# Giaccardi Inequality fors-Convex Functions in the Second Sense for Isotonic Linear Functionals and Associated Results

Authors: Dong Chen; Dina Abuzaid; Atiq Ur Rehman; Aqsa Rani
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, a well-known inequality called Giaccardi inequality is established for isotonic linear functionals by applyings-convexity in the second sense, which leads to notable Petrović inequality. As a special case, discrete and integral versions of Giaccardi inequality are derived along with the Petrović inequality as a particular case. In application point of view, newly established inequalities are derived for different time scales.


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## Body

## 1. Introduction

Many important areas of mathematics have been developed just because of convex functions. Due to important properties and characterizations of convex functions supreme branches of pure and applied mathematics are studied extensively. A convex function inℝ is a function, whose epigraph is a convex set. A convex function is always continuous and has left and right derivatives in the interior of its domain. More generally, a function ψ is convex on an interval U if for any two points x and y of U and any λ, where 0≤λ≤1, it satisfies the following inequality:(1)ψλx+1−λy≤λψx+1−λψy.Many researchers had generalized the notion of convex functions by inducting different parameters and functions provided the inequality (1) must retain preserved. For instance s-convex functions [1], m-convex functions [2], α−m-convex functions [3] and h-convex functions [4, 5] can be seen in this regard. One of the important generalizations of convex functions is s-convex functions [6]. The s-convex function in the second sense was introduced by H. Hudzik and L. Maligranda in [1].Definition 1.
Lets∈0,1andU⊆ℝbe an interval. A functionψ:U⟶ℝ is said to be s-convex in the second sense if(2)ψλx+μy≤λsψx+μsψy,for all x,y∈U and λ,μ≥0 with λ+μ=1.
The class ofs-convex functions in the second sense is usually denoted by Ks2.Example 1 (see [7]).
Lets∈0,1anda,b,c∈ℝ. Define the functionψ:0,∞⟶ℝ as follows:(3)ψζ=a,ifζ=0,bζs+c,ifζ>0.
It can be easily checked.(i)
ifb≥0 and 0≤c≤a, then ψ∈Ks2(ii)
ifb>0 and c<0, then ψ∉Ks2.
There are many publications in the literature that explore this form of convexity. Theories and inequalities underlyings-convexity by analytical methods are currently being discussed. It can be easily seen that in case s=1, s-convexity is reduced to the ordinary convexity of function ψ.
Let us recall the definition of isotonic linear functionals from [8].Definition 2.
LetEbe a nonempty set andLbe a class of real-valued functions defined onE having the properties:L1: ifψ,ϕ∈L, then aψ+bϕ∈L for all a,b∈ℝL2:1∈L, that is, if ψζ=1 for ζ∈E, then ψ∈L.
An isotonic linear functional is a functionalA:L⟶ℝ that satisfies the following axioms:A1:Aaψ+bϕ=aAψ+bAϕ for ψ,ϕ∈L,a,b∈ℝA2:ψ∈L,ψζ≥0 for ζ∈E⇒Aψ≥0.
Isotonic linear functionals contribute in a wide area of pure and applied mathematics and play a key role in resolving many problems and inequalities. One can note the common examples of such isotonic functionalsA are given in [8] as follows:(4)Aϕ=∑n∈Epnϕn,whereE⊆1,2,3,…,pn>0,or,Aϕ=∫Eϕdμ,where μ is a positive measure on some suitable set E.
Giaccardi inequality for isotonic linear functionals for convex functions is stated in the following theorem.Theorem 1 (see [8]).
LetL,Asatisfy the conditionsL1,L2, and A1, A2. Also, letx0∈Eandϕ∈Lsuch thatx0≠Aϕand eitherx0≤ϕζ≤Aϕfor allζ∈EorAϕ≤ϕζ≤x0for allζ∈E. Ifψis convex onx0,Aϕ (or on Aϕ,x0) and ψϕ∈L, then the following is valid:(5)Aψϕ≤Aϕ−A1x0ψAϕ+A1−1Aϕψx0Aϕ−x0.The aim of this paper is to prove the Giaccardi inequality fors-convex functions in the second sense for isotonic linear functionals. The well-known Petrović inequality is derived as a particular case of the newly established Giaccardi inequality. As a special case, discrete and integral versions of these inequalities are derived. As applications on the time scale, Giaccardi and Petrović inequalities are discussed for different time scales.


## 2. Main Results

In the following theorem, we prove Giaccardi inequality fors-convex functions in the second sense for isotonic linear functionals.Theorem 2.
LetL,Asatisfy the conditionsL1, L2,andA1, A2,respectively. Also, letϕ∈Lsuch thatx0≠Aϕand either(6)x0−ϕζϕζ−Aϕ≤0,for allζ∈E,or,x0−ϕζϕζ−Aϕ≥0,for allζ∈E.
Ifψ is s-convex in second sense on x0,Aϕ or on Aϕ,x0, and ψϕ∈L, then,(7)Aψϕ≤AψAϕ+ℬψx0,where(8)A=Aϕζ−x0sAϕ−x0s,ℬ=AAϕ−ϕζsAϕ−x0s.Proof.
Letu,v and w be points from the interval having end points x0 and Aϕ such that u<v<w. It is easy to check that(9)λ≔w−vw−u∈0,1,μ≔v−uw−u∈0,1,λ+μ=w−vw−u+v−uw−u=1.
If we takex=u and w=y, then λx+μy=v. As ψ is given to be s-convex in second sense so using the value of x and y along with the values of λ and μ defined above in inequality (2), one has(10)ψv≤w−vw−usψu+v−uw−usψw.
Substitutingu=m, v=ϕζ and w=M in above inequality to get(11)ψϕ≤M−ϕζsM−msψm+ϕζ−msM−msψM.
Now settingm=x0 and M=Aϕ (or) m=Aϕ and M=x0 to get(12)ψϕ≤ϕζ−x0sAϕ−x0sψAϕ+Aϕ−ϕζsAϕ−x0sψx0.
That is,(13)ϕζ−x0sAϕ−x0sψAϕ+Aϕ−ϕζsAϕ−x0sψx0−ψϕ≥0.
Applying the property A2 of the isotonic linear functionalA, one has(14)Aϕζ−x0sAϕ−x0sψAϕ+Aϕ−ϕζsAϕ−x0sψx0−ψϕ≥0.
Using the linearity of the functionalA, one get(15)Aϕζ−x0sAϕ−x0sψAϕ+AAϕ−ϕζsAϕ−x0sψx0≥Aψϕ.
A simplification of the inequality (15) leads us to the required result.Remark 1.
If we takes=1in Theorem2, then one gets Giaccardi’s inequality for isotonic linear functional stated in[8] (as given in Theorem1).
In the following, a discrete version of the above theorem has been derived.Theorem 3.
Letx1,x2,…,xn∈I, whereIis an interval,p1,p2,…,pn be positive numbers and x0,x˜n:=∑i=1npixi∈I such that(16)xi−x0x˜n−xi≥0,orxi−x0x˜n−xi≤0,where x˜n≠x0. If ψ is s-convex in second sense on I, then(17)∑i=1npiψxi≤A′ψx˜n+ℬ′ψx0,where(18)A′=∑i=1npixi−x0sx˜n−x0s,ℬ′=∑i=1npix˜n−xisx˜n−x0s.Proof.
LetE=1,2,…,n, L=ϕ:E⟶I|ϕi=xi,i∈E and(19)Aϕ=∑i∈Epiϕi=∑i=1npixi.
Then,L satisfies conditions L1, L2 and A satisfies conditions A1, A2 of Definition 2. Substituting above values of A and ϕ in Theorem 2, we get(20)ϕζ−x0Aϕ−ϕζ=xi−x0x˜n−xi≥0or≤0,Aϕ=∑i=1npixi≠x0.
Finally, inequality (7) becomes our required result.
The following result provides us the integral version of Theorem2.Theorem 4.
LetΩ,Λ,ρbe a measurable space, whereρΩis positive finite measure. Also letϕ:Ω⟶Ibe a measurable function andx0,∫Ωϕζdρ∈I such that ∫Ωϕζdρ≠x0 and(21)ϕζ−x0∫Ωϕζdρ−ϕζ≥0or≤0.
Ifψ is s-convex on second sense on I, then an inequality(22)∫Ωψϕdρ≤Cψ∫Ωϕζdρ+Dψx0,
is valid, provided the integrals exist and(23)C≔∫Ωϕζ−x0sdρ∫Ωϕζdρ−x0s,D≔∫Ω∫Ωϕζdρ−ϕζsdρ∫Ωϕζdρ−x0s.Proof.
Assume that(24)L=ϕ:Ω⟶I|∫Ωϕζdρexists,Aϕ=∫Ωϕζdρ.
Then,L satisfies conditions L1, L2 and A satisfies conditions A1, A2 of Definition 1. Substituting above values of Aϕ in Theorem 2, we get(25)ϕζ−x0Aϕ−ϕζ=ϕζ−x0∫Ωϕζdρ−ϕζ≥0or≤0,Aϕ=∫Ωϕζdρ≠x0.Ultimately, inequality (7) becomes (22) as our required result.
In the following theorem, Petrović’s inequality fors-convex functions in second sense for isotonic linear functionals is derived.Theorem 5.
Letψbes-convex in second sense on0,Aϕor onAϕ,0, then the inequality,(26)Aψϕ≤AϕsAsϕψAϕ+AAϕ−ϕζsAsϕψ0,holds for either 0≤ϕζ≤Aϕ Thwor all ζ∈E or Aϕ≤ϕζ≤0 for all. ζ∈E.Proof.
Putx0=0 in (7) to get (26).Remark 2.
If we puts=1in Theorem5, one get Petrović’ inequality for isotonic linear functionals stated in[8].
The famous Giaccardi’s inequality given in [9] can be deduced directly from Theorem 2 as stated in the following corollary.Corollary 1.
Letx1,x2,…,xn∈I, whereIis an interval,p1,p2,…,pnbe positive numbers andx0,x˜n:=∑i=1npixi∈Isuch that condition (16) is valid. Ifψis a convex onI(27)∑i=1npiψxi≤∑i=1npixi−x0x˜n−x0ψx˜n+∑i=1npi−1x˜n−x0x˜nψx0.Proof.
Takes=1 in (17) to get (27).
In the following corollary, Petrović’s inequality fors−convex functions has been deduced.Corollary 2.
Letx1,x2,…,xn∈0,aora,0,p1,p2,…,pnbe positive numbers such that,(28)x˜n≤xiorxi≤x˜n,fori=1,2,…,n,where x˜n:=∑i=1npixi≠0. If ψ is s-convex in second sense on 0,a (or) a,0, then(29)∑i=1npiψxi≤∑i=1npixix˜nsψx˜n+∑i=1npi1−xix˜nsψ0.Proof.
Putx0=0 in (17) to get (29).
In the following corollary, Petrović’s inequality [9] has been derived.Corollary 3.
Letx1,x2,…,xn∈0,aora,0,p1,p2,…,pnbe positive numbers such that condition (28) is valid. Ifψis convex on0,aora,0, then(30)∑i=1npiψxi≤ψx˜n+∑i=1npi−1ψ0.Proof.
Putx0=0 and s=1 in (17) to get (30).Corollary 4.
LetΩ,Λ,ρbe a measurable space, whereρΩis a positive finite measure. Also letϕ:Ω⟶Ibe a measurable function andx0,∫Ωϕζdρ∈Isuch that∫Ωϕζdρ≠x0and condition(21)is valid. Ifψis convex onI, then the following is valid:(31)∫Ωψϕdρ≤∫Ωϕζ−x0dρ∫Ωϕζdρ−x0ψ∫Ωϕζdρ+∫Ω∫Ωϕζdρ−ϕζdρ∫Ωϕζdρ−x0ψx0,Provided the integrals exist.Proof.
It is a simple consequence of Theorem4 for s=1.Corollary 5.
LetΩ,Λ,ρbe a measurable space, whereρΩis positive finite measure. Also letϕ:Ω⟶Ibe a measurable function andx0,∫Ωϕζdρ∈Isuch that∫Ωϕζdρ≠0and(32)∫Ωϕζdρ≤ϕζor∫Ωϕζdρ≥ϕζ.
Ifψ is s-convex in second sense on I, then the following is valid:(33)∫Ωψϕdρ≤∫Ωϕζsdρ∫Ωϕζdρsψ∫Ωϕζdρ+∫Ω∫Ωϕζdρ−ϕζsdρ∫Ωϕζdρsψ0,Provided the integrals exist.Proof.
Putx0=0 in (22) to get (33).
The following result holds as the integral version of the famous Petrović’s inequality.Corollary 6.
LetΩ,Λ,ρbe a measurable space, whereρΩis a positive finite measure. Also letϕ:Ω⟶Ibe a measurable function and∫Ωϕζdρ∈Isuch that(32)is valid. Ifψis a convex onI, then the following is valid:(34)∫Ωψϕdρ≤ψ∫Ωϕζdρ+∫Ω∫Ωϕζdρ−ϕζdμ∫Ωϕζdρψ0.Provided the integrals exist.Proof.
Puts=1 in Corollary 5 to get the required result.


## 3. Applications in Time Scale Calculus

The theory of time scale was introduced by S. Hilger in his Ph.D. Thesis in 1988 (see [10]). A time scale is an arbitrary nonempty closed subset of the real numbers ℝ, and we usually denote it by the symbol T. The real numbers, the integers, the nonnegative integers, and the natural numbers are examples of time scales. In this section, we give Giaccardi and Petrović inequalities for s-convex functions in the second sense for isotonic linear functional defined on different time-scales taken from the literature (for example, see [11]).(a)
Forp,q∈T=ℤ with p<q, suppose L consists of all real-valued functions defined on p,q−1∩ℤ and Aϕ=∑ζ=pq−1ϕζ. Let ϕ∈L such that the assumptions of Theorem 2 are satisfied. If ψ is an s-convex in second sense on x0,∞ (or −∞,x0) such that ψϕ∈L, then(35)∑ζ=pq−1ψϕζ≤∑ζ=pq−1ϕζ−x0s∑ζ=pq−1ϕζ−x0sψ∑ζ=pq−1ϕζ+∑ζ=pq−1∑ζ=pq−1ϕζ−ϕζs∑ζ=pq−1ϕζ−x0sψx0.Under the same conditions, Petrović’s inequality fors-convex in second sense for time-scale T can be derived by putting x0=0 in (35), that is,(36)∑ζ=pq−1ψϕζ≤∑ζ=pq−1ϕζs∑ζ=pq−1ϕζsψ∑ζ=pq−1ϕζ+∑ζ=pq−1∑ζ=pq−1ϕζ−ϕζs∑ζ=pq−1ϕζsψ0.(b)
Forh>0 and p,q∈T=hℤ with p<q, suppose L consists of all real-valued functions defined on p,q−h∩hℤ and Aϕ=h∑ζ=p/hq/h−1ϕζ. Also let ϕ∈L such that the assumptions of Theorem 2 are satisfied. If ψ is s-convex in second sense on x0,∞ (or −∞,x0) such that ψϕ∈L, then(37)∑ζ=p/hq/h−1ψϕζ≤∑ζ=p/hq/h−1ϕζ−x0sh∑ζ=p/hq/h−1ϕζ−x0sψh∑ζ=p/hq/h−1ϕζ+∑ζ=p/hq/h−1h∑ζ=p/hq/h−1ϕζ−ϕζsh∑ζ=p/hq/h−1ϕζ−x0sψx0.Under the same conditions, Petrović’s inequality fors-convex functions in second sense for time-scale T can be derived by putting x0=0 in (37), that is,(38)∑ζ=p/hq/h−1ψϕζ≤∑ζ=p/hq/h−1ϕζsh∑ζ=p/hq/h−1ϕζsψh∑ζ=p/hq/h−1ϕζ+∑ζ=p/hq/h−1h∑ζ=p/hq/h−1ϕζ−ϕζsh∑ζ=p/hq/h−1ϕζsψ0.(c)
Forp, q∈T=ℝ with p<q, suppose(39)L=Cp,q,ℝandAϕ=∫pqϕζdζ.Also, letϕ∈L such that the assumptions of Theorem 2 are satisfied. If ψ is s-convex in second sense on x0,∞ (or −∞,x0) such that ψϕ∈L, then(40)∫pqψϕζdζ≤∫pqϕζ−x0sdζ∫pqϕζdζ−x0sψ∫pqϕζdζ+∫pq∫pqϕζdζ−ϕζsdζ∫pqϕζdζ−x0sψx0.Under the same conditions, Petrović’s inequality fors-convex functions in the second sense for time-scale T can be derived by putting x0=0 in (40), that is,(41)∫pqψϕζdζ≤∫pqϕζsdζ∫pqϕζdζsψ∫pqϕζdζ+∫pq∫pqϕζdζ−ϕζsdζ∫pqϕζdζsψ0.(d)
Forp, q∈T with p<q, suppose(42)E=p,q∩T,L=Crdp,q,ℝ,Aϕ=∫pqϕζΔζ,whereCrd represents rd-continuous functions [12] and the integral is the Cauchy delta time-scale integral [13]. Also, let ϕ∈L such that the assumptions of Theorem 2 are satisfied. If ψ is s-convex in second sense on x0,∞ (or −∞,x0) such that ψϕ∈L, then(43)∫pqψϕζΔζ≤∫pqϕζ−x0sΔζ∫pqϕζΔζ−x0sψ∫pqϕζΔζ+∫pq∫pqϕζΔζ−ϕζsΔζ∫pqϕζΔζ−x0sψx0.Under the same conditions, Petrović’s inequality fors-convex functions in second sense for time-scale T can be derived by putting x0=0 in (43), that is,(44)∫pqψϕζΔζ≤∫pqϕζsΔζ∫pqϕζΔζsψ∫pqϕζΔζ+∫pq∫pqϕζΔζ−ϕζsΔζ∫pqϕζΔζsψ0.(e)
Forp, q∈T with p<q, suppose(45)E=p,q∩T,L=Cldp,q,ℝ,Aϕ=∫pqϕζ∇ζ,whereCld represents ld-continuous functions [14] and the integral is the Cauchy nabla time-scale integral [15]. Also, let ϕ∈L such that the assumptions of Theorem 2 are satisfied. If ψ is s-convex in second sense on x0,∞ (or −∞,x0) such that ψϕ∈L, then,(46)∫pqψϕζ∇ζ≤∫pqϕζ−x0s∇ζ∫pqϕζ∇ζ−x0sψ∫pqϕζ∇ζ+∫pq∫pqϕζ∇ζ−ϕζs∇ζ∫pqϕζ∇ζ−x0sψx0.Under the same assumptions, Petrović’s inequality fors-convex functions in second sense for time-scale T can be derived by putting x0=0 in (46), that is,(47)∫pqψϕζ∇ζ≤∫pqϕζs∇ζ∫pqϕζ∇ζsψ∫pqϕζ∇ζ+∫pq∫pqϕζ∇ζ−ϕζs∇ζ∫pqϕζ∇ζsψ0.(f)
Forp, q∈T with p<q, suppose(48)E=p,q∩T,L=Cp,q,ℝ,Aϕ=∫pqϕζ⋄αζ,where the integral is the Cauchyα-diamond time-scale integral [16]. Also, let ϕ∈L such that the assumptions of Theorem 2 are satisfied. If ψ is s-convex in second sense on x0,∞ (or −∞,x0) such that ψϕ∈L, then(49)∫pqψϕζ⋄αζ≤∫pqϕζ−x0s⋄αζ∫pqϕζ⋄αζ−x0sψ∫pqϕζ⋄αζ+∫pq∫pqϕζ⋄αζ−ϕζs⋄αζ∫pqϕζ⋄αζ−x0sψx0.Under the same assumptions, Petrović’s inequality fors-convex functions in second sense for given time-scale can be derived by putting x0=0 in (49), that is,(50)∫pqψϕζ+αζ≤∫pqϕζs+αζ∫pqϕζ+αζsψ∫pqϕζ+αζ+∫pq∫pqϕζ+αζ−ϕζs+αζ∫pqϕζ+αζsψ0.(g)
Assume thatT1, T2, …, Tn are time scales and pi, qi∈Ti such that pi<qi, for i=1,2,…,n. Also, suppose(51)E⊂p1,q1∩T1×⋯×pn,qn∩Tn,be JordanΔ-measurable and let L be the set of all bounded Δ-integrable functions from E to ℝ. Moreover, let(52)Aϕ=∫EϕζΔζ,where the integral is the multiple Riemann delta time-scale integral [11]. Also, let ϕ∈L such that the assumptions of Theorem 2 are satisfied. If ψ is s-convex in second sense on x0,∞ (or −∞,x0) such that ψϕ∈L, then(53)∫EψϕΔζ≤∫Eϕζ−x0sΔζ∫EϕζΔζ−x0sψ∫EϕζΔζ+∫E∫EϕζΔζ−ϕζsΔζ∫EϕζΔζ−x0sψx0.Under the same assumptions, Petrović’s inequality fors-convex functions in second sense for a given time-scale that can be derived by putting x0=0 in (53), that is(54)∫EψϕΔζ≤∫EϕζsΔζ∫EϕζΔζsψ∫EϕζΔζ+∫E∫EϕζΔζ−ϕζsΔζ∫EϕζΔζsψ0.(h)
Under the time-scale defined in the above-given clause 7, letE be Lebesgue Δ-measurable, L be the set of all Δ-integrable functions from E to ℝ and Aϕ, for ϕ∈L, be multiple Lebesgue delta time-scale integral (see [17]). If the assumptions of Theorem 2 are satisfied under these conditions, then for s-convex function in the second sense ψ on x0,∞ (or −∞,x0, the inequalities (53) and (54) are valid.


## 4. Discussion

The most famous Giaccardi and Petrović inequalities fors-convex functions in the second sense for isotonic linear functionals are derived. It has been shown that these new findings generate many classical results for different particular cases of the isotonic linear functionals. While understanding the significance of time-scale calculus, these inequalities have been derived for various time-scale integrals. It is important to note that the results derived in this paper for isotonic linear functional can be particularized for other branches of science.



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