# Strongly Reciprocallyp-Convex Functions and Some Inequalities

Authors: Hao Li; Muhammad Shoaib Saleem; Ijaz Hussain; Muhammad Imran
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we generalize the concept of strong and reciprocal convexity. Some basic properties and results will be presented for the new class of strongly reciprocallyp-convex functions. Furthermore, we will discuss the Hermite–Hadamard-type, Jensen-type, and Fejér-type inequalities for the strongly reciprocally p-convex functions.


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## Body

## 1. Introduction

The importance of convex functions and convex sets cannot be ignored, especially in nonlinear programing [1–5] and optimization theory [6], see, for instance, [7–14]. Generalization in the convexity is always appreciable. Also, many generalizations and extensions have been made in the theory of inequalities as well as in convexity. Several inequalities have been studied and established for the convexity of functions, and many generalizations, applications, and refinements take place, see [7, 9, 13, 15–18], for further study.In the theory of inequalities, the famous inequality, Hermite–Hadamard inequality was established by Jaques Hadamard [19]. If σ:L⟶R is a convex function, then(1)σc1+c22≤1c2−c1∫c1c2σxdx≤σc1+σc22holds for all c1,c2∈L with c1≤c2.In [10], Lipot Fejér established the weighted version of the Hermite–Hadamard inequality.Ifσ:L⟶R is a convex function, then the inequality(2)σc1+c22∫c1c2wxdx≤1c2−c1∫c1c2σxwxdx≤σc1+σc22∫c1c2wxdxholds for all c1,c2∈L with c1≤c2 and w:L⟶R is integrable, nonnegative, and symmetric about c1+c2/2.For more details on the Fejér inequality, see [8, 9, 11, 20–22]. The main motivation of this article is based on [18].Mathematically, Jensen-type inequality is stated as ifσ is a convex function defined on L⊂R, then(3)σ∑i=1nμixi≤∑i=1nμi∑i=1nσxiholds for all n∈R, x1,x2,…,xn∈L and μ1,μ2,…,μn≥0 with μ1+μ2+⋯+μn=1.This inequality has applications in probability and statistics.The article is organized as follows: Section2 is devoted to preliminaries and basic results, whereas in the last section, we will develop the main results for strongly reciprocally p-convex functions.


## 2. Preliminaries

This section concerns preliminaries and basic results for the strongly reciprocallyp-convex functions.Definition 1 (p-convex set; see [23]).
An intervalL is called the p-convex set if rc1p+1−rc2p1/p∈L for all c1,c2∈L and r∈0,1, where p=2u+1 or p=d/c, d=vr+1, c=2w+1, and u,v,w∈N.Definition 2 (p-convex function; see [24]).
A functionσ:L⟶R is called p-convex function if(4)σrc1p+1−rc2p1/p≤rσc1+1−rσc2,for all c1,c2∈L and r∈0,1, where L is the p-convex set.Definition 3 (strongly convex function; see [14]).
Letμ be a positive number. A function σ:L⟶R is called a strongly convex function if(5)σrc1+1−rc2≤rσc1+1−rσc2−μr1−rc2−c12,for all c1,c2∈L and r∈0,1.Definition 4 (stronglyp-convex function; see [25]).
Letμ be a positive number. A function σ:L⟶R is called strongly p-convex function if(6)σrc1p+1−rc2p1/p≤rσc1+1−rσc2−μr1−rc2p−c1p2,for all c1,c2∈L and r∈0,1.Definition 5 (harmonic convex function; see [22]).
LetL=c1,c2⊂R be an interval. A function σ:L⟶R is harmonic convex if(7)σc1c2rc1+1−rc2≤rσc1+1−rσc2,for all c1,c2∈L and r∈0,1.Definition 6 (harmonicp-convex function; see [26]).
A functionσ:L⟶R is called a harmonic p-convex function if(8)σc1pc2prc1p+1−rc2p1/p≤rσc1+1−rσc2,for all c1,c2∈L and r∈0,1.Definition 7.
(strongly reciprocally convex function; see [18]). Let L⊆R and μ∈0,∞. A function σ:L⟶R is said to be strongly reciprocally convex with modulus μ on L if the inequality(9)σc1c2rc1+1−rc2≤rσc1+1−rσc2−μr1−r1c1−1c22,holds for all c1,c2∈L and r∈0,1.
Now, we are ready to introduce a new class of convexity named as strongly reciprocallyp-convex function.Definition 8 (strongly reciprocallyp-convex function).
A functionσ:L⟶R is called strongly reciprocally p-convex with modulus μ on L if the inequality(10)σxpyprxp+1−ryp1/p≤rσx+1−rσy−μr1−r1yp−1xp2holds, for all x,y∈L=c1,c2 and r∈0,1.Remark 1.
(1)
If we insertp=1 in inequality (10), then we retrace the strong and reciprocal convexity [18](2)
If we insertμ=0 in inequality (10), then we retrace the harmonic p-convexity [26](3)
If we insertp=1 and μ=0 in inequality (10), then we retrace the harmonic convexity [22]
The following proposition expresses the algebraic property of strongly reciprocallyp-convex functions.Proposition 1.
Letσ,φ:L⟶R be two strongly reciprocally p-convex functions; then, the following statements hold:(i)
σ+φ:L⟶R is strongly reciprocally p-convex(ii)
For anyλ≥0, λσ:L⟶R is strongly reciprocally p-convex corresponding to λμ=μ∗Proof.
(i)
Choosev=xpyp/rxp+1−ryp1/p; then, by the definition of σ and φ, we obtain(11)σ+φxpyprxp+1−ryp1/p=σxpyprxp+1−ryp1/p+φxpyprxp+1−ryp1/p≤rσx+1−rσy−μr1−r1yp−1xp2+rφx+1−rφy−μr1−r1yp−1xp2,=rσ+φx+1−rσ+φy−2μr1−r1yp−1xp2≤rσ+φx+1−rσ+φy−μr1−r1yp−1xp2,whereμ≥0.(ii)
Letλ≥0; then, by definition, we obtain(12)λσxpyprxp+1−ryp≤λrσx+1−rσy−μr1−r1yp−1xp2=rλσx+1−rλσy−μ∗r1−r1yp−1xp2,where μ∗=λμ and μ≥0.
The next lemma establishes the connection between the strong and reciprocalp-convexity and harmonic p-convexity.Lemma 1.
Letσ:L⟶R be a function; σ is strongly reciprocally p-convex iff the function φ:L⟶R, defined by φx=σx−μ/x2p, is harmonically p-convex.Proof.
Letσ be strongly reciprocally p-convex; then, we have(13)φxpyprxp+1−ryp1/p=σxpyprxp+1−ryp1/p−μrxp+1−rypxpyp2≤rσy+1−rσx−μr1−r1xp−1yp2−μrxp+1−rypxpyp2=rσy+1−rσx−μr1−r1xp−1yp2−μr2y2p+1−r2x2p+2r1−rxpyp=rσy+1−rσx−μry2p−2rxpyp+rx2p−r2y2p+2rxpyp−r2x2p+r2y2p+2rxpyp−2r2xpyp+1x2p−2rx2p+r2x2p=rσy+1−rσx−μry2p+1x2p−rx2p=rσy+1−rσx−μry2p+1−rx2p=rσy−μy2p+1−rσx−μx2p=rφy+1−rφx.
This shows thatφ is a harmonic p-convex function.
Conversely, ifφ is harmonically p-convex, then(14)σxpyprxp+1−ryp1/p=φxpyprxp+1−ryp1/p+μrxp+1−rypxpyp2≤rφy+1−rφx+μrxp+1−rypxpyp2=rφy+1−rφx+μr2y2p+2r1−rxpyp+1−r2x2p≤rφy+1−rφx+μr1−1+ry2p+2r1−rxpyp+μ1−r1−rx2p=rφy+1−rφx+μry2p−r1−ry2p+μ2r1−rxpyp+1−rx2p−r1−rx2p=rφy+μ1y2p+1−rφx+μ1x2p−μr1−r1y2p−2xpyp+1x2p=rσy+1−rσx−μr1−r1yp−1xp2.
This implies thatσ is a strongly reciprocally p-convex function for all x,y∈L and r∈0,1.


## 3. Main Results

In this section, Hermite–Hadamard-, Fejér-, and Jensen-type inequalities are investigated. The next theorem gives the generalization of the Hermite–Hadamard inequality for strongly reciprocallyp-convex functions.Theorem 1.
(Hermite–Hadamard-type inequality). LetL⊂R/0 be an interval on the real line. If σ:L⟶R is a strongly reciprocally p-convex function with modulus μ≥0 and x∈L=c1,c2, then(15)σ2c1pc2pc1p+c2p1/p+μ12c2p−c1pc1pc2p2≤pc1pc2pc2p−c1p∫c1c2σxxp+1dx≤σc1+σc22−μ6c2p−c1pc1pc2p2,for all c1,c2∈L with c1≤c2.Proof.
We start by the definition; setr=1/2 in inequality (10), and we have(16)σ2xpypxp+yp1/p≤12σx+12σy−μ41yp−1xp2.
Letx=c1pc2p/rc1p+1−rc2p1/p and y=c1pc2p/rc2p+1−rc1p1/p, and by integrating w.r.t r over [0, 1], the above inequality yields(17)∫01σ2c1pc2pc1p+c2p1/pdr≤12∫01σc1pc2prc1p+1−rc2p1/pdr+∫01σc1pc2prc2p+1−rc1p1/pdr−μ4c2p−c1pc1pc2p2∫011−2r2dr∫01σc1pc2prc1p+1−rc2p1/pdr=pc1pc2pc2p−c1p=∫c1c2σxxp+1dx=∫01σc1pc2prc2p+1−rc1p1/pdr,and then inequality (17) is reduced to(18)σ2c1pc2pc1p+c2p1/p≤pc1pc2pc2p−c1p∫c1c2σxxp+1dx−μ12c2p−c1pc1pc2p2σ2c1pc2pc1p+c2p1/p+μ12c2p−c1pc1pc2p2≤pc1pc2pc2p−c1p∫c1c2σxxp+1dx,which is the left side of the inequality.
For the right side of inequality (15), set x=c1 and y=c2 in (10); we have(19)σc1pc2prc1p+1−rc2p1/p≤rσc1+1−rσc2−μr1−r1c1p−1c2p2.
Integrating w.r.tr over [0, 1], the above inequality yields(20)∫01σc1pc2prc1p+1−rc2p1/pdr≤12σc1+12σc2−μc2p−c1pc1pc2p2∫01r1−rdr∫01σc1pc2prc1p+1−rc2p1/pdr≤σc1+σc22−μ6c2p−c1pc1pc2p2.
Since(21)∫01σ2c1pc2prc1p+1−rc2p1/pdr=pc1pc2pc2p−c1p∫c1c2σxxp+1dx,then we obtain(22)pc1pc2pc2p−c1p∫c1c2σxxp+1dx≤σc1+σc22−μ6c2p−c1pc1pc2p2.
From (18) and (22), we get (15).Remark 2.
(1)
Forp=1 in (15), Hermite–Hadamard inequality for strongly reciprocally convex functions is obtained [18].(2)
If we allowμ⟶0+ in inequalities (15), we obtain the Hermite–Hadamard-type inequalities for harmonically convex functions [22].
For further details on Hermite–Hadamard inequities, see [27–30].Theorem 2.
(Fejér-type inequality). Assumeσ:L⟶R is a strongly reciprocally p-convex function with modulus μ on L; then,(23)σ2c1pc2pc1p+c2p1/p∫c1c2dx+μ2c1pc2p2∫c1c22c1pc2p−c1p+c2pxp2wxx3p+1dx≤∫c1c2σxwxxp+1dx≤c1pc2p−c1pσc1+σc2∫c1c2c2p−xpwxx2p+1dx−μc1pc2p∫c1c2wxxp+1dx+μ2c1pc2p2∫c1c22c1pc2p−c1p+c2pxp2wxx3p+1dxholds for c1,c2∈L with c1≤c2 and x∈L=c1,c2, where w:L⟶R is a nonnegative integrable function that satisfies(24)wc1pc2pxp1/p=wc1pc2pc1p+c2p−xp1/p.Proof.
Sinceσ:L⟶R is a strongly reciprocally p-convex function, then by definition for r=1/2 in (10), we have(25)σ2xpypxp+yp1/p≤σx+σy2−μ41yp−1xp2,for all x,y∈L; suppose x=c1pc2p/rc1p+1−rc2p1/p and y=c1pc2p/rc2p+1−c1p1/p in the above inequality; then, we obtain(26)σ2c1pc2pc1p+c2p1/p≤12σc1pc2prc1p+1−rc2p1/p+σc1pc2prc2p+1−rc1p1/p−μ4rc2p+1−rc1pc1pc2p−rc1p+1−rc2pc1pc2p2.
Sincew is nonnegative and symmetric, we have(27)σ2c1pc2pc1p+c2p1/pwc1pc2prc1p+1−rc2p1/p≤12σc1pc2prc1p+1−rc2p1/p+σc1pc2prc2p+1−rc1p1/pwc1pc2prc1p+1−rc2p1/p−μ4rc2p+1−rc1pc1pc2p−rc1p+1−rc2pc1pc2p2wc1pc2prc1p+1−rc2p1/p.
The above inequality is integrated with respect tor over [0, 1], and then putting x=c1pc2p/rc1p+1−rc2p1/p, we obtain(28)pc1pc2pc2p−c1pσ2c1pc2pc1p+c2p1/p∫c1c2wxxp+1dx≤pc1pc2pc2p−c1p∫c1c2σxwxxp+1dx−μp4c2p−c1pc1pc2p∫c1c22c1pc2p−c1p+c2pxp2x3p+1wxdx.
After simplification, the above inequality becomes(29)σ2c1pc2pc1p+c2p1/p∫c1c2wxxp+1dx+μ2c1pc2p2∫c1c22c1pc2p−c1p+c2pxp2x3p+1wxdx≤∫c1c2σxwxxp+1dx.
For the right-hand side of (23), set x=c1 and y=c2 in (10); we have(30)σc1pc2prc1p+1−rc2p1/pwc1pc2prc1p+1−rc2p1/p,≤rσc1+1−rσc2wc1pc2prc1p+1−rc2p1/p−μr1−r1c1p−1c2p2wc1pc2prc1p+1−rc2p1/p.
Integrating with respect tor over [0, 1] and then putting x=c1pc2p/rc1p+1−rc2p1/p, we obtain(31)pc1pc2pc2p−c1p∫c1c2σxwxxp+1dx≤pc1pc1pc2pc2p−c1p2σc1+σc2∫c1c2c2p−xpwxx2p+1dx−μpc2p−c1p∫c1c2c2p−xpxp−c1pwxx3p+1dx.
After simplification, we have(32)∫c1c2σxwxxp+1dx≤c1pc2p−c1pσc1+σc2∫c1c2c2p−xpwxx2p+1dx−μc1pc2p∫c1c2c2p−xpxp−c1pwxx3p+1dx.
From (32) and (27), we get (23).Remark 3.
If we setp=1 in (23), the Fejér-type inequality for strongly reciprocally convex functions is obtained.
Jensen-type inequality for the aforementioned inequality is described in the next theorem.Theorem 3.
(Jensen-type inequality). Ifσ:L⟶R is a reciprocally strongly p-convex function with modulus μ, then(33)σ∑i=1nr1xipi1/p≤∑i=1nriσ1xi−μ∑i=1nri1xip−1x¯p2holds for all 1/x1p,1/x2p,…,1/xnp∈L, r1,r2,…,rn≥0 with r1+r2+…+rn=1 and 1/x¯p=r11/x1p+r21/x2p+…+rn1/xnP.Proof.
Fix1/x1p,1/x2p,…,1/xnp∈L and r1,r2,…,rn>0 such that r1+r2+…+rn=1.
Put1/x¯p=r11/x1p+r21/x2p+⋯+rn1/xnp, and suppose a function w:L⟶R of the form(34)w1x=μ1xp−1x¯p2+a1xp−1x¯p+σx¯,supporting at x¯, satisfying w1/x¯=σ1/x¯ and wx≤σx,x∈L. Then, for every i=1,2,…,n, we have(35)σ1xi≥w1xi=μ1xip−1x¯p2+a1xip−1x¯p+σ1x¯.
Multiplying both sides byri and summing up to n, we have(36)∑i=1nriσ1xi≥μ∑i=1nri1xip−1x¯p2+a∑i=1nri1xip−1x¯p+σ1x¯.
Since∑i=1nri1/xip−1/x¯p=0, we have(37)σ1x¯1/p≤∑i=1nrσ1xii−μ∑i=1nri1xip−1x¯p2,which completes the proof.Remark 4.
In inequality (34), fixing p=1 and μ=0 yields the Jensen-type inequality for the harmonic convex function [22]. See [31–34] for more details on Jensen-type inequalities.



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