally -convex functions.

Proposition 1. Let be two strongly reciprocally -convex functions; then, the following statements hold:

(i) is strongly reciprocally -convex(ii)For any , is strongly reciprocally -convex corresponding to

Proof.

(i)Choose ; then, by the definition of and , we obtain

 where .(ii)Let ; then, by definition, we obtain

where and .
The next lemma establishes the connection between the strong and reciprocal -convexity and harmonic -convexity.

Lemma 1. Let be a function; is strongly reciprocally -convex iff the function , defined by , is harmonically -convex.

Proof. Let be strongly reciprocally -convex; then, we have

This shows that is a harmonic -convex function.
Conversely, if is harmonically -convex, then

This implies that is a strongly reciprocally -convex function for all and .

3. Main Results

In this section, Hermite–Hadamard-, Fejér-, and Jensen-type inequalities are investigated. The next theorem gives the generalization of the Hermite–Hadamard inequality for strongly reciprocally -convex functions.

Theorem 1. (Hermite–Hadamard-type inequality). Let be an interval on the real line. If is a strongly reciprocally -convex function with modulus and , then

for all with .

Proof. We start by the definition; set in inequality (10), and we have

Let and , and by integrating w.r.t over [0, 1], the above inequality yields

and then inequality (17) is reduced to

which is the left side of the inequality.
For the right side of inequality (15), set and in (10); we have

Integrating w.r.t over [0, 1], the above inequality yields

Since

then we obtain

From (18) and (22), we get (15).

Remark 2.

(1)For in (15), Hermite–Hadamard inequality for strongly reciprocally convex functions is obtained [18].(2)If we allow in inequalities (15), we obtain the Hermite–Hadamard-type inequalities for harmonically convex functions [22].

For further details on Hermite–Hadamard inequities, see [27–30].

Theorem 2. (Fejér-type inequality). Assume is a strongly reciprocally -convex function with modulus on ; then,

holds for with and where is a nonnegative integrable function that satisfies

Proof. Since is a strongly reciprocally -convex function, then by definition for in (10), we have

for all ; suppose and in the above inequality; then, we obtain

Since is nonnegative and symmetric, we have

The above inequality is integrated with respect to over [0, 1], and then putting , we obtain

After simplification, the above inequality becomes

For the right-hand side of (23), set and in (10); we have

Integrating with respect to over [0, 1] and then putting , we obtain

After simplification, we have

From (32) and (27), we get (23).

Remark 3. If we set in (23), the Fejér-type inequality for strongly reciprocally convex functions is obtained.
Jensen-type inequality for the aforementioned inequality is described in the next theorem.

Theorem 3. (Jensen-type inequality). If is a reciprocally strongly p-convex function with modulus , then

holds for all , with and .

Proof. Fix and such that .
Put , and suppose a function of the form

supporting at , satisfying and . Then, for every , we have

Multiplying both sides by and summing up to n, we have

Since , we have

which completes the proof.

Remark 4. In inequality (34), fixing and yields the Jensen-type inequality for the harmonic convex function [22]. See [31–34] for more details on Jensen-type inequalities.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

Hao Li analyzed all results and proofread and revised the paper, Muhammad Shoaib Saleem proposed the problem and supervised the work, Ijaz Hussain proved the results, and Muhammad Imran wrote the whole paper.

Acknowledgments

This research was supported by the Higher Education Commission of Pakistan.

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