# On Generalized Stronglyp-Convex Functions of Higher Order

Authors: Muhammad Shoaib Saleem; Yu-Ming Chu; Nazia Jahangir; Huma Akhtar; Chahn Yong Jung
Journal: Journal of Mathematics
 (2020)

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## Abstract

The aim of this paper is to introduce the definition of a generalized stronglyp-convex function for higher order. We will develop some basic results related to generalized strongly p-convex function of higher order. Moreover, we will develop Hermite–Hadamard-, Fejér-, and Schur-type inequalities for this generalization.


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## Body

## 1. Introduction

Some geometric properties of convex sets and functions have been studied before 1960 by great mathematicians Hermann, Minkowski, and Werner Fenchel. The classical convexity is defined as follows: a functionf:I⟶R is said to be convex function if(1)ftx+1−ty≤tfx+1−tfyfor all x,y∈I and t∈0,1.The notion of convexity is crucial to the solution of many real-world problems. Fortunately, many problems encountered in constrained control and estimation are convex. Since the convexity of sets and functions has been the main object of studies of recent years, in many new problems encountered in applied mathematics, the notion of classical convexity is not enough to reach favorite results [1–4]. Recently, several extensions have been considered for classical convexity such that some of these new concepts are based on extension of the domain of convex function or set to a generalized form [5–7]. Some new generalized concepts in this point of view are pseudoconvex function, quasiconvex functions, invex functions, preinvex functions, B-convex functions, strongly convex functions, and generalized strongly convex functions. There are several fundamental books devoted to different aspects of convex analysis optimization [8–13]. Among them, we mention convex analysis by Rickafaller [11], convex analysis and minimization algorithm by Hiriart-Urruty and Lemerechel [8], convex analysis and nonlinear optimization by Browan and Lewin [9], and introducing lectures on convex optimization by Nesterov [10].Several inequalities, including some famous inequalities are of Schur-type, Hermite–Hadamard-type, and Fejér-type inequalities, are satisfied by convex functions. The following inequality is known as Hermite–Hadamard inequality [14–20].Letf:I⊂R⟶R be a convex function and let c,d∈I with c<d, then the following holds:(2)fc+d2≤1d−c∫cdfxdx≤fc+fd2.The present paper is organized as follows: in the first section, we will give some basic definitions and some basic properties related to our work. Next, we will derive Hermite–Hadamard-type, Fejér-type and Schur-type inequalities for our definition.


## 2. Basic Definitions

Let us recall few definitions related to our work [21–25].A convex function is called strictly convex if the above inequality holds strictly wheneverx and y are distinct points and t∈0,1.Definition 1.
(see [21]). A function f:Rn⟶R is said to be quasiconvex if for any x,y∈Rn and t∈0,1, we have(3)fx1−t+tfy≤maxfx,fy.Definition 2.
The intervalI is said to be p-convex set if(4)txp+1−typ1/p∈Ifor all x,y∈I and t∈0,1.
In [26], Zhang et al. introduced p-convexity as follows.Definition 3.
LetI be convex set. A function f is said to be p-convex function if the following inequality holds:(5)ftxp+1−typ1/p≤tfx+1−tfy,wherever x,y∈a,b and t∈0,1,p>0.
It can be easily seen that, forp=1, p-convexity reduced to classical convexity of functions defined on I⊂0,∞.
In [27], Polyak introduced the strongly convex function as follows.Definition 4.
SupposeI is a convex set. A function f:I⟶R is said to be strongly convex function with modulus μ>0, if(6)ftx+1−ty≤tfx+1−tfy−μt1−ty−x2for all x,y∈I and t∈0,1.Definition 5.
(see [28]). A function f:I⟶R is said to be strongly p-convex function, if(7)ftxp+1−typ1/p≤tfx+1−tfy−μt1−ty−x2for all x,y∈I and t∈0,1.Definition 6.
(see [29]). A function f:I⟶R is said to be generalized convex function with respect to η:A×A⟶B for appropriate A,B⊆R, if(8)ftx+1−ty≤fy+tηfx,fyfor all x,y∈I and t∈0,1.Definition 7.
(see [26]). A function f:I⟶R is said to be generalized p-convex function, if(9)ftxp+1−typ1/p≤fy+tηfx,fyfor all x,y∈I and t∈0,1.
The generalized stronglyp-convex function [30] is defined as follows.Definition 8.
A functionf is said to be a generalized strongly p-convex function, if(10)ftxp+1−typ1/p≤fy+tηfx,fy−μt1−typ−xp2holds for t∈0,1.
In [31], Lin and Fukushima gave the concept of higher order strongly convex functions and also used it many mathematical programs. Mishra and Sharma [32] derived the Hermite–Hadamard-type inequalities of higher order strongly convex function.Definition 9.
(see [31]). A function f on the convex and closed set I is said to be strongly convex function of higher order if(11)f1−tx+ty≤1−tfy+tfx−μϕty−xqfor all x,y∈I,t∈0,1 with μ≥0, where ϕt=t1−t and q is any positive real number. If q=2, then higher order strongly convex function becomes strongly convex functions with ϕt=t1−t.
Now, in the view of above definitions, we are in the position to introduce new generalization of convexity as follows:Definition 10.
A functionf is said to be generalized strongly convex of higher order if(12)ftx+1−ty≤fy+tηfx,fy−μϕty−xqfor x,y∈I, with μ≥0 and q is any positive real number
Some generalizations of stronglyp-convex function of higher order are given in [11] for bifunctions.Definition 11.
A functionf is said to be a generalized strongly p-convex of higher order if(13)ftxp+1−typ1/p≤fy+tηfx,fy−μϕtyp−xpqfor x,y∈I, with μ≥0.Remark 1.
(1)
If we takeq=2 and ϕt=t1−t in (13), then we obtain (10).(2)
If we takep=1, then we get (12).(3)
Insertingp=1 and ηx,y=x−y, we obtain (11).


## 3. Basic Results

This section is to introduce some basic results related to a generalized stronglyp-convex function of higher order.Definition 12.
A functionη is said to be nonnegatively homogeneous if nλx,λy=ληx,y for all x,y∈R and λ≥0.Definition 13.
(see [33]). A function η is said to be additive if ηa,b+ηc,d=ηa+b,c+d for all a,b,c,d∈R.Proposition 1.
Letf,g:I⟶R be two generalized strongly p-convex functions of higher order, then the following statements hold:(i)
Ifη is additive, then f+g:I⟶R is a generalized strongly p-convex function of higher order.(ii)
Ifη is nonnegatively homogeneous, then for any λ≥0,λf:I⟶R is a generalized strongly p-convex function of higher order.(iii)
Iff1,f2,f3,…fn be generalized strongly p-convex function of higher order with μ≥0 on I and αi≥0, then f=α1f1+α2f2+⋯+αnfn=∑i=1nαifi. is also generalized strongly p-convex functions of higher order.Proof.
(i)
Takeu=txp+1−typ1/p, then by definition of f and g, we obtain(14)f+gtxp+1−typ1/p=ftxp+1−typ1/p+gtxp+1−typ1/p≤fy+tηfx,fy−μϕtyp−xpq+gy+tηgx,gy−μϕtyp−xpq=f+gy+tηf+gx,f+gy−μ∗ϕtyp−xpq,whereμ∗=2μ and μ≥0.(ii)
Letλ≥0, then by definition, we obtain(15)λftxp+1−typ1/p≤λfy+tηfx,fy−μϕtyp−xpqλfy+tηλfx,λfy−μ∗ϕtyp−xpq,whereμ∗=λμ and μ≥0.(iii)
It directly follows from (i) and (ii).Proposition 2.
Letfi:Rn⟶R¯ for i∈I be collection of generalized strongly p-convex functions of higher order. Then, supremum function(16)fx=supfix,i∈I,is also a generalized strongly p-convex function of higher order.Proof.
Fixx,y∈Rn and t∈0,1. For every i∈I, we have(17)fitxp+1−typ1/p≤fiy+tηfix,fiy−μϕtyp−xpq≤fy+tηfx,fy−μϕtyp−xpq,which implies in turn that(18)ftxp+1−typ1/p=supi∈Ifitxp+1−typ1/p≤fy+tηfx,fy−μϕtyp−xpq.
This justifies the convexity of supremum function.Proposition 3.
Letfi:Rn⟶R be a generalized strongly p-convex function of higher order, then(19)f=maxfi,i=1,2,3,…,n,is also a generalized strongly p-convex function.Proof.
Take anyx,y∈Rn and t∈0,1. Denote f=maxfi, where i=1,2,3,…,n, then(20)ftxp+1−typ1/p=maxfitxp+1−typ1/p,i=1,2,3,…,n=fwtxp+1−typ1/p≤fwy+tηfwx,fwy−μϕtyp−xpq=maxfiy+tηmaxfix,maxfiy−μϕtyp−xpq=fy+tηfx,fy−μϕtyp−xpq,implying that f=maxfi,i=1,2,3,…,n is also a generalized strongly p-convex function of higher order.


## 4. Main Results

In this section, we will develop the Hermite–Hadamard-, Fejér-, and Schur-type inequalities for generalized stronglyp-convex function of higher order.Theorem 1.
(Hermite–Hadamard-type inequality). Letf:I⟶R be a generalized strongly p-convex function of higher order with μ≥0 and p>0 such that η.,. is bounded above Mη, then the function satisfies the following:(21)fap+bp21/p−Mη2+μϕ12bp−apq1+−1q+12q+1≤pbp−ap∫abxp−1fxdx≤fa+fb2+Mη2−μbp−apq∫01ϕtdt.Proof.
We begin the proof by proving the left-hand side of theorem.
Letf:I⟶R be a generalized strongly p-convex function of higher order for all x,y∈I, we set t=1/2, and we have(22)fxp+yp21/p≤fy+12ηfx,fy−μϕ12yp−xpq.
Integrating above equation with respect tot over [0, 1], we have(23)∫01fap+bp21/pdt≤∫01fydt+12∫01ηfx,fydt−μϕ12∫01yp−xpq.
Takex=tap+1−tbp1/p and y=tbp+1−ta,p1/p, we obtain(24)∫01fap+bp2dt≤∫01ftbp+1−tap1/pdt+12∫01ηftap+1−tbp1/p,ftbp+1−tap1/p−μϕ12∫012t−1qbp−apqdt.
We obtain(25)fap+bp21/p−Mη2+μϕ12bp−apq1+−1q+12q+1≤pbp−ap∫abxp−1fxdx,which is the left-hand side of theorem.
To prove right-hand side, we takex=tap+1−tbp1/p for all t∈0,1. By definition, we have(26)ftxp+1−typ1/p≤fy+tηfx,fy−μϕtyp−xpq.
Integrating the above inequality with respect tot over [0, 1], we get(27)∫01ftap+1−tbp1/pdt≤fb+tηfa,fb∫01tdt−μbp−apq∫01ϕtdt.
After solving this, we get(28)pbp−ap∫abxp−1fxdx≤fb+η2fa,fb−μbp−apq∫01ϕtdt=A.
Also in the similar way, we obtain(29)pbp−ap∫abxp−1fxdx≤fa+η2fb,fa−μbp−apq∫01ϕtdt=B.
From (A) and (B), we have(30)pbp−ap∫abxp−1fxdx≤minA,B.
This implies that(31)pbp−ap∫abxp−1fxdx≤A+B2,(32)pbp−ap∫abxp−1fxdx≤fa+fb2+ηfa,fb+ηfb,fa4−μbp−apq∫01ϕtdt≤fa+fb2+Mη2−μbp−apq∫01ϕtdt,which is the right-hand side of theorem. This completes the proof.Remark 2.
Imposing some conditions on Theorem1, we obtain different versions of Hermite–Hadamard inequality [30, 33]:(1)
If we takeϕt=t1−t, q=2 gives the Hermite–Hadamard inequality for generalized strongly p-convex functions.(2)
If we takeμ=0 and p=1, Theorem 1 gives Hermite–Hadamard inequality for η-convex functions.(3)
If we takeηx,y=x−y, then we obtain the Hermite–Hadamard-type inequality of strongly convex function of higher order [34].Theorem 2.
(Schur-type inequality). Letf:I⟶R be a generalized strongly convex function of higher order with μ≥0 and η.,. , be a bifunction, then for all x1,x2,x3∈I such that x1<x2<x3 and x3p−x1p,x3p−x2p,x2p−x1p∈0,1, the following inequality holds:(33)fx3x3p−x1p−fx2x3p−x1p+ηfx,fyx3p−x1p−μϕx3p−x2px3p−x1q≥0.Proof.
Letf be a generalized strongly p-convex function of higher order with μ≥0 and x1,x2,x3∈I, then(34)x3p−x2px3p−x1p,x2p−x1px3p−x1p∈0,1.x3p−x2px3p−x1p+x2p−x1px3p−x1p=1.
Taket=x3p−x2p/x3p−x1p,x=x1, and y=x3 in inequality (26), we have(35)x2p=tx1p+1−tx3p,(36)fx2≤fx3+x3p−x2px3p−x1pηfx1,fx3−μϕx3p−x2px3p−x1px3p−x1pq.
Multiplying (36) by x3p−x1p on both sides, we get(37)fx3x3p−x1p−fx2x3p−x1p+ηfx,fyx3p−x1p−μϕx3p−x2px3p−x1q≥0.Remark 3.
If we takeq=2,ϕt=tt−1, then we obtain the Schur-type inequality of a generalized strongly p-convex function [30].
Now we will derive the weighted version of Hermite–Hadamard inequality via a generalized strongly convex function of higher order.Theorem 3.
(Fejér-type inequality). Letf:I⟶R be a generalized strongly function of higher order with μ≥0 and provided η.,. is bounded from above on fI×fI. Also suppose that w:a,b⟶R is nonnegative, integrable, and symmetric with respect to ap+bp/21/p, then(38)fap+bp21/p∫abxp−1wxdx+μϕ12∫ab2xp−ap−bpqwxxp−1dx−12∫abηfap+bp−xp1/p,fxxp−1fxwxdx≤∫abxp−1fxwxdx≤fa+fb2∫abwxxp−1dx+Mηap−bp∫abxp−bpwxxp−1dx−μbp−apq∫abϕxp−bpap−bpwxxp−1dx.Proof.
Let us prove the left-hand side of theorem. Takep>0 and f:I⟶R be a generalized strongly p-convex function of higher order. We take t=1/2 for all x,y∈a,b in (26):(39)fap+bp21/p≤fy+12ηfx,fy−μϕ12yp−xpq.
Take(40)x=tap+1−tbp1/p,y=tbp+1−tap1/p.fap+bp21/p≤ftbp+1−tap1/p+12ηftap+1−tbp1/p,ftbp+1−tap1/p−μϕ12bp−apq2t−1q.
Sincew:a,b⟶R be a nonnegative, integrable, and p-symmetric with respect ap+bp/2, then(41)wx=wap+bp−xp1/p,(42)wtap+1−tbp1/p=wtbp+1−tap1/p,for all x,y∈a,b. Multiplying (40) by w=tbp+1−tap1/p,(43)fap+bp21/p∫01wtbp+1−tap1/pdt≤∫01ftbp+1−tap1/pwtbp+1−tap1/pdt+12∫01ηftap+1−tbp,ftbp+1−tap1/p−μϕ12bp−apq∫012t−1qwtbp+1−tap1/pdt.
Choosex=tbp+1−tap1/p and rearranging the above equation yields that(44)fap+bp21/p∫abxp−1wxdx−12∫01ηfap+bp−xp,fxxp−1wxdx−μϕ12∫ab2xp−ap−bpqwxxp−1dx≤∫abxp−1fxwxdx.
For the right-hand side of theorem, we choosex=tap+1−tbp1/p:(45)ftap+1−tbp1/p≤fb+tηfx,fy−μϕtyp−xpq.
Multiplying on both sides bywtbp+1−tap1/p and integrating w.r.t t over [0, 1], we get(46)pbp−ap∫abxp−1fxwxdx≤fb∫abwxxp−1dx+ηfa,fb∫ab1ap−bp∫abxp−bpwxxp−1dx−μbp−apq∫abϕxp−bpap−bpwxxp−1dx.
Similarly, we get(47)pbp−ap∫abxp−1fxwxdx≤fa∫abwxxp−1dx+ηfb,fa∫ab1ap−bp∫abxp−bpwxxp−1dx−μbp−apq∫abϕxp−bpap−bpwxxp−1dx.
By adding (46) and (47), we get(48)pbp−ap∫abxp−1fxwxdx≤fa+fb2∫abwxxp−1dx+ηfb,fa+ηfa,fb2ap−bp∫abxp−bpwxxp−1dx−μbp−apq∫abϕxp−bpap−bpwxxp−1dx.
We get(49)fap+bp21/p∫abxp−1wxdx+μϕ12∫ab2xp−ap−bpqwxxp−1dx−12∫abηfap+bp−xp1/p,fxxp−1fxwxdx≤∫abxp−1fxwxdx≤fa+fb2∫abwxxp−1dx+Mηap−bp∫abxp−bpwxxp−1dx−μbp−apq∫abϕxp−bpap−bpwxxp−1dx.
This completes the proof.Remark 4.
Imposing some conditions on Theorem3, we get different versions of Hermite–Fejér-type inequality:(1)
If we putq=2 and ϕt=t1−t, then we obtain the Fejér-type inequality of a generalized strongly p-convex function [30].(2)
Forwx=1 in the Theorem 3, we obtain Theorem 1.



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