Giaccardi Inequality for Modified -Convex Functions and Mean Value Theorems

Abstract

In this article, we consider the class of modified convex functions and derive the famous Giaccardi and Petrovi type inequalities for this class of functions. The mean value theorems for the functionals due to Giaccardi and Petrovi type inequalities are formulated. Some special cases are discussed by taking different examples of function .

1. Introduction and Preliminaries

Convex functions have played an important role in the development of various fields of pure and applied sciences. Convexity theory describes a broad spectrum of very interesting developments involving a link among various fields of mathematics, physics, economics, and engineering sciences.

Convexity theory is developing rapidly in recent years by utilising fresh and inventive methodologies. Toader [1] developed convex functions, which seemed like a nice generalization of the convex functions. Varošanec [2] gave the definition of convex functions.

It is important to note that convex functions and convex functions are clearly two distinct types of convex functions. It is only reasonable to group these classes together. Özdemir et al. [3] used these facts to introduce convex functions and derive some Hermite-Hadamard type inequalities. Orlicz [4] introduced convex functions, which was used in the theory of Orlicz spaces. Motivated by this, Dragomir and Fitzpatrick (see [5, 6]) introduced the class of convex functions in the first and second sense.

Here we recall some basic definitions.

Definition 1. A function is convex, if Varošanec [2] gave the definition of convex function and derived several results by imposing the conditions on which seemed like a nice generalization of the convex functions.

Definition 2. Let be a nonnegative function such that A function is convex, if

Toader defined a new class of nonconvex functions, known as convex functions. Toader looked into the fundamental features of this type of nonconvex function. Here, we recall the definition of modified convex, which is basically a special case of convex functions defined by Toader in [7]. Many researchers and mathematicians have explored the modified convex functions in the literature in recent years. Noor et al. [8] generalized the Hermite-Hadamard inequality for modified convex functions. Zhao et al. [9] discussed Schur-type, Hermite-Hadamard-type, and Fejér-type inequalities for the class of generalized strongly modified convex functions.

Definition 3. Let be a nonnegative functions. A function is modified convex, if Here, we discuss Definition 3 in some detail. (1)Substituting with an identity function in (3), one gets the convex function.(2)By taking in (3), one gets convex function given in [8]

Definition 4. Let A function is convex in the first sense, if The most famous inequality given by Giaccardi is known as the Giaccardi inequality [10] given in the following theorem.

Theorem 5. Let be an interval, , and such that If is a convex function, then

Many scholars have contributed to the understanding of Giaccardi inequality by publishing results linked to it. In [11] Peari and Peri derived an elegant method of producing exponentially and exponentially convex functions when the Giaccardi and Petrovi differences are applied. Rehman et al. [12] generalized the Giaccardi inequality to coordinates in plane. Also, the authors defined the nonnegative linear functional due to Giaccardi inequality and find the mean value theorems related to that functional. For further information on the Giaccardi and Petrovi inequalities, see [10, 13, 14].

Giaccardi inequality is generalization of famous Petrovi’s inequality introduced by Petrovi [15]. It is a particular case of (6) when . In [16], authors considered the following functional due to Petrovi’s inequality: where and be positive -tuples such that

The following mean value theorems of Lagrange and Cauchy type for above functional was proved in [17].

Theorem 6. Consider a functional defined in (7), if , then there exists , such that which provided that is nonzero and , where .

Theorem 7. Let and be positive -tuples such that the condition given in (8) is valid. If , then there exists , such that which provided that the denominators are nonzero and .

The special case of above functional for a certain class of convex function has been considered in [18]. Many properties of this functional including its particular cases have been discussed in [16, 18].

This paper is organized as follows: in the section 2, the authors give important lemmas for modified convex functions. With the help of these lemmas, the authors derive the Giaccardi and Petrovi inequalities for modified convex functions. Some special cases are discussed. In the section 3, the authors define the nonnegative linear functional due to Giaccardi inequality for modified convex and convex. Also, define the nonnegative linear functional for the Petrovi’s inequality for modified convex and derived the mean values theorems related to these functionals.

2. Main Results

For convenience, we assume that is a positive function in the rest of the paper.

Lemma 8. A function is modified convex on an interval if and only if is increasing for , where

Proof. Assume that is modified convex function and such that where then (11) gives us As is multiplicative, so we have It follows that This shows that is increasing on .
Conversely, let such that and That is, Take where and then one has Using the fact that is multiplicative and then simplifying, one gets the definition of modified convex functions.

Remark 9. One can note that if the function is modified convex function, then the mapping defined in Lemma 8 is increasing if and only if , provided that the derivatives exist. This is equivalent to Substituting with an identity function in (18), one gets the result for convex functions given in [19], p. 09].
The Giaccardi inequality for modified convex functions is given in the following theorem.

Theorem 10. Let be an interval, , and such that the condition (8) is valid. Also, let be a modified convex function with the condition that is a multiplicative function. Then, where

Proof. To prove the main result, first assume that is increasing for such that . As we have given , so one has This gives Multiplying above inequality by and taking sum from one has This leads to Since is modified convex function, so by Lemma 8, is increasing for .
Substituting by in (24), one has This is equivalent to From above inequality, one can deduce (19).

Remark 11. By taking in Theorem 10, one gets Theorem 5.
A Giaccardi inequality for convex functions is given in the following corollary.

Corollary 12. Let the conditions of Theorem 10 be valid. Also, let be a convex function. Then,

Proof. A function , , satisfied all the condition of Theorem 10. So, substituting this value of in Theorem 10 gives the required result.
A Petrovi’s inequality for convex functions is given in the following corollary.

Corollary 13. Let the conditions of Theorem 10 be valid. Also, let be a convex function. Then,

Proof. Take and in Theorem 10 with the restriction that to get the required result.
A Petrovi’s inequality for modified convex functions is given in the following corollary.

Corollary 14. Let the conditions of Theorem 10 be valid for . Then,

Proof. Take in Theorem 10 with the restriction that to get the required result.

Remark 15. If one take , , and in Theorem 10, one gets the result given by the Petrovi for convex function in [15].

3. Mean Value Theorems

To give the mean value theorems (MVTs) for the nonnegative difference of inequality (19), we define the linear functional as follows:

Let be a closed interval, , , and such that condition (8) is valid. Then, for and multiplicative positive function , we define a functional. where is defined in (20).

By taking in (31), one gets the linear functional for Giaccardi inequality for convex function given as follows:

By taking , in (31), one gets the linear functional for Petrovi’s inequality for modified convex functions given as follows:

In the following lemma, two modified convex functions are introduced under certain condition to prove MVT of Lagrange type.

Lemma 16. Let and be differentiable functions such that The functions are modified convex function on , if

Proof. First consider that After differentiating, one has From (34), one has This leads to This implies Hence,
In a similar way, one can prove
It means and are increasing for Hence, by Lemma 8, and are modified convex functions.

Theorem 17. Consider a functional defined in (31). If and , are bounded, then there exists in the interior of such that where , provided that is nonzero.

Proof. Since and , are bounded, there exists real numbers and such that Consider the function defined in Lemma 16. As is modified convex function on , therefore, That is, This implies In similar way, if one consider the function defined in Lemma 16, then Combining inequalities (46) and (47), one has So, there exists in the interior of such that This is equivalent to (52).
In the following corollary, Largrange type MVT related to functional due to Giaccardi inequality for convex functions is given.

Corollary 18. Consider a functional defined in (32). Also, let and be convex function. Then, where , provided that is nonzero.

Proof. By taking , where , in (31), one has Using it in Theorem 17, one gets the required result.
Lagrange type MVT for functional due to Petrovi’s inequality for modified convex function has been stated in the following corollary.

Corollary 19. Consider a functional defined in (33). If and and are bounded, then there exists in the interior of such that where , provided that is nonzero.

Proof. It is a simple consequence of the fact that as stated in (33). Using this fact in Theorem 17 gives the required result.

Remark 20. By taking in Theorem 17, one gets the result for Giaccardi inequality for convex function. A similar result for Petrovi’s inequality for convex function was given by Rehman et al. in [17], Corollary 13].

Theorem 21. Consider a functional defined in (31) If , and then there exists in the interior of such that provided that the denominators are nonzero.

Proof. Let be a function, defined as Replace with in Theorem 17, then one has This gives Putting the values of and , one gets the required result.
In the following corollary, Cauchy type MVT related to functional due to Petrovi’s inequality for modified convex functions is given.

Corollary 22. Let the conditions of Theorem 17 be valid. If , then there exists in the interior of such that provided that the denominators are nonzero.

Proof. It is just a natural result of the fact that as stated in (33). The desired result is obtained by applying this fact to Theorem 17.
Cauchy type MVT related to functional due to Giaccardi inequality for convex functions is given in the following corollary.

Corollary 23. Let the conditions given in Theroem 6 are valid and be convex function. Then, provided that the denominators are nonzero.

Proof. If one take , where , in (31), then This information is used in Theorem 17 to get the required result.
Cauchy type MVT related to functional due to Petrovi’s inequality for convex functions is given in the following corollary.

Corollary 24. Let the conditions given in Theroem 6 be valid and be convex function. Then, provided that the denominators are nonzero.

Proof. If one take , where , in (27), then This information is used in Theorem 17 to get the required result.

4. Conclusion

In this paper, the authors considered the modified convex function and derived the most important Giaccardi and Petrovi inequalities for this class of functions. A linear functional due to the newly defined inequalities is considered to give the MVTs of Lagrange and Cauchy type. It is shown that the results of this article for some examples of functions give us previously known results published in [16–18]. This is an interesting direction for future research.

Data Availability

There is no external data used.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Thanks are due to the National Key Research and Development Program under Grant 2018YFB0904205 and Science & Technology Bureau of Chengdu 2020-YF09-00005-SN supported by Sichuan Science and Technology Program 2021YFH0107 and Erasmus+ SHYFTE Project 598649-EPP-1-2018-1-FR-EPPKA2-CBHE-JP. The research work of Atiq ur Rehman is supported by the Higher Education Commission of Pakistan with NRPU No. 7962.