# Fractional Versions of Hermite-Hadamard, Fejér, and Schur Type Inequalities for Strongly Nonconvex Functions

Authors: Wenbo Xu; Muhammad Imran; Faisal Yasin; Nazia Jahangir; Qunli Xia
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In modern world, most of the optimization problems are nonconvex which are neither convex nor concave. The objective of this research is to study a class of nonconvex functions, namely, strongly nonconvex functions. We establish inequalities of Hermite-Hadamard and Fejér type for strongly nonconvex functions in generalized sense. Moreover, we establish some fractional integral inequalities for strongly nonconvex functions in generalized sense in the setting of Riemann-Liouville integral operators.


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## Body

## 1. Introduction

The integral and differential operators have remarkable impact on applied sciences, and the interest of researchers is increasing day by day in this research area [1, 2]. Consider a convex function f:I⊂R⟶R defined on the interval I⊂ℝ with a,b∈I being constants and a<b. The inequality
(1)fa+b2≤1b−a∫abfxdx≤fa+fb2is Hermite-Hadamard’s (see [3, 4]).The notion of convexity is very old, and it appears in Archimedes treatment of orbit length. Nowadays, convex geometry is a mathematical subject in its own right. There are several modern works on convexity that are for the studies of real analysis, linear algebra, geometry, and functional analysis. The theory of convexity helps us to solve many applied problems. In recent years, the theory of convex analysis gains huge attention of researchers due to its interesting applications in optimizations, geometry, and engineering [5, 6].The present paper deals with a new class of convex functions and establishes inequalities of Hermite-Hadamard and Fejér. Moreover, we develop some fractional integral inequalities. See [7, 8] for more general inequalities via convexity of functions.The classical definition of convex functions was given in [3]. Another concept which is used widely in convex analysis is p-convex sets and p-convexity (see [4]). By taking p=1 in the above definition, we get classical notion of convexity. After that, the strongly convex with modulus μ>0 was introduced in [9]. And in [10], the notion of the strongly p-convex function had been introduced. The notion of generalized convex functions had been introduced in [11, 12].Motivated by the above researches, [13] introduced the following class of functions.The functionf is strongly nonconvex in generalized sense if
(2)ftxp+1−typ1/p≤fy+tηfx,fy−μt1−typ−xp2holds fort∈0,1.Definition 1 (see [13, 14]).
Considerf∈La,b, then the RHS and the LHS Riemann-Liouville fractional integral (RL) of order α>0 with b>a>0 are defined by
(3)Ja+αfx=1Γα∫axx−tα−1ftdt,x>a,Jb−αfx=1Γα∫axt−xα−1ftdt,x<b,
respectively, whereΓα is the Gamma function defined as
(4)Γα=∫0∞e−ttα−1.
It is to be noted thatJα+0fx=Jb−0fx=fx.The Riemann integral is reduced to classical integral forα=1 [15–18].The definition of strongp-convexity was studied in [13]. The aim of this paper is to establish the inequalities of Schur, Fejér, and Hermite-Hadamard type for the strongly nonconvex functions via RL fractional integrals.


## 2. Inequality of Hermite-Hadamard Type

In order to prove the inequality of Hermite-Hadamard type, the following lemma is very important.Lemma 2 (see [19]).
Letp be any nonzero real number and α be any positive constant. Further consider an integrable function w:A⟶ℝ, where A=a,b⊂0,∞ which is p-symmetric w.r.t. ap+bp/21/p; then, we have the following:
(i)
Ifp>0,(5)Jap+αwogbp=Jbp−αwogap=12Jap+αwogbp+Jbp−αwogap,with gx=x1/p, x∈ap,bp(ii)
Ifp<0,(6)Jbp+αwogap=Jap−αwogap=12Jbp+αwogbp+Jap−αwogbp,with gx=x1/p, x∈bp,apTheorem 3.
Let the strongly generalizedp-convex function f:I⟶ℝ with magnitude μ>0 and η (·) be bounded above in fI×fIyf∈La,b. Then, if p is any positive real number, we have
(7)fap+bp2−Mη+μbp−ap26Γα+1Γα+3≤2α−1Γα+1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap≤fa+fb2+αMη2α+1−μαα+3bp−ap24α+1α+2.Proof.
We begin the proof by insertingx=tap+1−tbp1/p and y=tbp+1−tap1/p(8)fxp+yp21/p−Mη2−μxp−yp212≤fx+fy2+Mη2−μxp−yp26.
Takex=tap+1−tbp1/p and y=tbp+1−tap1/p, then (8) yields
(9)fap+bp21/p−Mη2−μ2t−12bp−ap212≤12ftap+1−tbp1/p+12ftbp+1−tap1/p+Mη2−μ2t−12bp−ap26.
Multiplying (9) by tα−1 and then integrating w.r.t. t over the interval 0,1/2,(10)∫01/2fap+bp21/ptα−1dt−∫01/2Mη2tα−1dt−μbp−ap212∫01/22t−12tα−1dt≤12∫01/2tα−1ftap+1−tbpdt+12∫01/2tα−1ftbp+1−tapdt+∫01/2Mη2tα−1dt−μap−bp26∫01/22t−12tα−1dt,(11)fap+bp21/p−Mη2+μbp−ap2Γα+16Γα+3≤Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap,which is the left side of Theorem 3
Now, to obtain the left-hand side of Theorem 3, we have forx=tap+1−tbp1/p,
(12)ftap+1−tbp1/p≤fb+tηfa,fb−μt1−tbp−ap2,and for y=tbp+1−tap1/p,(13)ftbp+1−tap1/p≤fa+tηfb,fa−μt1−tbp−ap2.
Combining (12) and (13), we have
(14)ftap+1−tbp1/p+ftbp+1−tap1/p≤fa+tηfa,fb+fb+tηfb,fa−2μt1−tbp−ap2.
Multiplying (14) by 2tα−1 and then integrating w.r.t. t over the interval 0,1/2, we have
(15)2∫01/2ftap+1−tbp1/ptα−1+ftbp+1−tap1/ptα−1dt≤2∫01/2fa+fbtα−1dt+4Mη∫01/2tαdt−4μbp−ap2∫01/2t1−ttα−1dt,(16)Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap≤fa+fb2+αMη2α+1−μαα+3bp−ap24α+1α+2.
Together (11) and (16) give the required result.Remark 4.
(i)
Fixingp=1 in Theorem 3 gives Hermite-Hadamard inequality in the sense of the strongly generalized convexity(ii)
Fixingp=1 and μ=0 in Theorem 3, we obtain [20] (Theorem 2.1)(iii)
Fixingηx,y=x−y and μ=0 in Theorem 3 yields [21] (Theorem 2.1)(iv)
Applying both (ii) and (iii) on Theorem3, we obtain classical fractional version of H-H inequalityDefinition 5 (see [22]).
Letp be any nonzero real number; then, the function w:a,b⟶ℝ is p-symmetric w.r.t. ap+bp/21/p if wx=wap+bp−xp1/p for all x∈a,b.Theorem 6 (inequality of Fejér type).
Suppose thatf is a function as in Theorem 3 and an integrable, nonnegative function w:a,b⟶R is symmetric w.r.t. ap+bp/21/p, then
(17)Γα2fap+bp21/pJap+αwogbp+Jbp−αwogap−MηΓα2Jap+αwogbp+Jbp−αwogap+μ2∫apbp2x−bp−ap2bp−xα−1wogxdx≤Γα2Jap+αfwogbp+Jbp−αfwogapfa+fb2Γα2Jap+αwogbp+Jbp−αwogap+Mηbp−ap∫apbpbp−xαwogxdx−μ∫apbpbp−x2x−apwogxdx.Proof.
Settingt=1/2 in (2),
(18)fxp+yp21/p≤fy+12ηfx,fy−μ4yp−xp2.
Substitutey=tap+1−tbp1/p and x=tbp+1−tap1/p in (18),
(19)fxp+yp21/p≤ftap+1−tbp1/p+Mη2−μ42t−12bp−ap2.
According to the given conditions ofw, we have
(20)wx=wap+bp−xp1/p,wtap+1−tbp1/p=wtbp+1−tap1/p
∀x,y∈a,b. Multiplying (19) by 2tα−1wtbp+1−tap1/p and then integrating w.r.t. t over the interval 0,1,
(21)∫012tα−1wtbp+1−tap1/p×fap+bp21/p≤∫012tα−1wtbp+1−tap1/p×ftap+1−tbp1/pdt+Mη2∫012tα−1wtbp+1−tap1/pdt−μ4∫012tα−12t−12bp−ap2×wtbp+1−tap1/pdt,(22)fap+bp21/p∫bpap2x−bpap−apα−1wx1/pdxap−bp≤∫bpap2fx1/pwx1/px−bpap−apα−1dxap−bp+∫bpapMηx−bpap−apα−1wx1/pdxap−bp−μ2∫bpap2x−bp−ap2x−bpap−apα−1×wx1/pdxap−bp,(23)fap+bp21/p∫apbp2bp−xα−1wx1/pdxbp−apα≤∫apbp2fx1/pwx1/pbp−xα−1dxbp−apα+Mη∫apbpbp−xα−1wx1/pdxbp−apα−μ2∫bpap2x−bp−ap2bp−xα−1×wx1/pdxbp−apα.
Letgx=x1/p, then (23) becomes
(24)fap+bp21/p1bp−apα∫apbpbp−xα−1wogxdx2bp−apα∫apbpfwogxbp−xα−1dx+Mηbp−apα∫apbpbp−xα−1wogxdx−μ2bp−apα∫apbp2x−bp−ap2×xbp−α−1wogxdx,(25)Γα2fap+bp21/pJap+αwogbp+Jbp−αwogap−Mη2ΓαJap+αwogbp+Jbp−αwogap+μ2∫apbp2x−bp−ap2bp−xα−1wogxdx≤Γα2Jap+αfwogbp+Jbp−αfwogap.
Now, takex=tap+1−tbp∀t∈0,1, then by Def of f,
(26)ftap+1−tbp1/p≤fb+tηfa,fb−μt1−tbp−ap2.
Multiply on both sides of (26) by 2tα−1wtbp+1−tap1/p and then integrate w.r.t. t over the interval 0,1,
(27)∫012ftap+1−tbp1/ptα−1wtbp+1−tap1/pdt≤∫012tα−1wtbp+1−tap1/pfbdt+∫012tαMηwtbp+1−tap1/pdt−2μ∫012tα1−tbp−ap2×wtbp+1−tap1/pdt,(28)∫bpapfx1/pwx1/px−bpap−bpα−1dxap−bp≤∫bpapfbx−bpap−bpα−1wx1/pdxap−bp+Mη∫bpapx−bpap−bpα−1wx1/pdxap−bp−μ∫bpapbp−xαx−apwx1/pdxbp−apα.
Takegx=x1/p in (28), then we have
(29)∫apbpbp−xα−1wfogxdx≤∫apbpbp−xα−1wogxfbdx+Mηbp−ap∫apbpbpbp−xαwogxdx−μ∫apbpbpbp−xαx−apwogxdx.
Similarly, we have(30)∫apbpbp−xα−1wfogxdx≤∫apbpbp−xα−1wogxfadx+Mηbp−ap∫apbpbpbp−xαwogxdx−μ∫apbpbpbp−xαx−apwogxdx,
from definition off by fixing x=tbp+1−tap.
Combining (29) and (30), we obtain
(31)∫apbpbp−xα−1wfogxdx≤fa+fb2∫apbpbp−xα−1wogxdx+Mηbp−ap∫apbpbpbp−xαwogxdx−μ∫apbpbp−xαx−apwogxdx,(32)Γα2Jap+αfwogbp+Jbp−αfwogap≤fa+fb2Γα2Jap+αwogbp+Jbp−αwogap+Mηbp−ap∫apbpbpbp−xαwogxdx−μ∫apbpbp−xαx−apwogxdx.
Combining (32) and (25) completes the theorem (17).


## 3. Fractional Integral Inequalities for Strongly Generalizedp-Convex Function

Lemma 7.
Consider a differentiable functionf:I⊂0,∞⟶R on Io, with f′∈La,b, where a, b∈I and a<b. If w:a,b⟶R is integrable, then
(33)fap+bp21/p−Γα+121−αbp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη2+μbp−ap2Γα+12Γα+3=121−αbp−apα∫apap+bp/2t−apαfog′t−∫ap+bp/2bpbp−tαfog′tdt−Mη2+μbp−ap2Γα+12Γα+3
holds withgx=x1/p.Proof.
Letp>0, and x∈ap,bp, then for generalized strongly p-convex function, we have
(34)K=121−αbp−apα∫apap+bp/2t−apαfog′tdt−121−αbp−apα∫ap+bp/2bpbp−tαfog′tdt−Mη2+μbp−ap2Γα+12Γα+3,(35)K=K1−K2−Mη2+μbp−ap2Γα+!2Γα+3,where
(36)K1=121−αbp−apα∫apap+bp/2t−apαfog′tdt,K2=121−αbp−apα∫ap+bp/2bpbp−tαfog′tdt.
By integration by parts, we have(37)K1=121−αbp−apαbp−ap2αfap+bp21/p−α21−αbp−apα∫apap+bp/2t−apα−1fogtdt=12fap+bp21/p−Γα+121−αbp−apαΓα×∫apap+bp/2t−apα−1fogtdt,(38)K2=−12fap+bp21/p+Γα+121−αbp−apα×Jap+bp/2+αfogbp.
By combining (34), (37), and (38), we have (33). This completes the proof.Remark 8.
Settingμ=0 and η=x−y in Lemma 7 gives us [21] (Lemma 2.1).Theorem 9.
Let the functionf be as in Theorem 3.1. If f′ is a strongly generalized p-convex function on a,b for positive p and α, then
(39)fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη2+μbp−ap2Γα+12Γα+3≤bp−ap21−αC1α,pf′b+C2α,pηf′b,f′a−C3α,pμbp−ap2−Mη2+μbp−ap2Γα+1Γα+3,where
(40)C1α,p=∫01/2uαpuap+1−ubp1−1/pdu+∫1/211−uαpuap+1−ubp1−1/pdu,C2α,p=∫01/2uα+1puap+1−ubp1−1/pdu+∫1/211−uαupuap+1−ubp1−1/pdu,C3α,p=∫01/2uα+11−ubp−ap2puap+1−ubp1−1/pdu+∫1/21uα+11−ubp−ap2puap+1−ubp1−1/pdu.Proof.
Theorem (3) gives
(41)fap+bp21/p−Mη2+μbp−ap2Γα+12Γα+3−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap≤121−αbp−apα∫apap+bp/2t−apαf′t1/p1t1−1/pdt+121−αbp−apα∫ap+bp/2bpbp−tαf′t1/p1t1−1/pdt−Mη2+μbp−ap2Γα+12Γα+3≤bp−ap21−α∫apap+bp/2t−apαf′t1/p1t1−1/pdt+bp−ap21−α∫ap+bp/2bpbp−tαf′t1/p1t1−1/pdt−Mη2+μbp−ap2Γα+12Γα+3.
Settingt=uap+1−ubp, dt=ap−bpdu, we have
(42)fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη2+μbp−ap2Γα+1Γα+3≤bp−ap21−α∫01/2uαpuap+1−ubp1−1/pf′uap+1−ubp1/pdu+bp−ap21−α∫1/211−uαpuap+1−ubp1−1/pf′uap+1−ubp1/pdu−Mη2+μbp−ap2Γα+1Γα+3.
Sincef′ is a strongly generalized p-convex function on a,b, we have
(43)f′uap+1−ubp1/p≤f′b+μηf′a,f′b−μ1−ubp−ap2.
After combining (48) and (43), we have
(44)fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη2+μbp−ap2Γα+1Γα+3≤bp−ap21−α∫01/2uαpuap+1−ubp1−1/pf′b+uηf′a,f′b−μu1−ubp−ap2du+bp−ap21−α∫1211−uαpuap+1−ubp1−1/pf′b+uηf′a,f′b−μu1−ubp−ap2du−Mη2+μbp−ap2Γα+1Γα+3,fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη2+μbp−ap2Γα+1Γα+3=bp−ap21−α∫01/2uαpuap+1−ubp1−1/pf′b+bp−ap21−α∫1/211−uαpuap+1−ubp1−1/pf′bdu+bp−ap21−α∫01/2uα+1puap+1−ubp1−1/pηf′a,f′b+bp−ap21−α∫1/211−uαupuap+1−ubp1−1/pηf′a,f′bdu−Mη2+μbp−ap2Γα+1Γα+3+bp−ap21−α∫01/2uα+11−ubp−ap2μpuap+1−ubp1−1/p+bp−ap21−α∫1/21u1−uα+1μpuap+1−ubp1−1/p,fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη2+μbp−ap2Γα+1Γα+3≤bp−ap21−αC1α,pf′b+C2α,pηf′b,f′a−C3α,pμbp−ap2−Mη2+μbp−ap2Γα+1Γα+3.Remark 10.
If one takesη=x−y and μ=0, then we get [21] (Theorem 2.2).Theorem 11.
Let the functionf be as in Theorem 3.1. If f′ is as in Theorem 9, then
(45)fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη+μbp−ap2Γα+16Γα+3≤C5α,p1−1/qC5α,pf′b+C6α,pMη−C7α,pμbp−ap2+C8α,p1−1/qC8f′b+C9Mη+C10μbp−ap2−Mη+μbp−ap2Γα+16Γα+3,where
(46)C5α,p=∫01/2uαpuap+1−ubp1−1/pdu,C6α,p=∫01/2uα+1puap+1−ubp1−1/pdu,C7α,p=∫01/2uα1−upuap+1−ubp1−1/pdu,C8α,p=∫1/211−uαpuap+1−ubp1−1/pdu,C9α,p=∫1/21u1−uαpuap+1−ubp1−1/pdu,C10α,p=∫1/21u1−uα+1puap+1−ubp1−1/pdu.Proof.
Letp>0:
(47)fap+bp21/p−Mη+μbp−ap2Γα+16Γα+3−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap≤121−αbp−apα∫apap+bp/2t−apαf′t1/p1t1−1/pdt+121−αbp−apα∫ap+bp/2bpbp−tαf′t1/p1t1−1/pdt−Mη+μbp−ap2Γα+16Γα+3≤bp−ap21−α∫apap+bp/2t−apαf′t1/p1t1−1/pdt+bp−ap21−α∫ap+bp/2bpbp−tαf′t1/p1t1−1/pdt−Mη+μbp−ap2Γα+16Γα+3.
Settingt=uap+1−ubp, dt=ap−bpdu, we have
(48)fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη+μbp−ap2Γα+16Γα+3≤bp−ap21−α∫01/2uαpuap+1−ubp1−1/pf′uap+1−ubp1/pdu+bp−ap21−α×∫1/211−uαpuap+1−ubp1−1/pf′uap+1−ubp1/pdu−Mη+μbp−ap2Γα+16Γα+3.
Using the inequality of power mean the definition off′q,
(49)fap+bp21/p−Γα+12α−1bp−apαJap+bp/2+αfogbp+Jap+bp/2−αfogap−Mη+μbp−ap2Γα+16Γα+3≤bp−ap21−α∫01/2uαpuap+1−ubp1−1/p1−1/q×∫01/2uαpuap+1−ubp1−1/pf′uap+1−ubp1/pqdu1/q+bp−ap21−α∫1/211−uαpuap+1−ubp1−1/p1−1/q×∫1/211−uαpuap+1−ubp1−1/pf′uap+1−ubp1/pqdu1/q≤bp−ap21−α∫01/2uαpuap+1−ubp1−1/p1−1/q×∫01/2uαpuap+1−ubp1−1/pduf′b+∫01/2uα+1Mηpuap+1−ubp1−1/pdu+bp−ap21−α∫01/2uαpuap+1−ubp1−1/p1−1/q×∫01/2uα+11−ubp−ap2puap+1−ubp1−1/pdu+bp−ap21−α∫1/21u1−uαpuap+1−ubp1−1/p1−1/q×∫1/211−uαpuap+1−ubp1−1/pduf′b+∫01/21−uαuMηpuap+1−ubp1−1/pdu+bp−ap21−α∫1/211−uαpuap+1−ubp1−1/p1−1/q×∫1/211−uα+1ubp−ap2puap+1−ubp1−1/pdu=C5α,p1−1/qC5α,pf′b+C6α,pMη−C7α,pμbp−ap2+C8α,p1−1/qC8α,pf′b+C9α,pMη+C10α,pμbp−ap2−Mη+μbp−ap2Γα+16Γα+3.
This completes the proof.Remark 12.
(i)
Settingp=1 and μ=0,η=xy in Theorem 11 gives H-H type inequality for convex functions(ii)
Settingp=1 and α=1,μ=0,η=xy in Theorem 11 gives H-H inequality for convex functions


## 4. Conclusion

In this paper, we established inequalities of Hermite-Hadamard and Fejér type for strongly generalizedp-convex functions. We also established some fractional integral inequalities for this class of function in the setting of RL fractional integrals. We also related our results with the existing results and proved that by fixing involved parameters, we get many previous results.



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