Journal of Function Spaces
Volume 2019 (2019), Article ID 9487823, 11 pages
https://doi.org/10.1155/2019/9487823

Research Article

Integral Majorization Type Inequalities for the Functions in the Sense of Strong Convexity

Syed Zaheer Ullah,¹ Muhammad Adil Khan,¹ Zareen Abdulhameed Khan,² and Yu-Ming Chu³

¹Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
²Department of Mathematics, Princess Nora Bint Abdulrahman University, Riyadh 11538, Saudi Arabia
³Department of Mathematics, Huzhou University, Huzhou 313000, China

Correspondence should be addressed to Yu-Ming Chu; chuyuming@zjhu.edu.cn

Received 24 January 2019; Accepted 27 March 2019; Published 2 May 2019

Guest Editor: Vita Leonessa

Copyright © 2019 Syed Zaheer Ullah 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 article, we establish several integral majorization type and generalized Favard’s inequalities for the class of strongly convex functions. Our results generalize and improve the previous known results.

1. Introduction

It is well known that convex functions are a class of important functions in the fields of mathematics and other natural sciences; they have been studied for more than one hundred years. In recent years there is a growing interest in generalized convex functions (such as quasi-convex function [1], strongly convex function [2–4], -convex function [5], approximately convex function [6], logarithmically convex function [7, 8], midconvex function [9], pseudo-convex function [10], -convex function [11], -convex function [12], -convex function [13], delta-convex function [14], Schur convex function [15–21], and other convex functions [22–29]) among the researchers of applied mathematics due to the fact that mathematical models with these functions are more suitable to describe problems of the real world than models using conventional convex functions. Recently, a large number of remarkable results and applications for the generalized convex functions can be found in the literature [30–49].

In the article, our focus is on the integral majorization type inequalities for the strongly convex functions.

Definition 1. Let be a real-valued function defined on the interval and a positive real number. Then is said to be strongly convex with modulus if the inequality

holds for all and . From (1) we clearly see that

The following Lemma 2 for strongly convex function is given in [2] (see also [50, Proposition 1.1.2]).

Lemma 2. A real-valued function is a strongly convex function with modulus if and only if the function defined by is a convex function.

Every strongly convex function is convex, but the converse is not true in general. Strongly convex functions have been utilized for showing the convergence of a gradient type algorithm for minimizing a function. They play a significant role in mathematical economics, approximation theory, and optimization theory; many applications and properties for strongly functions can be found in [2–4, 13, 30].

Next we are going to present some basic theories of majorization.

There is a natural description of the indefinite notion that the entries of -tuple are more nearly equal, or less spread out than, to the entries of -tuple . The applicable assertion is that majorizes ; it means that the sum of largest entries of does not exceed the sum of largest entries of for all with equality for . That is, let and be two real tuples and let

be their ordered entries. Then the tuple is said to majorize (or is said to be majorized by ), in symbol , if

holds for and

The theory of majorization is a very significant topic in mathematics; a remarkable and complete reference on the majorization subject is the book by Olkin and Marshall [51]. For example, the theory of majorization is an essential tool that permits us to transform nonconvex complicated constrained optimization problems that involve matrix valued variables into simple problems with scalar variables that can be easily solved [52–55].

The definition of majorization for integrable functions can be stated as follows (see [7]).

Definition 3. Let and be two decreasing real-valued integrable functions on the interval . Then is said to majorize (or is said to be majorized by ), in symbol, , if the inequality

holds for all and

Theorem 4 (See [56]). Let and be two continuous and increasing real-valued functions defined on , and let be a bounded variation function. Then the following statements are true.

(a)If

 for all and

 then

 holds for every continuous convex function .(b)If (8) and (9) hold, then (10) holds for every continuous increasing convex function .

Theorem 5 (See [57]). Let be a convex function, , and be three positive and integrable functions defined on such that

for all and

Then the following statements are true:

(a)If is decreasing on , then

(b)If is increasing on , then

Let , be a positive and continuous concave function defined on , and let be a convex function defined on with

Then Favard [58] proved that the inequalities

and

hold.

The main purpose of the article is to establish several integral majorization type and generalized Favard’s inequalities for strongly convex functions.

2. Main Results

Theorem 6. Let , be a continuous strongly convex function with modulus , and let , , and be three positive and integrable functions defined on such that

for all and

Then the following statements are true.

(a)If is decreasing on , then we have

(b)If is increasing on , then one has

Proof. (a) Let and . Then it follows from (2) and the proof of Lemma 2 given in [57] that

Let . Then (18) and (19) lead to for all , , and

Since is decreasing on , therefore inequality (20) can be deduced easily from the above inequality. Similarly, we can prove part (b) for increasing function defined on .

Theorem 7. Suppose that all the assumptions of Theorem 6 hold. Then the following statements are true.

(a)If is decreasing on , then

(b)If is increasing on , then

Proof. Since is a strongly convex function with modulus , therefore is a convex function, and inequalities (24) and (25) follow easily from the convexity of the function and Lemma 2 given in [57].

Remark 8. Inequalities (13) and (14) can be obtained by (24) and (25) immediately.

Remark 9. Generally, the assumptions of both the functions and are monotonic in majorization theorem, but in Theorems 6 and 7 we only need one of the functions and to be monotonic.

Theorem 10. Suppose that is a continuous strongly convex function with modulus , and , , and are three integrable functions on . If and are nondecreasing (nonincreasing) functions on and , then

Proof. Since is a strongly convex function, therefore using (2) for and , we have

It follows from the Čebyšev inequality [59] that

Therefore, inequality (26) follows from (27) and (28).

Making use of the similar idea as in the proof of Theorem 10, we can obtain the following Theorem 11 immediately.

Theorem 11. Suppose that is a continuous strongly convex function with modulus , and , , and are three integrable functions on . If and are nondecreasing (nonincreasing) functions on , and , then inequality (26) holds.

Theorem 12. The inequality

holds if all the assumptions of Theorem 10 are satisfied.

Proof. Since is a strongly convex function with modulus , therefore is a convex function and inequality (29) can be deduced by applying this convex function in Theorem 6 of [59].

Using strongly convex function we can give an extension of [60, Theorem 2] in the following form.

Theorem 13. Let be two functions such that is a strictly increasing and is strongly convex with modulus , , , and being three positive and integrable functions on such that

for all and

Then the following statements are true.

(a)If is decreasing on , then inequality (20) holds.(b)If is increasing on , then inequality (21) holds.

Proof. We clearly see that it is sufficient to prove the case of , but this case is already proved in Theorem 6.

Similarly, we have Theorem 14 as follows.

Theorem 14. Suppose that all the assumptions of Theorem 13 are satisfied. Then the following statements are true.

(a)If is decreasing on , then inequality (24) holds.(b)If is increasing on , then inequality (25) holds.

The following Lemma 15 was given in [57].

Lemma 15. Let be a positive integrable function and let be an increasing function on ; then

If is a decreasing function on , then inequality (32) holds in reverse directions.

Lemma 16. Let be a real-valued function defined on . Then the following statements are true.

(i)If is a strongly concave function with modulus , then

(a)the function is decreasing on ;(b)the function is increasing on .

(ii)If is a strongly convex function with modulus , then

(c)the function is increasing on if ;(d)the function is decreasing on if .

Proof. (i) Suppose that is a strongly concave function with modulus .
(a) To show that the function is decreasing on , in fact, for we have

which shows that the function is decreasing on .
(b) To show that the function is increasing on , in fact, for we have

which shows that the function is increasing on .
(ii) Suppose that is a strongly convex function with modulus .
(c) Since , by similar method of (a) we can easily prove that the function is increasing on .
(d) Since , by similar method of (b) we can prove that the function is decreasing on .

Next, we establish several Favard type inequalities for strongly convex functions.

Theorem 17. Let be a strongly concave function with modulus on such that is a positive increasing function, let be a strongly convex function with modulus on , , and

Then

If is a strongly convex function with modulus on such that is a positive increasing function and , then the reverse inequality in (36) holds.
Let be a strongly concave function with modulus on such that is a positive decreasing function, let be a strongly convex function with modulus on , , and

Then

If is a strongly convex function with modulus on such that is a positive decreasing function and , then the reverse inequality in (38) holds.

Proof. (a) From Lemma 16(a) we know that the function is decreasing; then using Lemma 15 to the functions and , we obtain

It follows from (35) that inequality (39) can be rewritten as

for all .
As is an increasing function, and by use of Theorem 6(b), we have

Note that

Therefore, we get

If is a strongly convex function with modulus on such that is a positive increasing function and , then the reverse inequality in (36) can be proved by using a similar method as in the proof of part (a) and Lemma 16(c).
(b) From Lemma 16(b) we know that the function is increasing; then using Lemma 15 to the functions and , we obtain

From (37) we clearly see that inequality (44) can be rewritten as

for all .
As is decreasing function, and by using Theorem 6(a) we have

Note that

Therefore,

If is a strongly convex function with modulus on such that is a positive decreasing function and , then the reverse inequality in (38) can be proved by using a similar method as in the proof of part (b) and Lemma 16(d).

Theorem 18. The following statements are true under the assumptions of Theorem 17.

(a)If is a positive increasing function, then

(b)If is a positive decreasing function, then

Proof. (a) We clearly see that is convex function due to being a strongly convex function with modulus . Therefore, inequality (49) follows easily from [57, Theorem 1(i)] and the strong convexity of together with the fact that is a strongly concave function with modulus on such that is positive increasing function.
(b) Similarly, inequality (50) follows easily from [57, Theorem 1(ii)] and the strong convexity of together with the fact that is a strongly concave function with modulus on such that is positive decreasing function.

Theorem 19. Let be an increasing function on , let be a decreasing function on , let , , and be three positive functions on , and let and be integrable on such that

And let be a strongly convex function with modulus . Then the inequality holds

for all .

Proof. From and (51), applying Lemma 15 to the function and the decreasing function we get

Since is increasing, therefore by using Theorem 6 we have

Theorem 20. Let be a strongly convex function with modulus . Then the inequality

holds for all if all the assumptions of Theorem 19 are satisfied.

Proof. We clearly see that is a convex function due to being a strongly convex function with modulus . Therefore, inequality (55) follows easily from [60, Theorem 3] and the convexity of the function .

Remark 21. Clearly, [60, Theorem 3] can be deduced from (52) due to

or from (55) due to

for convex function .

The following Theorem 22 is an extension of Theorem 19.

Theorem 22. Let be a continuous strongly convex function with modulus , ( is a nonnegative real number), and two positive integrable functions on , , and . Then the following statements are true.

(a)If is increasing on and is decreasing on , then

(b)If is increasing on and is increasing on , then

Proof. (a) Let ; then applying Lemma 15 to the function and the decreasing function , we have

Therefore, inequality (58) follows from Theorem 6 and the fact that is an increasing function on .
(b) Let ; then applying Lemma 15 to the function and the increasing function , we get

Therefore, inequality (59) follows from Theorem 6 and the fact that is an increasing function on .

Theorem 23. The following statements are true under the assumptions of Theorem 22.

(a)If is an increasing function on and is a decreasing function on , then

(b)If is increasing on and is an increasing function on , then

Proof. We clearly see that is a convex function due to being a strongly convex function with modulus . Therefore, inequalities (62) and (63) follow from [61, Theorem 2.3] and the convexity of the function .

Remark 24. Clearly, Theorem 2.3(1) and Theorem 2.3(2) given in [57] can be deduced by (62) and (63), respectively.

Remark 25. Theorem 22 is an extension of Favard’s inequality given in Theorem 17. Indeed, let be a strongly convex function with modulus , then is also a strongly convex function with modulus for any . Substituting in (58), one has

Since is a strongly concave function with modulus on such that is a positive increasing function, taking and using (35) and in (64), we obtain the Favard’s inequalities given in Theorem 17.

Remark 26. From (64) we can easily obtain Remark 2.4 given in [61] due to

For an application of Theorem 22, we get Corollary 27 as follows.

Corollary 27. Let , , , and , , , , and be stated as in Theorem 22. Then the following statements are true.

(a)If is increasing on and is decreasing on , then

(b)If is increasing on and is an increasing function on , then

Proof. (a) Since is increasing on and is a decreasing function on , using (58) given in Theorem 22 and substituting , we get

Therefore, inequality (66) follows from (68).
(b) Since is increasing on and is an increasing function on , using (59) given in Theorem 22 and substituting we get inequality (67).

Remark 28. Let , , and be a strongly concave function with modulus on such that is a positive increasing function. Then inequality (66) leads to the classical Favard’s inequality for strongly convex functions with modulus :

Remark 29. From (69) we get the classical Favard’s inequality given in [58] due to

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant nos. 61673169, 11301127, 11701176, 11626101, 11601485).

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