# Giaccardi Inequality for Modifiedh-Convex Functions and Mean Value Theorems

Authors: Yonghong Liu; Wasim Iqbal; Atiq Ur Rehman; Ghulam Farid; Kamsing Nonlaopon
Journal: Journal of Function Spaces
 (2022)

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## Abstract

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


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## Body

## 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 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 thatm−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 h,m−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.Definition 1.
A functionϕ:Ω⊆ℝ⟶ℝ is convex, if
(1)ϕτυ+1−τw≤τϕυ+1−τϕw,∀υ,w∈Ω,τ∈0,1.
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.
Leth:J⟶ℝ be a nonnegative function such that 0,1⊆J. A function ϕ:Ω⊆ℝ⟶ℝ is h−convex, if
(2)ϕτυ+1−τw≤hτϕυ+h1−τϕw,∀υ,w∈Ω,τ∈0,1.Toader defined a new class of nonconvex functions, known ash,λ,μ−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 h,λ,μ−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.Definition 3.
Leth:J⊂ℝ⟶ℝ be a nonnegative functions. A function ϕ:Ω⊆ℝ⟶ℝ is modified h−convex, if
(3)ϕτυ+1−τw≤hτϕυ+1−hτϕw,∀υ,w∈Ω,τ∈0,1.
Here, we discuss Definition3 in some detail.
(1)
Substitutingh with an identity function in (3), one gets the convex function.(2)
By takinghτ=τα in (3), one gets α,1−convex function given in [8]Definition 4.
Letα∈0,1. A function ϕ:Ω⊆ℝ⟶ℝ is α,1−convex in the first sense, if
(4)ϕτυ+1−τw≤ταϕυ+1−ταϕw,∀υ,w∈Ω,τ∈0,1.
The most famous inequality given by Giaccardi is known as the Giaccardi inequality [10] given in the following theorem.Theorem 5.
LetΩ⊆ℝ be an interval, υ0∈Ωυ1,⋯,υn∈Ωn, and w1,⋯,wn∈ℝ+nn≥2 such that
(5)υ~n≔∑τ=1nwτυτ∈Ωandυτ−υ0υ~n−υτ≥0forτ=1,⋯,n.
Ifϕ:ω⟶ℝ is a convex function, then
(6)∑τ=1nwτϕυτ≤∑τ=1nwτυτ−υ0υ~n−υ0ϕυ~n+υ~n∑τ=1nwτ−1υ~n−υ0ϕυ0.Many scholars have contributed to the understanding of Giaccardi inequality by publishing results linked to it. In [11] Pec˘aric′ 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].Giaccardi inequality is generalization of famous Petrovic′’s inequality introduced by Petrovic′ [15]. It is a particular case of (6) when υ0=0. In [16], authors considered the following functional due to Petrovic′’s inequality:
(7)Pϕ=ϕ∑τ=1nwτυτ−∑τ=1nwτϕυτ−1−∑τ=1nwτϕ0,where υ1,⋯,υn∈0,b and w1,⋯,wn be positive n-tuples such that
(8)υ~n≔∑τ=1nwτυτ≥υτ≥υ0forτ=1,⋯,n.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 ϕ∈C10,b, then there exists η∈0,b, such that
(9)Pϕ=η−υ0ϕ′η−ϕη+ϕυ0η−υ02Pφ,
which provided thatPφ is nonzero and υ0∈0,b, where φυ=υ2.Theorem 7.
Letυ1,⋯,υn∈0,b and w1,⋯,wn be positive n-tuples such that the condition given in (8) is valid. If ϕ1,ϕ2∈C10,b, then there exists η∈0,b, such that
(10)Pϕ1;υ0Pϕ2;υ0=η−υ0ϕ′η−ϕη+ϕυ0η−υ0ϕ′η−ϕη+ϕυ0,
which provided that the denominators are nonzero andυ0∈0,b.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 section2, 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 α,1−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.


## 2. Main Results

For convenience, we assume thath:0,∞⟶ℝ is a positive function in the rest of the paper.Lemma 8.
A functionϕ is modified h−convex on an interval Ω if and only if Φϕ;hυ is increasing for υ>υ0, where
(11)Φϕ;hυ=ϕυ−ϕυ0hυ−υ0forallυ,υ0∈Ω.Proof.
Assume thatϕ is modified h−convex function and η>υ>υ0 such that υ=τη+1−τυ0, where τ∈0,1, then (11) gives us
(12)Φϕ;hυ=ϕτη+1−τυ0−ϕυ0hτη+1−τυ0−υ0≤hτϕη+1−hτϕυ0−ϕυ0hτη−υ0.
Ash is multiplicative, so we have
(13)Φϕ;hυ≤hτϕη−hτϕυ0hτhη−υ0.
It follows that(14)Φϕ;hυ≤ϕη−ϕυ0hη−υ0=Φϕ;hη.
This shows thatΦϕ;hυ is increasing on Ω.
Conversely, letυ,η∈Ω such that υ<η and
(15)Φϕ;hυ≤Φϕ;hη.
That is,(16)ϕυ−ϕυ0hυ−υ0≤ϕη−ϕυ0hη−υ0.
Takeυ=τη+1−τυ0, where τ∈0,1, and then one has
(17)ϕτη+1−τυ0−ϕυ0hτη−τυ0≤ϕη−ϕυ0hη−υ0.
Using the fact thath is multiplicative and then simplifying, one gets the definition of modified h−convex functions.Remark 9.
One can note that if the functionϕ is modified h−convex function, then the mapping Φϕ;hυ defined in Lemma 8 is increasing if and only if Φϕ;h′υ≥0, provided that the derivatives exist. This is equivalent to
(18)ϕ′υ≥ϕυ−ϕυ0hυ−υ0h′υ−υ0.
Substitutingh with an identity function in (18), one gets the result for convex functions given in [19], p. 09].
The Giaccardi inequality for modifiedh−convex functions is given in the following theorem.Theorem 10.
LetΩ⊆ℝ be an interval, υ0∈Ω,υ1,⋯,υn∈Ωn, and w1,⋯,wn∈ℝ+nn≥2 such that the condition (8) is valid. Also, let ϕ:Ω⟶ℝ be a modified h−convex function with the condition that h is a multiplicative function. Then,
(19)∑τ=1nwτϕυτ≤Aϕυ~n+∑τ=1nwτ−Aϕυ0,
where(20)A=∑τ=1nwτhυτ−υ0hυ~n−υ0.Proof.
To prove the main result, first assume thatϕυ/hυ−υ0 is increasing for υ∈Ω such that υ>υ0. As we have given υ0≤υ~n, so one has
(21)ϕυτhυτ−υ0≤ϕυ~nhυ~n−υ0.
This gives(22)hυ~n−υ0ϕυτ≤hυτ−υ0ϕυ~n.
Multiplying above inequality bywτ and taking sum from τ=1,⋯,n, one has
(23)hυ~n−υ0∑τ=1nwτϕυτ≤∑τ=1nwτhυτ−υ0ϕυ~n.
This leads to(24)∑τ=1nwτϕυτ≤∑τ=1nwτhυτ−υ0hυ~n−υ0ϕυ~n.
Sinceϕ is modified h−convex function, so by Lemma 8,
(25)Φϕ;hυ=ϕυ−ϕυ0hυ−υ0
is increasing forυ>υ0.
Substitutingϕυ by ϕυ−ϕυ0 in (24), one has
(26)∑τ=1nwτϕυτ−ϕυ0≤∑τ=1nwτhυτ−υ0hυ~n−υ0ϕυ~n−ϕυ0.
This is equivalent to(27)∑τ=1nwτϕυτ≤∑τ=1nwτhυτ−υ0hυ~n−υ0ϕυ~n−ϕυ0−ϕυ0∑τ=1nwτ.
From above inequality, one can deduce (19).Remark 11.
By takinghυ=υ in Theorem 10, one gets Theorem 5.
A Giaccardi inequality forα,1−convex functions is given in the following corollary.Corollary 12.
Let the conditions of Theorem10 be valid. Also, let ϕ:Ω⟶ℝ be a α,1−convex function. Then,
(28)∑τ=1nwτϕυτ≤∑τ=1nwτυτ−υ0αυ~n−υ0αϕυ~n+∑τ=1nwτ1−υτ−υ0αυ~n−υ0αϕυ0.Proof.
A functionhτ=τα, α∈0,1, satisfied all the condition of Theorem 10. So, substituting this value of h in Theorem 10 gives the required result.
A Petrovic′’s inequality for α,1−convex functions is given in the following corollary.Corollary 13.
Let the conditions of Theorem10 be valid. Also, let ϕ:Ω⟶ℝ be a α,1−convex function. Then,
(29)∑τ=1nwτϕυτ≤∑τ=1nwτυταυ~nαϕυ~n+∑τ=1nwτ1−υταυ~nαϕ0.Proof.
Takehτ=τα and υ0=0 in Theorem 10 with the restriction that Ω=0,a 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 Theorem10 be valid for Ω=0,a. Then,
(30)∑τ=1nwτϕυτ≤∑τ=1nwτhυτhυ~nϕυ~n+∑τ=1nwτ−∑τ=1nwτhυτhυ~nϕ0.Proof.
Takeυ0=0 in Theorem 10 with the restriction that Ω=0,a to get the required result.Remark 15.
If one takeΩ=0,a, hτ=τ, and τ0=0 in Theorem 10, one gets the result given by the Petrovic′ 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:LetΩ~⊆ℝ be a closed interval, υ0∈Ω~, υ1,⋯,υn∈Ω~n, and w1,⋯,wn∈ℝ+nn≥2 such that condition (8) is valid. Then, for ϕ:Ω~⟶ℝ and multiplicative positive function h, we define a functional.
(31)Gϕ;h,υ0=Aϕυ~n−∑τ=1nwτϕυτ+∑τ=1nwτ−Aϕυ0,where A is defined in (20).By takinghτ=τα in (31), one gets the linear functional for Giaccardi inequality for α,1−convex function given as follows:
(32)Jϕ;υ0=∑τ=1nwτυτ−υ0αυ~n−υ0αϕυ~n−∑τ=1nwτϕυτ+∑τ=1nwτ1−υτ−υ0αυ~n−υ0αϕυ0.By takingυ0=0, in (31), one gets the linear functional for Petrovic′’s inequality for modified h−convex functions given as follows:
(33)Pϕ;h=∑τ=1nwτhυτhυ~nϕυ~n−∑τ=1nwτϕυτ+∑τ=1nwτ1−hυτhυ~nϕ0.In the following lemma, two modifiedh−convex functions are introduced under certain condition to prove MVT of Lagrange type.Lemma 16.
Letϕ:Ω~⟶ℝ and h:0,∞⟶ℝ+ be differentiable functions such that
(34)n⩽hυ−υ0ϕ′υ−ϕυ−ϕυ0h′υ−υ02υhυ−υ0−υ2−υ02h′υ−υ0⩽N∀υ,υ0∈Ω~.
The functionsϕ1,ϕ2:Ω~⟶ℝ are modified h−convex function on Ω~, if
(35)ψ1υ=Nυ2−ϕυ,(36)ψ2υ=ϕυ−nυ2.Proof.
First consider that(37)Φψ1;hυ=ψ1υ−ψ1υ0hυ−υ0=Nυ2−ϕυ−Nυ02+ϕυ0hυ−υ0=Nυ2−υ02hυ−υ0−ϕυ−ϕυ0hυ−υ0.
After differentiating, one has(38)Φ′ψ1;hυ=Nhυ−υ02υ−υ2−υ02h′υ−υ0h2υ−υ0−hυ−υ0ϕ′υ−ϕυ−ϕυ0h′υ−υ0h2υ−υ0.
From (34), one has
(39)hυ−υ0ϕ′υ−ϕυ−ϕυ0h′υ−υ0≤N2υhυ−υ0−υ2−υ02h′υ−υ0.
This leads to(40)hυ−υ0ϕ′υ−ϕυ−ϕυ0h′υ−υ0h2υ−υ0≤N2υhυ−υ0−υ2−υ02h′υ−υ0h2υ−υ0.
This implies(41)N2υhυ−υ0−υ2−υ02h′υ−υ0h2υ−υ0−hυ−υ0ϕ′υ−ϕυ−ϕυ0h′υ−υ0h2υ−υ0≥0.
Hence,Φψ1;h′υ≥0.
In a similar way, one can proveΦψ2;h′υ≥0.
It meansΦψ1;hυ and Φψ2;hυ are increasing for υ>υ0. Hence, by Lemma 8, ψ1 and ψ2 are modified h−convex functions.Theorem 17.
Consider a functionalG defined in (31). If ϕ∈C1Ω~ and h, h′ are bounded, then there exists η in the interior of Ω~ such that
(42)Gϕ;h,υ0=hη−υ0ϕ′η−ϕη−ϕυ0h′η−υ02ηhη−υ0−η2−υ02h′η−υ0Gφ;h,υ0,where φυ=υ2, provided that Gφ;h,υ0 is nonzero.Proof.
Sinceϕ∈C1Ω~ and h, h′ are bounded, there exists real numbers n and N such that
(43)n⩽hυ−υ0ϕ′υ−ϕυ−ϕυ0h′υ−υ02υhυ−υ0−υ2−υ02h′υ−υ0⩽N,∀υ,υ0∈Ω~.
Consider the functionψ1 defined in Lemma 16. As ψ1 is modified h−convex function on Ω~, therefore,
(44)Gψ1;h,υ0≥0.
That is,(45)GNυ2−ϕυ;h,υ0≥0.
This implies(46)NGφ;h,υ0≥Gϕ;h,υ0.
In similar way, if one consider the functionψ2 defined in Lemma 16, then
(47)nGφ;h,υ0≤Gϕ;h,υ0.
Combining inequalities (46) and (47), one has
(48)n≤Gϕ;h,υ0Gφ;h,υ0≤N.
So, there existsη in the interior of Ω~ such that
(49)Gϕ;h,υ0Gφ;h,υ0=hη−υ0ϕ′η−ϕη−ϕυ0h′η−υ02ηhη−υ0−η2−υ02h′η−υ0.
This is equivalent to (52).
In the following corollary, Largrange type MVT related to functional due to Giaccardi inequality forα,1−convex functions is given.Corollary 18.
Consider a functionalG defined in (32). Also, let ϕ∈C1Ω~ and ϕ:Ω~⟶ℝ be α,1−convex function. Then,
(50)Jϕ;υ0=η−υ0ϕ′η−αϕη−ϕυ0η−υ02η−αη+υ0Jφ;υ0,where φυ=υ2, provided that Jφ;υ0 is nonzero.Proof.
By takinghτ=στ=τα, where α∈0,1, in (31), one has
(51)Jϕ;υ0=Gφ;σ,υ0.
Using it in Theorem17, 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 functionalH defined in (33). If ϕ∈C1Ω~ and h and h′ are bounded, then there exists η in the interior of Ω~ such that
(52)Pϕ;h=hηϕ′η−ϕη−ϕ0h′η2ηhη−η2h′ηPφ;h,where φυ=υ2, provided that Pφ;h is nonzero.Proof.
It is a simple consequence of the fact that(53)Pϕ;h=Gφ;h,0,
as stated in (33). Using this fact in Theorem 17 gives the required result.Remark 20.
By takinghτ=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 functionalG defined in (31) If ϕ1,ϕ2∈C1Ω~, and then there exists η in the interior of Ω~ such that
(54)Gϕ1;h,υ0Gϕ2;h,υ0=hη−υ0ϕ1′η−ϕ1ηh′η−υ0+ϕ1υ0h′η−υ0hη−υ0ϕ2′η−ϕ2ηh′η−υ0+ϕ2υ0h′η−υ0,
provided that the denominators are nonzero.Proof.
LetF∈C1Ω~ be a function, defined as
(55)F=t1ϕ1−t2ϕ2,wheret1=Gϕ2;h,υ0andt2=Gϕ1;h,υ0.
Replaceϕ with F in Theorem 17, then one has
(56)0=hη−υ0t1ϕ1−t2ϕ2η′−t1ϕ1−t2ϕ2ηh′η−υ0+t1ϕ1−t2ϕ2υ0h′η−υ0=hη−υ0t1ϕ1′η−t2ϕ2′η−t1ϕ1ηh′η−υ0+t2ϕ2ηh′η−υ0+t1ϕ1υ0h′η−υ0−t2ϕ2υ0h′η−υ0=t1hη−υ0ϕ1′η−ϕ1ηh′η−υ0+ϕ1υ0h′η−υ0−t2hη−υ0ϕ2′η−ϕ2ηh′η−υ0+ϕ2υ0h′η−υ0.
This gives(57)t2t1=hη−υ0ϕ1′η−ϕ1ηh′η−υ0+ϕ1υ0h′η−υ0hη−υ0ϕ2′η−ϕ2ηh′η−υ0+ϕ2υ0h′η−υ0.
Putting the values oft1 and t2, 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 Theorem17 be valid. If ϕ1,ϕ2∈C1Ω~, then there exists η in the interior of Ω~ such that
(58)Pϕ1;hPϕ2;h=hηϕ1′η−ϕ1ηh′η+ϕ10h′ηhηϕ2′η−ϕ2ηh′η+ϕ20h′η,
provided that the denominators are nonzero.Proof.
It is just a natural result of the fact that(59)Pϕ;h=Gφ;h,0,
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 forα,1−convex functions is given in the following corollary.Corollary 23.
Let the conditions given in Theroem6 are valid and ϕ:Ω⟶ℝ be α,1−convex function. Then,
(60)Jϕ1,υ0Jϕ2,υ0=η−υ0ϕ1′η−αϕ1η+αϕ1υ0η−υ0ϕ2′η−αϕ2η+αϕ2υ0,
provided that the denominators are nonzero.Proof.
If one takehτ=στ=τα, where α∈0,1, in (31), then
(61)Jϕ;υ0=Gφ;σ,υ0.
This information is used in Theorem17 to get the required result.
Cauchy type MVT related to functional due to Petrovic′’s inequality for α,1−convex functions is given in the following corollary.Corollary 24.
Let the conditions given in Theroem6 be valid and ϕ:Ω⟶ℝ be α,1−convex function. Then,
(62)Jϕ1Jϕ2=ηϕ1′η−αϕ1η+αϕ10ηϕ2′η−αϕ2η+αϕ20,
provided that the denominators are nonzero.Proof.
If one takehτ=στ=τα, where α∈0,1, in (27), then
(63)Jϕ;υ0=Gφ;σ,υ0.
This information is used in Theorem17 to get the required result.


## 4. Conclusion

In this paper, the authors considered the modifiedh−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 [16–18]. This is an interesting direction for future research.



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