# Some Properties and Inequalities for theh,s-Nonconvex Functions

Authors: Chengli Wang; Muhammad Shoaib Saleem; Hamood Ur Rehman; Muhammad Imran
Journal: Journal of Mathematics
 (2020)

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## Abstract

The purpose of this paper is to introduce the notion of stronglyh,s-nonconvex functions and to present some basic properties of this class of functions. We present Schur inequality, Jensen inequality, Hermite–Hadamard inequality, and weighted version of the Hermite–Hadamard inequality.


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## Body

## 1. Introduction

Convexity, the study of convex functions, has scope in various fields of science and pure mathematics, as well as applied mathematics. From the last few decades, many extensions and generalizations of convexity have been expressed to support different research ideas in mathematics, see, for instance, [1–12].Supposef:I=μ1,μ2⟶ℝ be a convex function and μ1<μ2, where μ1,μ2∈I=μ1,μ2, then the inequality(1)fμ1+μ22≤1μ2−μ1∫μ1μ2fudu≤fμ1+fμ22,is named as the Hermite–Hadamard equality. The above inequality is a necessary condition for a function to be convex.The generalization of inequality (1) was given by Fejér as f:I=μ1,μ2⟶ℝ is a convex function and w:I=μ1,μ2⟶ℝ is nonnegative, integrable, and symmetric about μ1+μ2/2, and then(2)fμ1+μ22wudu≤1μ2−μ1∫μ1μ2fuduwudu≤fμ1+fμ22∫μ1μ2wudu.Obeidat [13] generalized the class of h,s-convex functions, which is called strongly h,s-convex functions.In this article, we generalized the concept ofh,s-convex functions and defined strongly h,s-convex functions. Moreover, we present some basic properties and results of strongly h,s-nonconvex functions, and Jensen inequality, Schur inequality, Hermite–Hadamard inequality, and weighted version of Hermite–Hadamard-type inequalities are obtained for this class of functions.


## 2. Preliminaries and Some Properties

In this section, we recall some definitions from the literature which are helpful for the study of stronglyh,s-nonconvex functions.Definition 1.
(convex function). A functionf:I=μ1,μ2⟶ℝ is said to be a convex function if(3)fξu+1−ξv≤ξfu+1−ξfv,for all u,v∈I=μ1,μ2 and ξ∈0,1.Definition 2.
(h-convex function, see [2]). A function f:I=μ1,μ2⟶ℝ is called a h-convex function if(4)fξu+1−ξv≤hξfu+h1−ξfv,for all u,v∈I=μ1,μ2 and ξ∈0,1.Definition 3.
(strongly convex function, see [14]). A function f:I=μ1,μ2⟶ℝ is said to be a strongly convex function with modulus ϑ≥0 if(5)fξu+1−ξv≤ξfu+1−ξfv−ϑξ1−ξu−v2,for all u,v∈I=μ1,μ2 and ξ∈0,1.Definition 4.
(stronglyh-convex function, see [15]). A function f:I=μ1,μ2⟶ℝ is said to be a strongly h-convex function with modulus ϑ≥0 if(6)fξu+1−ξv≤hξfu+h1−ξfv−ϑξ1−ξu−v2,for all u,v∈I=μ1,μ2 and ξ∈0,1.Definition 5.
(s-convex function). A function f:I=μ1,μ2⟶ℝ is said to be a s-convex function in the second sense if(7)f1−ξu+ξv≤1−ξsfv+ξsfu,for all u,v∈I=μ1,μ2 and ξ∈0,1, where I⊂ℝ+.Definition 6.
(h,s-convex function, see [1]). A function f:I=μ1,μ2⟶ℝ is said to be a h,s-convex function if(8)f1−ξu+ξv≤h1−ξsfu+hξsfv,for all u,v∈I=μ1,μ2 and ξ∈0,1.Now, we give our definition of a stronglyh,s-convex function.Definition 7.
(stronglyh,s-convex function, see [13]). A function f:I=μ1,μ2⟶ℝ is said to be a strongly h,s-convex function with modulus ϑ≥0 if(9)f1−ξu+ξv≤h1−ξsfu+hξsfv−ϑξ1−ξu−v2,for all u,v∈I=μ1,μ2 and ξ∈0,1.Definition 8.
(nonconvex set). A setI is a p-convex set if λup+1−λvp1/p∈I=μ1,μ2 for all u,v∈I, p=2k+1 (or p=2r+1/2t+1), and λ∈0,1, where k,t∈ℕ.Definition 9.
(nonconvex function). A functionf:I=μ1,μ2⟶ℝ is called a nonconvex function if(10)fξup+1−ξvp1/p≤ξfu+1−ξfv,for all u,v∈I=μ1,μ2 and ξ∈0,1, where I is a p-convex set.Definition 10.
(h,s-nonconvex function). A function f:I=μ1,μ2⟶ℝ is called a h,s-nonconvex function if(11)fξup+1−ξvp1/p≤hξsfu+h1−ξsfv,for all u,v∈I=μ1,μ2 and ξ∈0,1.Definition 11.
(stronglyh,s-nonconvex function). A function f:I=μ1,μ2⟶ℝ is called a strongly h,s-nonconvex function with modulus ϑ≥0 if(12)fξup+1−ξvp1/p≤hξsfu+h1−ξsfv−ϑξ1−ξup−vp2,for all u,v∈I=μ1,μ2 and ξ∈0,1.
Clearly, a stronglyh,s-nonconvex function covers h,s-convex function, h,s-nonconvex function, h-convex function, and s-convex function.Remark 1.
Allowingp=1, strong h,s-convexity is obtained, and with s=1 and p=1, strong h-convexity is obtained. Similarly, replacing hξ=1, s=1, p=1, and ϑ=0, the classical convexity is retraced.In the following, we present some properties of stronglyh,s-nonconvex functions.Proposition 1.
Leth1andh2 be nonnegative functions defined on interval I=μ1,μ2 such that h2ξs≤h1ξs where ξ∈0,1 and s∈0,1. If f is a strongly h2−s-nonconvex function, then f is a strongly h1−s-nonconvex function.Proof.
Iff is a strongly h2−s-nonconvex function, then for every u,v∈μ1,μ2 and ξ∈0,1, we have(13)fξup+1−ξvp1/p≤h2ξsfu+h21−ξsfv−ϑξ1−ξup−vp2≤h1ξsfu+h11−ξsfv−ϑξ1−ξup−vp2.
This completes the proof.Proposition 2.
Iff and g are strongly h,s-nonconvex functions and λ>0, then(i)
f+g is a strongly h,s-nonconvex function(ii)
λf is a strongly h,s-nonconvex functionProof.
(i)
Sincef and g are strongly h,s-nonconvex functions, then we have(14)f+gξup+1−ξvp1/p=fξup+1−ξvp1/p+gξup+1−ξvp1/p≤hξsfu+h1−ξsfv−ϑξ1−ξup−vp2+hξsgu+h1−ξsgv−ϑξ1−ξup−vp2=hξsf+gu+h1−ξsf+gv−ϑμ1∗t1−ξup−vp2,whereϑμ1∗=2ϑ, and this completes the proof.(ii)
Sinceλ>0 and f is a strongly h,s-nonconvex function, then we have(15)λfξup+1−ξvp1/p≤λhξsfu+λh1−ξsfv−ϑμ1∗t1−ξup−vp2,whereϑμ1∗=λϑ.Theorem 1.
Supposehξs≤ξ for each ξ∈0,1, then f:I=μ1,μ2⟶ℝ is a strongly h,s-nonconvex function with modulus ϑ, and g:I=μ1,μ2⟶ℝ is defined as(16)gu=fu−ϑup2,then g is a h,s-nonconvex function.Proof.
We start the proof with(17)gξvp+1−ξup1/p=fξvp+1−ξup1/p−ϑξvp+1−ξup2≤hξsfv+h1−ξsfu−ϑξ1−ξu−v2−ϑξvp+1−ξup2=hξsgv+h1−ξsgu+ϑh1−ξsup2+ϑhξsvp2−ϑξ1−ξup−vp2−ϑξvp+1−ξup2≤hξsgu+h1−ξsgv+ϑh1−ξs−1−ξup2+ϑhξs−tvp2.
Ashξs≤ξ and h1−ξs≤1−ξ, then we obtain(18)gξvp+1−ξup1/p≤hξsgv+h1−ξsgu.Remark 2.
Theorem1 is not true for arbitrary h, see Example 1 in [13].Theorem 2.
Supposehξs≥ξ for each ξ∈0,1, then f:I=μ1,μ2⟶ℝ is a h,s-nonconvex function, and g:I=μ1,μ2⟶ℝ is defined as(19)gu=fu+ϑup2,then g is a strongly h,s-nonconvex function.Proof.
We start the proof with(20)gξvp+1−ξup1/p=fξvp+1−ξup1/p+ϑξvp+1−ξup2≤hξsfv+h1−ξsfu+ϑξ1−ξu−v2=hξsgv+h1−ξsgu−ϑh1−ξsup2−ϑhξsvp2+ϑξvp+1−ξup2=hξsgv+h1−ξsgu−ϑh1−ξsup2−ϑhξsvp2+ϑ1−ξ2up2+ϑξ2vp2+2ϑξ1−ξupvp≤hξsgv+h1−ξsgu−ϑξ1−ξup−vp2+ϑ1−ξ−h1−ξsup2+ϑt−hξsvp2.
Ashξs≥ξs≥ξ and h1−ξs≥1−ξs≥1−ξ, then the above inequality yields(21)gξvp+1−ξup1/p≤hξsgv+h1−ξsgu−ϑξ1−ξup−vp2.
This completes the proof.Remark 3.
Theorem2 is not true for arbitrary h, see Example 2 in [13].


## 3. Main Results

In this section, we intend to make the reformulations of the Jensen-type inequality, Hermite–Hadamard type inequalities, Fejér type inequality, and Schur type inequality for stronglyh,s-nonconvex functions.Theorem 3.
(Jensen inequality). Supposef:I=μ1,μ2⟶ℝ be a strongly h,s-nonconvex function, then the inequality(22)f∑i=1nhξisuip1/p≤∑i=1nhuisfuip−ϑ∑i=1nhξisuip−u¯p2,holds for all u1p,…,unp∈I=μ1,μ2 and hξ1s,…,hξns>0 with hξ1s+…+hξns=1 and u¯p=hξ1su1p+…+hξnsunp.Proof.
Takeu1,…,un∈I=μ1,μ2 and hξ1s,…,hξns>0, where ξ1,…,ξn>0, such that hξ1s+…+hξns=1.
Setu¯p=hξ1su1p+…+hξnsunp and take a function(23)gu=ux−u¯p2+ax−u¯p2+fu¯p,supporting f at u¯p, that is, gu¯p=fu¯p and gu≤fu, u∈I=μ1,μ2.
Then, we have(24)fuip≥guip=uuip−u¯p2+auip−u¯p2+fu¯p,for every ξ∈1,2,…,n.
Multiplying on both sides of the above inequality byhξis and summing up, we obtain(25)∑i=1nhξisfuip≥∑i=1nguip=u∑i=1nhξisuip−u¯p2+a∑i=1nhξisuip−u¯p2+fu¯p.
Since∑i=1nhξisuip−u¯p2=0, we have(26)fu¯p≤∑i=1nhξisfuip−u∑i=1nhξisuip−u¯p2,which completes the proof.Now, the following results are obtained.Corollary 1.
Supposef:I=μ1,μ2⟶ℝ be a strongly s,p-convex function, then the inequality(27)f∑i=1nξisuip1/p≤∑i=1nξisfuip−u∑i=1nξisuip−u¯p2,holds for all u1p,…,unp∈I=μ1,μ2, ξ1s,…,ξns>0, and s∈0,1 with ξ1s+…+ξns=1 and u¯p=ξ1su1p+…+ξnsunp.Corollary 2.
Supposef:I=μ1,μ2⟶ℝ be a strongly h,s-convex function, then the inequality(28)f∑i=1nξisuip1/p≤∑i=1nξisfui−u∑i=1nξisuip−u¯p2,holds for all u1,…,un∈I=μ1,μ2, ξ1s,…,ξns>0, and s∈0,1 with ξ1s+…+ξns=1 and u¯p=ξ1u1+…+ξnun.Fixinghξs=ξs in inequality (22), we obtain Corollary 1. Similarly, for p=1, the inequality (22) yields Corollary 2. If we impose the above conditions with s=1 on inequality (22), we obtain the Jensen inequality for a strongly convex function.Theorem 4.
(Schur inequality). Assume thatf:I=μ1,μ2⟶ℝ be a strongly h,s-nonconvex function with modulus ϑ≥0, then for all u1p,u2p,u3p∈I=μ1,μ2 such that u1p<u2p<u3p and u3p−u1p,u3p−u2p,u2p−u1p∈0,1, the inequality(29)hu3p−u2psfu1+hu3p−u1psfu2+hu2p−u1psfu3−uu3p−u2pu2p−u1p≥0,holds for s∈0,1.Proof.
The proof starts with the assumption thatf:I=μ1,μ2⟶ℝ be a strongly h,s-nonconvex function with modulus ϑ≥0 and u1p,u2p,u3p∈I=μ1,μ2, then u3p−u2p/u3p−u1p, u3p−u1p/u3p−u1p∈0,1, and u3p−u2p/u3p−u1p+u3p−u1p/u3p−u1p=1.
Substitutingξ=u3p−u2p/u3p−u1p, up=u1p, and vp=u3p in inequality (12) yields(30)fu2≤hu3p−u2pu3p−u1psfu1+hu3p−u1pu3p−u1psfu3−uu3p−u2pu2p−u1p.
Using the property ofh and multiplying on both sides of the above inequality by hu3p−u1p, we obtain(31)hu3p−u2psfu1+hu3p−u1psfu2+hu2p−u1psfu3−uu3p−u2pu2p−u1p≥0,which completes the proof.Substitutingp=1 in (29), we obtain the following corollary.Corollary 3.
Assume thatf:I=μ1,μ2⟶ℝ be a strongly h,s-convex function with modulus ϑ≥0, then for all u1p,u2p,u3p∈I=μ1,μ2 such that u1p<u2p<u3p and u3p−u1p,u3p−u2p,u2p−u1p∈0,1, the inequality(32)hu3−u2sfu1+hu3−u1sfu2+hu2−u1sfu3−uu3−u2u2−u1≥0,holds for s∈0,1.Theorem 5.
(Hermite–Hadamard inequality). Letf:I=μ1,μ2⟶ℝ be a strongly h,s-nonconvex function with modulus ϑ≥0 and h1/2s>0, then the inequality(33)12h1/2sfμ1p+μ2p21/p+ϑμ2p−μ1p212≤pμ2p−μ1p∫μ1μ2up−1fudu≤fμ1+fμ2∫01hξsdξ−ϑμ2p−μ1p26,holds.Proof.
We begin the proof by using the definition of the stronglyh,s-nonconvex function:(34)fξμ1p+1−ξμ2p1/p≤hξsfμ1+h1−ξsfμ2−ϑξ1−ξμ2p−μ1p2,for all ξ∈0,1 and s∈0,1.
Integrating the above inequality with respect to “t” over 0,1 yields(35)∫01fξμ1p+1−ξμ2p1/pdξ≤fμ1∫01hξsdξ+fμ2∫01h1−ξsdξ−ϑμ2p−μ1p2∫01t1−ξdξpμ2p−μ1p∫μ1μ2up−1fudu≤fμ1+fμ2∫01hξsdξ−ϑμ2p−μ1p26.
Now, for the formulation of the left-hand side of inequality (33), we have(36)fμ1p+μ2p21/p=fξμ1p+1−ξμ2p+ξμ2p+1−ξμ1p21/p≤h12sfξμ1p+1−ξμ2p1/p+fξμ2p+1−ξμ1p1/p−ϑ42ξ−12μ2p−μ1p2,where μ1=ξμ1p+1−ξμ2p1/p and μ2=ξμ2p+1−ξμ1p1/p.
Since(37)∫01fξμ1p+1−ξμ2p1/pdξ=∫01fξμ2p+1−ξμ1p1/pdξ=pμ2p−μ1p∫μ1μ2up−1fudu,then by integrating inequality (36) with respect to “t” over 0,1, we obtain(38)fμ1p+μ2p21/p≤2ph1/2sμ2p−μ1p∫μ1μ2up−1fudu−ϑ12μ2p−μ1p2.
Hence,(39)12h1/2sfμ1p+μ2p21/p+ϑμ2p−μ1p212≤pμ2p−μ1p∫μ1μ2up−1fudu,which completes the proof.Remark 4.
If we setp=1 in (33), we obtain the Hermite–Hadamard inequality for a strongly h,s convex function, see [13]. And, allowing s=1, hξs=ξs, and ϑ=0, it yields a classical Hermite–Hadamard inequality.Theorem 6.
(Fejér inequality). Letf:I=μ1,μ2⟶ℝ be a strongly h,s-nonconvex function, h1/2s>0, and w:I=μ1,μ2⟶ℝ be a nonnegative symmetric function about μ1p+μ2p/2, then(40)p2h1/2sμ2p−μ1pfμ1p+μ2p21/p∫μ1μ2wudu+ϑ4∫μ1μ2μ2p+μ1p−2up2wudu≤pμ2p−μ1p∫μ1μ2up−1wufudu≤pfμ1+fμ2μ2p−μ1p∫μ1μ2up−1hup−μ1pμ2p−μ1pswudu−upμ2p−μ1p∫μ1μ2up−μ1pμ2p−upup−1wudu.Proof.
Supposef is a strongly h,s-nonconvex function, then for u=ξμ2p+1−ξμ1p1/p∈I=μ1,μ2, we have(41)fξμ2p+1−ξμ1p1/p≤hξsfμ2+h1−ξsfμ1−uξ1−ξμ2p−μ1p2,where ξ∈0,1. Since w is a nonnegative symmetric function about μ1p+μ2p/2, then we have(42)wξμ2p+1−ξμ1p1/pfξμ2p+1−ξμ1p1/p≤hξsfμ2+h1−ξsfμ1−uξ1−ξμ2p−μ1p2wξμ2p+1−ξμ1p1/p−uξ1−ξμ2p−μ1p2wξμ2p+1−ξμ1p1/p.
Integrating the above inequality with respect to “ξ” over 0,1 yields(43)∫01wξμ2p+1−ξμ1p1/pfξμ2p+1−ξμ1p1/pdξ≤∫01hξsfμ2+h1−ξsfμ1−uξ1−ξμ2p−μ1p2wξμ2p+1−ξμ1p1/pdξ−∫01ut1−ξμ2p−μ1p2wξμ2p+1−ξμ1p1/pdξ,and since w is symmetric about μ1p+μ2p/2, we obtain(44)∫01wξμ2p+1−ξμ1p1/pfξμ2p+1−ξμ1p1/pdξ=pμ2p−μ1p∫μ1μ2up−1fuwudu∫01h1−ξswξμ2p+1−ξμ1p1/pdξ=pμ2p−μ1p∫μ1μ2up−1hμ2p−upμ2p−μ1pswudu=pμ2p−μ1p∫μ1μ2up−1hμ2p−upμ2p−μ1pswμ1p+μ2p−updu=pμ2p−μ1p∫μ1μ2up−1hup−μ1pμ2p−μ1pswudu=∫01hξswξμ2p+1−ξμ1p1/pdξ.
Hence, we obtain(45)pμ2p−μ1p∫μ1μ2up−1wufudu≤pfμ1+fμ2μ2p−μ1p∫μ1μ2up−1hμ2p−upμ2p−μ1pswudu−upμ2p−μ1p∫μ1μ2up−μ1pμ2p−upup−1wudu.
For the left-hand side of inequality (40), we observe that for all ξ∈0,1, we have(46)fμ1p+μ2p21/p=fξμ2p+1−ξμ1p+ξμ1p+1−ξμ2p21/p≤h12sfξμ2p+1−ξμ1p1/p+fξμ1p+1−ξμ2p1/p−u4μ2p−μ1p2∫011−2ξ2wξμ2p+1−ξμ1p1/pdξ.
After integrating the above inequality with respect to “ξ” over 0,1 and multiplying both sides by w, which is a nonnegative and symmetric function about μ1p+μ2p/2, we obtain(47)fμ1p+μ2p21/p∫01wξμ2p+1−ξμ1p1/pdξ≤h12s∫01fξμ2p+1−ξμ1p+fξμ1p+1−ξμ2pwξμ2p+1−ξμ1p1/pdξ−uμ2p−μ1p24∫011−2ξ2wξμ2p+1−ξμ1p1/pdξ,∫01fξμ1p+1−ξμ2p1/pwξμ2p+1−ξμ1p1/pdξ=pμ2p−μ1p∫μ1μ2up−1wudu=pμ2p−μ1p∫μ1μ2up−1wμ1p+μ2p−updu=∫01fξμ2p+1−ξμ1p1/pwξμ2p+1−ξμ1p1/pdξ.
Thus, we obtain(48)pμ2p−μ1pfμ1p+μ2p21/p∫μ1μ2wudu≤2ph1/2sμ2p−μ1p∫μ1μ2up−1fuwudu−up4μ2p−μ1p∫μ1μ2up−1μ1p+μ2p−2up2wudu,since h1/2s>0, which yields(49)pμ2p−μ1p∫μ1μ2up−1wufudu≥p2h1/2sμ2p−μ1pfμ1p+μ2p21/p∫μ1μ2wudu+ϑ4∫μ1μ2μ2p+μ1p−2up2wududu.
This completes the proof.Remark 5.
If we allowp=1 in Theorem 6, then it reduces to the Fejér-type inequality for a strongly h,s-convex function, and for p=1 and ϑ=0, we obtain the inequalities generalized in [1].


## 4. Conclusion

In this paper, we have introduced a stronglyh,s-nonconvex function, which is the generalization of many existing definitions. We also proved several inequalities, for example, Schur inequality, Jensen inequality, and Hermite–Hadamard inequality, for a newly defined strongly h,s-nonconvex function. This definition can also be used to develop inequalities presented in [16–18] and references therein.


## 5. Future Directions

In future, we are interested to work on generalizations of stochastich-convex processes and stochastic h,s processes. We will develop Schur inequality, Jensen inequality, and Hermite–Hadamard inequality for these generalizations.



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