Giaccardi Inequality for Modified h -Convex Functions and Mean Value Theorems

In this article, we consider the class of modi ﬁ ed h − convex functions and derive the famous Giaccardi and Petrovi c ′ type inequalities for this class of functions. The mean value theorems for the functionals due to Giaccardi and Petrovi c ′ type inequalities are formulated. Some special cases are discussed by taking di ﬀ erent examples of function h .


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 m − convex functions, which seemed like a nice generalization of the convex functions. Varošanec [2] gave the definition of h − convex functions.
It is important to note that m − convex functions and h − 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 ðh, mÞ − convex functions and derive some Hermite-Hadamard type inequalities. Orlicz [4] introduced ϕ − convex functions, which was used in the theory of Orlicz spaces. Motivated by this, Dragomir and Fitzpatrick (see [5,6]) introduced the class of s − convex functions in the first and second sense.
Here we recall some basic definitions.
Þϕ w ð Þ,∀υ, w ∈ Ω, τ ∈ 0, 1 ð Þ: Varošanec [2] gave the definition of h − convex function and derived several results by imposing the conditions on h, which seemed like a nice generalization of the convex functions.
Definition 2. Let h : J ⟶ ℝ be a nonnegative function such that ð0, 1Þ ⊆ J: A function ϕ : Toader defined a new class of nonconvex functions, known as ðh, λ, μÞ − convex functions. Toader looked into the fundamental features of this type of nonconvex function. Here, we recall the definition of modified h − convex, which is basically a special case of ðh, λ, μÞ − convex functions defined by Toader in [7]. Many researchers and mathematicians have explored the modified h − convex functions in the literature in recent years. Noor et al. [8] generalized the Hermite-Hadamard inequality for modified h − convex functions. Zhao et al. [9] discussed Schur-type, Hermite-Hadamard-type, and Fejér-type inequalities for the class of generalized strongly modified h − convex functions.
Here, we discuss Definition 3 in some detail.
(1) Substituting h with an identity function in (3), one gets the convex function.

Theorem 5.
Let Ω ⊆ ℝ be an interval, υ 0 ∈ Ωðυ 1 , ⋯, υ n Þ ∈ Ω n , and ðw 1 , ⋯, w n Þ ∈ ℝ n + ðn ≥ 2Þ such that If ϕ : ω ⟶ ℝ is a convex function, then Many scholars have contributed to the understanding of Giaccardi inequality by publishing results linked to it. In [11] Pecaric′ and Peric′ derived an elegant method of producing n − exponentially and exponentially convex functions when the Giaccardi and Petrovic′ 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 Petrovic ′ inequalities, see [10,13,14].
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 h − convex functions. With the help of these lemmas, the authors derive the Giaccardi and Petrovic′ inequalities for modified h − convex functions. Some special cases are discussed. In the section 3, the authors define the nonnegative linear functional due to Giaccardi inequality for modified h − convex and ðα, 1Þ − convex. Also, define the nonnegative linear functional for the Petrovic ′ 's inequality for modified h − convex and derived the mean values theorems related to these functionals.

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Main Results
For convenience, we assume that h : ð0,∞Þ ⟶ ℝ is a positive function in the rest of the paper.
Proof. Assume that ϕ is modified h − convex function and As h is multiplicative, so we have It follows that This shows that Φ ðϕ;hÞ ðυÞ is increasing on Ω.
Conversely, let υ, η ∈ Ω such that υ < η and That is, Take υ = τη + ð1 − τÞυ 0 , where τ ∈ ð0, 1Þ, and then one has Using the fact that h is multiplicative and then simplifying, one gets the definition of modified h − convex functions.
Remark 9. One can note that if the function ϕ is modified h − convex function, then the mapping Φ ðϕ;hÞ ðυÞ defined in Lemma 8 is increasing if and only if Φ ðϕ;hÞ ′ ðυÞ ≥ 0, provided that the derivatives exist. This is equivalent to Substituting h with an identity function in (18), one gets the result for convex functions given in [19], p. 09].
The Giaccardi inequality for modified h − convex functions is given in the following theorem. where Proof. To prove the main result, first assume that ϕðυÞ/h ðυ − υ 0 Þ is increasing for υ ∈ Ω such that υ > υ 0 . As we have given υ 0 ≤υ n , so one has This gives Multiplying above inequality by w τ and taking sum from τ = 1, ⋯, n, one has This leads to Since ϕ is modified h − convex function, so by Lemma 8, is increasing for υ > υ 0 . Substituting ϕðυÞ by ϕðυÞ − ϕðυ 0 Þ in (24), one has

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This is equivalent to From above inequality, one can deduce (19).
Remark 11. By taking hðυÞ = υ in Theorem 10, one gets Theorem 5. A Giaccardi inequality for ðα, 1Þ − convex functions is given in the following corollary.
Proof. Take hðτÞ = τ α and υ 0 = 0 in Theorem 10 with the restriction that Ω = ½0, aÞ to get the required result. A Petrovic′'s inequality for modified h − convex functions is given in the following corollary.

Corollary 14.
Let the conditions of Theorem 10 be valid for Ω = ½0, aÞ. Then, Proof. Take υ 0 = 0 in Theorem 10 with the restriction that Ω = ½0, aÞ to get the required result.
where A is defined in (20). By taking hðτÞ = τ α in (31), one gets the linear functional for Giaccardi inequality for ðα, 1Þ − convex function given as follows: By taking υ 0 = 0, in (31), one gets the linear functional for Petrovic ′ 's inequality for modified h − convex functions given as follows: In the following lemma, two modified h − convex functions are introduced under certain condition to prove MVT of Lagrange type.
Proof. Since ϕ ∈ C 1 ðΩÞ and h, h′ are bounded, there exists real numbers n and N such that Consider the function ψ 1 defined in Lemma 16. As ψ 1 is modified h − convex function onΩ, therefore, That is, This implies In similar way, if one consider the function ψ 2 defined in Lemma 16, then Combining inequalities (46) and (47), one has So, there exists η in the interior ofΩ such that This is equivalent to (52). In the following corollary, Largrange type MVT related to functional due to Giaccardi inequality for ðα, 1Þ − convex functions is given.
Proof. By taking hðτÞ = σðτÞ = τ α , where α ∈ ½0, 1, in (31), one has Using it in Theorem 17, one gets the required result. Lagrange type MVT for functional due to Petrovic ′ 's inequality for modified h − convex function has been stated in the following corollary.

Corollary 19.
Consider a functional H defined in (33). If ϕ ∈ C 1 ðΩÞ and h and h ′ are bounded, then there exists η in the interior ofΩ such that where φðυÞ = υ 2 , provided that Pðφ ; hÞ 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 hðτÞ = τ in Theorem 17, one gets the result for Giaccardi inequality for convex function. A similar result for Petrovic ′ 's inequality for convex function was given by Rehman et al. in [17], Corollary 13].
Theorem 21. Consider a functional G defined in (31) If ϕ 1 , ϕ 2 ∈ C 1 ðΩÞ, and then there exists η in the interior ofΩ such that provided that the denominators are nonzero.
Proof. Let F ∈ C 1 ðΩÞ be a function, defined as Replace ϕ with F in Theorem 17, then one has This gives Putting the values of t 1 and t 2 , one gets the required result.
In the following corollary, Cauchy type MVT related to functional due to Petrovic′'s inequality for modified h − convex functions is given.

Corollary 22.
Let the conditions of Theorem 17 be valid. If ϕ 1 , ϕ 2 ∈ C 1 ðΩÞ, then there exists η in the interior ofΩ 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 ðα, 1Þ − convex functions is given in the following corollary.
Corollary 23. Let the conditions given in Theroem 6 are valid and ϕ : Ω ⟶ ℝ be ðα, 1Þ − convex function. Then, provided that the denominators are nonzero.

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Proof. If one take hðτÞ = σðτÞ = τ α , where α ∈ ½0, 1, in (31), then This information is used in Theorem 17 to get the required result.
Cauchy type MVT related to functional due to Petrovic ′ 's inequality for ðα, 1Þ − convex functions is given in the following corollary.
Corollary 24. Let the conditions given in Theroem 6 be valid and ϕ : Ω ⟶ ℝ be ðα, 1Þ − convex function. Then, provided that the denominators are nonzero.

Conclusion
In this paper, the authors considered the modified h − convex function and derived the most important Giaccardi and Petrovic′ 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 h give us previously known results published in [16][17][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.