# Estimates of Upper Bound for Differentiable Functions Associated withk-Fractional Integrals and Higher Order Strongly s-Convex Functions

Authors: Shanhe Wu; Muhammad Uzair Awan; Zakria Javed
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we establish two integral identities associated with differentiable functions and thek-Riemann-Liouville fractional integrals. The results are then used to derive the estimates of upper bound for functions whose first or second derivatives absolute values are higher order strongly s-convex functions.


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## Body

## 1. Introduction

Fractional calculus also known as noninteger calculus is a branch of mathematical analysis in which we discuss the integrals and derivatives of arbitrary order. The study of fractional calculus has a very long history, which can be traced back to the end of the 17th century; in 1695, L’Hospital wrote to Leibniz to discuss fractional derivative about a function. For hundreds of years, many mathematicians, such as Euler, Laplace, Fourier, Abel, Liouville, and Riemann, have carried out in-depth research on this subject (see [1]). Especially, in recent decades, the fractional calculus has found numerous applications in various fields such as mechanics, electricity, chemistry, biology, economics, notably control theory, and signal and image processing. Due to the backgrounds in practical applications, the fractional calculus has developed rapidly and has become a hot research topic (see [2–4]).Among several known forms of fractional integrals, the Riemann-Liouville fractional integral has been investigated extensively, which is defined as follows:Definition 1 ([3]).
Letf∈La,b. The Riemann-Liouville integrals Ja+αf and Jb−αf of order α>0 with a≥0 are defined by
(1)Ja+αfv=1Γα∫aυv−uα−1fudu,v>a,Jb−αfv=1Γα∫υbu−vα−1fudu,v<b,where
(2)Γα=∫0∞e−uuα−1duis the gamma function.In recent years, several researchers have utilized the concepts of fractional calculus to obtain the fractional analogues of classical inequalities. For example, Sarikaya et al. [5] established a generalized Hermite-Hadamard inequality via the Riemann-Liouville fractional integrals; Set [6] gave some fractional integral inequalities of the Ostrowski type through s-convex functions; Du et al. [7] deduced some variants of the fractional Hermite-Hadamard’s inequality using the class of generalized (α,m)-preinvex functions; Noor et al. [8] used the class of the s-Godunova-Levin convex functions to obtain refinements of the fractional Hermite-Hadamard inequalities; Peng et al. [9] obtained the Riemann-Liouville fractional Simpson inequalities through generalized (m,h1,h2)-preinvex functions; Wu and Awan [10] used the class of h-convex functions to derive the upper bound estimates of function involving fractional integrals; Wu et al. [11] established some fractional integral inequalities using k-th order differentiable strongly h-preinvex functions; Zhang et al. [12] provided some variations of the fractional Hermite-Hadamard’s inequalities. For more results related to this topic, we refer the interested reader to [13–17] and references cited therein.In [18], Mubeen and Habibullah introduced the k-fractional integral of the Riemann-Liouville type as follows:Definition 2 ([18]).
LetF∈La,b. The k-Riemann-Liouville fractional integrals Ja+αF and Jb−αF of order α>0 with a≥0, k>0 are defined by
(3)Jka+αFv=1kΓkα∫avv−uα/k−1Fudu,v>a,Jkb−αFv=1kΓkα∫vbu−vα/k−1Fudu,v<b,where
(4)Γkα=∫0∞e−uk/kuα−1duis the k-gamma function.Note that ifk⟶1, then the k-Riemann-Liouville fractional integrals reduces to the classical Riemann-Liouville fractional integral.Sarikaya et al. [19, 20] generalized the k-Riemann-Liouville fractional integrals and discussed their properties. Moreover, for the k-gamma function, they showed that Γkα=kα/k−1Γα/k and Γkα+k=αΓkα. For k-beta function, it is defined by
(5)Bkx,y=1k∫01ux/k−11−uy/k−1du,x>0,y>0,k>0,which implies that Bkx,y=1/kBx/k,y/k and Bkx,y=ΓkxΓky/Γkx+y.Motivated by the ideas of [19, 20], in this paper, we first establish two identities for the k-Riemann-Liouville fractional integrals associated with differentiable functions. We then apply the results to derive some estimates of the upper bound for differentiable functions involving k-fractional integrals via higher order strongly s-convex functions.


## 2. Preliminaries and Lemmas

Let us briefly summarize the concepts on generalized convex functions which are related to the contents of this paper.As a strengthening property of convexity, Polyak [21] introduced the strongly convex functions, as follows:Definition 3.
LetI⊂ℝ be an interval. A function f:I⟶ℝ is said to be strongly convex function with modulus μ>0, if
(6)f1−tx+ty≤1−tfx+tfy−μt1−ty−x2,∀x,y∈I,t∈0,1.Angulo et al. [22] presented an extension of strongly convex functions which is called strongly s-convex functions, i.e.:Definition 4.
LetI⊂ℝ be an interval, s∈0,1. A function f:I⟶ℝ is said to be strongly s-convex function with modulus μ>0, if
(7)f1−tx+ty≤1−tsfx+tsfy−μt1−ty−x2,∀x,y∈I,t∈0,1.Here we provide a further extension of stronglys-convex functions, as follows:Definition 5.
LetI⊂ℝ be an interval, s∈0,1, μ>0. A function f:I⟶ℝ is said to be strongly s-convex functions of order σ>0, if
(8)f1−tx+ty≤1−tsfx+tsfy−μtσ1−t+t1−tσy−xσ,∀x,y∈I,t∈0,1.Remark 6.
Ifσ=2, then the class of strongly s-convex functions of order σ>0 reduces to the class of strongly s-convex functions. For σ=2 and s=1, we have the class of classical strongly convex functions.In the following, we establish two integral identities associated with differentiable functions and thek-Riemann-Liouville fractional integrals. These integral identities play important role in dealing with subsequent results.Lemma 7.
Letf:a,b⟶ℝ be a differentiable function and f′∈La,b. Then for any α>0, k>0, and x∈a,b, we have
(9)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb=x−a2b−a2∫01atα/k+xf′1−ta+txdt−b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt.Proof.
Let(10)I=x−a2b−a2∫01atα/k+xf′1−ta+txdt−b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt≜I1−I2.Integrating by parts gives(11)I1=x−a2b−a2∫01atα/k+xf′1−ta+txdt=x−ax+afx−xfab−a2−aαx−akb−a2∫01tα/k−1f1−ta+txdt=x−ax+afx−xfab−a2−aΓkα+kx−aα/k−1b−a2kΓkα∫axu−aα/k−1fudu=x−ax+afx−xfab−a2−aΓkα+kx−aα/k−1b−a2Jkx−αfa.Similarly, we have(12)I2=b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt=b−xxfb−b+xfxb−a2−bαb−xkb−a2∫011−tα/k−1f1−tx+tbdt=b−xxfb−b+xfxb−a2+bΓkα+kb−xα/k−1b−a2kΓkα∫xbb−uα/k−1fudu=b−xxfb−b+xfxb−a2+bΓkα+kb−xα/k−1b−a2Jkx+αfb.Using (11) and (12) in (10) leads to (9). This completes the proof of Lemma 7.Lemma 8.
Letf:a,b⟶ℝ be twice differentiable function and f″∈La,b. Then for any α>0, k>0 and x∈a,b, we have
(13)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb=x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt.Proof.
Let(14)I=x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−ttα/k+1a+1−txf″tb+1−txdt≜I1+I2.Integrating by parts, we obtain(15)I1=x−ab−a2∫01ta/k+1x+taf″tx+1−tadt=x+af′xb−a2−1b−a2∫01αk+1tα/kx+af′tx+1−tadt=x+af′xb−a2+α+kx+kafx−kafaka−xb−a2+xαα+kk2x−ab−a2∫01tα/k−1ftx+1−tadt=x+af′xb−a2+α+kx+kafx−kafaka−xb−a2+xα+kΓkα+kΓkαk2x−aα/k+1b−a2∫axu−aα/k−1fudu=x+af′xb−a2+xα+k+kafx−kafaka−xb−a2+xα+kΓkα+kkx−aα/k+1b−a2Jkx−αfa.Similarly, we have(16)I2=b−xb−a2∫011−ta/k+1a+1−txf″tb+1−txdt=−x+af′xb−a2+1b−a2∫01αk+11−tα/ka+xf′tb+1−txdt=−x+af′xb−a2+α+ka+kxfx−kxfbkx−bb−a2+aαα+kk2b−xb−a2∫011−tα/k−1ftb+1−txdt=−x+af′xb−a2+α+ka+kxfx−kxfbkx−bb−a2+aα+kΓkα+kΓkαk2b−xα/k+1b−a2∫xbb−uα/k−1fudu=−x+af′xb−a2+aα+k+kxfx−kxfbkx−bb−a2+aα+kΓkα+kkb−xα/k+1b−a2Jkx+αfb.Combining (14), (15), and (16) leads to (13). Lemma 8 is proved.


## 3. Main Results

### 3.1. Estimates of the Upper Bound for Functions Whose First Derivatives Absolute Values Are Higher Order Stronglys-Convex Functions

Theorem 9.
Letf:a,b⊂0,∞⟶ℝ be a differentiable function and f′∈La,b. If f′is strongly s-convex functions of order σ>0, then for any α>0, k>0, μ>0, s∈0,1, and x∈a,b, we have
(17)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2L1f′a+L2f′x−μL3x−aσ+b−x2b−a2L4f′x+L5f′b−μL6b−xσ,where
(18)L1=akBkα+k,sk+k+xs+1,L2=akα+ks+k+xs+1,L3=ak2α+kσ+kα+kσ+2k+akBkα+2k,kσ+k+xσ+1σ+2+xkBk2k,kσ+k,L4=bkα+ks+k+xs+1,L5=bkBkks+k,α+k+xs+1,L6=bkBkkσ+k,α+2k−bkBkkσ+2k,α+k+bkBk2k,α+kσ+k+xσ+1σ+2+xkBk2k,kσ+k.Proof.
By Lemma7 and the property of absolute value, we have
(19)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2∫01atα/k+xf′1−ta+txdt+b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt.Utilizing the fact thatf′ is strongly s-convex functions of the order σ>0, we obtain
(20)x−a2b−a2∫01atα/k+xf′1−ta+txdt+b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt≤x−a2b−a2∫01atα/k+x1−tsf′a+tsf′x−μtσ1−t+t1−tσx−aσdt+b−x2b−a2∫01b1−tα/k+x1−tsf′x+tsf′b−μtσ1−t+t1−tσb−xσdt=x−a2b−a2akBkα+k,sk+k+xs+1f′a+akα+ks+k+xs+1f′x−μak2α+kσ+kα+kσ+2k+akBkα+2k,kσ+k+xσ+1σ+2+xkBk2k,kσ+kx−aσ+b−x2b−a2bkα+ks+k+xs+1f′x+bkBkks+k,α+k+xs+1f′b−μbkBkkσ+k,α+2k−bkBkkσ+2k,α+k+bkBk2k,α+kσ+k+xσ+1σ+2+xkBk2k,kσ+kb−xσ, which implies the desired inequality (17). The proof of Theorem 9 is complete. is complete.Theorem 10.
Letf:a,b⊂0,∞⟶ℝ be a differentiable function and f′∈La,b, and let p>1,q>1,1/p+1/q=1. If f′q is strongly s-convex functions of the order σ>0, then for anyα>0,
k>0, μ>0, s∈0,1, and x∈a,b, we have
(21)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2akα+k+x1/pL1f′aq+L2f′xq−μL3x−aσ1/q+b−x2b−a2bkα+k+x1/pL4f′xq+L5f′bq−μL6b−xσ1/q,where L1, L2, L3, L4, L5, and L6 are given by the same expressions as described in Theorem 9.Proof.
Using Lemma7, Hölder’s inequality, and the fact that f′q is strongly s-convex functions of the order σ>0, it follows that
(22)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2∫01atα/k+xf′1−ta+txdt+b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt≤x−a2b−a2∫01atα/k+xdt1/p∫01atα/k+xf′1−ta+txqdt1/q+b−x2b−a2∫01b1−tα/k+xdt1/p∫01b1−tα/k+xf′1−tx+tbqdt1/q≤x−a2b−a2∫01atα/k+xdt1/p×∫01atα/k+x1−tsf′aq+tsf′xq−μtσ1−t+t1−tσx−aσdt1/q+b−x2b−a2∫01b1−tα/k+xdt1/p×∫01b1−tα/k+x1−tsf′xq+tsf′bq−μtσ1−t+t1−tσb−xσdt1/q=x−a2b−a2akα+k+x1/p×akBkα+k,sk+k+xs+1f′aq+akα+ks+k+xs+1f′xq−μak2α+kσ+kα+kσ+2k+akBkα+2k,kσ+k+xσ+1σ+2+xkBk2k,kσ+kx−aσ1/q+b−x2b−a2bkα+k+x1/p×bkα+ks+k+xs+1f′xq+bkBkks+k,α+k+xs+1f′bq−μbkBkkσ+k,α+2k−bkBkkσ+2k,α+k+bkBk2k,α+kσ+k+xσ+1σ+2+xkBk2k,kσ+kb−xσ1/q=x−a2b−a2akα+k+x1/pL1f′aq+L2f′xq−μL3x−aσ1/q+b−x2b−a2bkα+k+x1/pL4f′xq+L5f′bq−μL6b−xσ1/q.
This completes the proof of Theorem10.


### 3.2. Estimates of the Upper Bound for Functions Whose Second Derivatives Absolute Values Are Higher Order Stronglys-Convex Functions

Theorem 11.
Letf:a,b⊂0,∞⟶ℝ be a differentiable function and f″∈La,b. Iff″is strongly s-convex functions of the order σ>0, then for any α>0, k>0, μ>0, s∈0,1, and x∈a,b, we have
(23)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb≤x−ab−a2M1f″x+M2f″a−μM3x−aσ+b−xb−a2M4f″b+M5f″x−μM6b−xσ,where
(24)M1=xkα+ks+2k+as+2,M2=xkBkα+2k,ks+k+akBk2k,ks+k,M3=xk2α+kσ+2kα+kσ+3k+xkBkα+3k,kσ+k+aσ+2σ+3+akBk3k,kσ+k,M4=akBkks+k,α+2k+xs+1s+2,M5=kaα+ks+2k+xs+2,M6=kaBkkσ+k,α+3k+kaBk2k,α+kσ+2k+xkBkkσ+k,3k+xkBk2k,kσ+2k.Proof.
By Lemma8 and the property of absolute value, we have
(25)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb=x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt≤x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt.
Utilizing the fact thatf″ is strongly s-convex functions of the order σ>0, we obtain
(26)x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt≤x−ab−a2∫01tα/k+1x+tatsf″x+1−tsf″a−μtσ1−t+t1−tσx−aσdt+b−xb−a2∫011−tα/k+1a+1−txtsf″b+1−tsf″x−μtσ1−t+t1−tσb−xσdt=x−ab−a2xkα+ks+2k+as+2f″x+xkBkα+2k,ks+k+akBk2k,ks+kf″a−μxk2α+kσ+2kα+kσ+3k+xkBkα+3k,kσ+k+aσ+2σ+3+akBk3k,kσ+kx−aσ+b−xb−a2akBkks+k,α+2k+xs+1s+2f″b+kaα+ks+2k+xs+2f″x−μkaBkkσ+k,α+3k+kaBk2k,α+kσ+2k+xkBkkσ+k,3k+xkBk2k,kσ+2kb−xσ,which implies the desired inequality (23). This completes the proof of Theorem 11 .Theorem 12.
Letf:a,b⊂0,∞⟶ℝ be twice differentiable function and f″∈La,b, and let p>1,q>1,1/p+1/q=1. If f″q is strongly s-convex functions of the order σ>0, then for any α>0, k>0, μ>0, s∈0,1, and x∈a,b, we have
(27)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb≤x−ab−a2xkα+2k+a21/pM1f″xq+M2f″aq−μM3x−aσ1/q+b−xb−a2akα+2k+x21/pM4f″bq+M5f″xq−μM6b−xσ1/q,where M1, M2, M3, M4, M5, and M6 are given by the same expressions as described in Theorem 11.Proof.
Using Lemma8, Hölder’s inequality, and the fact that f″q is strongly s-convex functions of the order σ>0, we obtain
(28)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb≤x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt≤x−ab−a2∫01tα/k+1x+tadt1/p×∫01tα/k+1x+taf″tx+1−taqdt1/q+b−xb−a2∫011−tα/k+1a+1−txdt1/p×∫011−tα/k+1a+1−txf″tb+1−txqdt1/q≤x−ab−a2∫01tα/k+1x+tadt1/p×∫01tα/k+1x+tatsf″xq+1−tsf″aq−μtσ1−t+t1−tσx−aσdt1/q+b−xb−a2∫011−tα/k+1a+1−txdt1/p×∫011−tα/k+1a+1−txtsf″bq+1−tsf″xq−μtσ1−t+t1−tσb−xσdt1/q=x−ab−a2xkα+2k+a21/pxkα+ks+2k+as+2f″xq+xkBkα+2k,ks+k+akBk2k,ks+kf″aq−μxk2α+kσ+2kα+kσ+3k+xkBkα+3k,kσ+k+aσ+2σ+3+akBk3k,kσ+kx−aσ1/q+b−xb−a2akα+2k+x21/pakBkks+k,α+2k+xs+1s+2f″bq+kaα+ks+2k+xs+2f″xq−μkaBkkσ+k,α+3k+kaBk2k,α+kσ+2k+xkBkkσ+k,3k+xkBk2k,kσ+2kb−xσ1/q.
The proof of Theorem12 is complete.


## 3.1. Estimates of the Upper Bound for Functions Whose First Derivatives Absolute Values Are Higher Order Stronglys-Convex Functions

Theorem 9.
Letf:a,b⊂0,∞⟶ℝ be a differentiable function and f′∈La,b. If f′is strongly s-convex functions of order σ>0, then for any α>0, k>0, μ>0, s∈0,1, and x∈a,b, we have
(17)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2L1f′a+L2f′x−μL3x−aσ+b−x2b−a2L4f′x+L5f′b−μL6b−xσ,where
(18)L1=akBkα+k,sk+k+xs+1,L2=akα+ks+k+xs+1,L3=ak2α+kσ+kα+kσ+2k+akBkα+2k,kσ+k+xσ+1σ+2+xkBk2k,kσ+k,L4=bkα+ks+k+xs+1,L5=bkBkks+k,α+k+xs+1,L6=bkBkkσ+k,α+2k−bkBkkσ+2k,α+k+bkBk2k,α+kσ+k+xσ+1σ+2+xkBk2k,kσ+k.Proof.
By Lemma7 and the property of absolute value, we have
(19)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2∫01atα/k+xf′1−ta+txdt+b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt.Utilizing the fact thatf′ is strongly s-convex functions of the order σ>0, we obtain
(20)x−a2b−a2∫01atα/k+xf′1−ta+txdt+b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt≤x−a2b−a2∫01atα/k+x1−tsf′a+tsf′x−μtσ1−t+t1−tσx−aσdt+b−x2b−a2∫01b1−tα/k+x1−tsf′x+tsf′b−μtσ1−t+t1−tσb−xσdt=x−a2b−a2akBkα+k,sk+k+xs+1f′a+akα+ks+k+xs+1f′x−μak2α+kσ+kα+kσ+2k+akBkα+2k,kσ+k+xσ+1σ+2+xkBk2k,kσ+kx−aσ+b−x2b−a2bkα+ks+k+xs+1f′x+bkBkks+k,α+k+xs+1f′b−μbkBkkσ+k,α+2k−bkBkkσ+2k,α+k+bkBk2k,α+kσ+k+xσ+1σ+2+xkBk2k,kσ+kb−xσ, which implies the desired inequality (17). The proof of Theorem 9 is complete. is complete.Theorem 10.
Letf:a,b⊂0,∞⟶ℝ be a differentiable function and f′∈La,b, and let p>1,q>1,1/p+1/q=1. If f′q is strongly s-convex functions of the order σ>0, then for anyα>0,
k>0, μ>0, s∈0,1, and x∈a,b, we have
(21)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2akα+k+x1/pL1f′aq+L2f′xq−μL3x−aσ1/q+b−x2b−a2bkα+k+x1/pL4f′xq+L5f′bq−μL6b−xσ1/q,where L1, L2, L3, L4, L5, and L6 are given by the same expressions as described in Theorem 9.Proof.
Using Lemma7, Hölder’s inequality, and the fact that f′q is strongly s-convex functions of the order σ>0, it follows that
(22)x−ax+afx−xfa+b−xb+xfx−xfbb−a2−Γkα+kb−a2ax−aα/k−1Jkx−αfa+bb−xα/k−1Jkx+αfb≤x−a2b−a2∫01atα/k+xf′1−ta+txdt+b−x2b−a2∫01b1−tα/k+xf′1−tx+tbdt≤x−a2b−a2∫01atα/k+xdt1/p∫01atα/k+xf′1−ta+txqdt1/q+b−x2b−a2∫01b1−tα/k+xdt1/p∫01b1−tα/k+xf′1−tx+tbqdt1/q≤x−a2b−a2∫01atα/k+xdt1/p×∫01atα/k+x1−tsf′aq+tsf′xq−μtσ1−t+t1−tσx−aσdt1/q+b−x2b−a2∫01b1−tα/k+xdt1/p×∫01b1−tα/k+x1−tsf′xq+tsf′bq−μtσ1−t+t1−tσb−xσdt1/q=x−a2b−a2akα+k+x1/p×akBkα+k,sk+k+xs+1f′aq+akα+ks+k+xs+1f′xq−μak2α+kσ+kα+kσ+2k+akBkα+2k,kσ+k+xσ+1σ+2+xkBk2k,kσ+kx−aσ1/q+b−x2b−a2bkα+k+x1/p×bkα+ks+k+xs+1f′xq+bkBkks+k,α+k+xs+1f′bq−μbkBkkσ+k,α+2k−bkBkkσ+2k,α+k+bkBk2k,α+kσ+k+xσ+1σ+2+xkBk2k,kσ+kb−xσ1/q=x−a2b−a2akα+k+x1/pL1f′aq+L2f′xq−μL3x−aσ1/q+b−x2b−a2bkα+k+x1/pL4f′xq+L5f′bq−μL6b−xσ1/q.
This completes the proof of Theorem10.


## 3.2. Estimates of the Upper Bound for Functions Whose Second Derivatives Absolute Values Are Higher Order Stronglys-Convex Functions

Theorem 11.
Letf:a,b⊂0,∞⟶ℝ be a differentiable function and f″∈La,b. Iff″is strongly s-convex functions of the order σ>0, then for any α>0, k>0, μ>0, s∈0,1, and x∈a,b, we have
(23)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb≤x−ab−a2M1f″x+M2f″a−μM3x−aσ+b−xb−a2M4f″b+M5f″x−μM6b−xσ,where
(24)M1=xkα+ks+2k+as+2,M2=xkBkα+2k,ks+k+akBk2k,ks+k,M3=xk2α+kσ+2kα+kσ+3k+xkBkα+3k,kσ+k+aσ+2σ+3+akBk3k,kσ+k,M4=akBkks+k,α+2k+xs+1s+2,M5=kaα+ks+2k+xs+2,M6=kaBkkσ+k,α+3k+kaBk2k,α+kσ+2k+xkBkkσ+k,3k+xkBk2k,kσ+2k.Proof.
By Lemma8 and the property of absolute value, we have
(25)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb=x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt≤x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt.
Utilizing the fact thatf″ is strongly s-convex functions of the order σ>0, we obtain
(26)x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt≤x−ab−a2∫01tα/k+1x+tatsf″x+1−tsf″a−μtσ1−t+t1−tσx−aσdt+b−xb−a2∫011−tα/k+1a+1−txtsf″b+1−tsf″x−μtσ1−t+t1−tσb−xσdt=x−ab−a2xkα+ks+2k+as+2f″x+xkBkα+2k,ks+k+akBk2k,ks+kf″a−μxk2α+kσ+2kα+kσ+3k+xkBkα+3k,kσ+k+aσ+2σ+3+akBk3k,kσ+kx−aσ+b−xb−a2akBkks+k,α+2k+xs+1s+2f″b+kaα+ks+2k+xs+2f″x−μkaBkkσ+k,α+3k+kaBk2k,α+kσ+2k+xkBkkσ+k,3k+xkBk2k,kσ+2kb−xσ,which implies the desired inequality (23). This completes the proof of Theorem 11 .Theorem 12.
Letf:a,b⊂0,∞⟶ℝ be twice differentiable function and f″∈La,b, and let p>1,q>1,1/p+1/q=1. If f″q is strongly s-convex functions of the order σ>0, then for any α>0, k>0, μ>0, s∈0,1, and x∈a,b, we have
(27)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb≤x−ab−a2xkα+2k+a21/pM1f″xq+M2f″aq−μM3x−aσ1/q+b−xb−a2akα+2k+x21/pM4f″bq+M5f″xq−μM6b−xσ1/q,where M1, M2, M3, M4, M5, and M6 are given by the same expressions as described in Theorem 11.Proof.
Using Lemma8, Hölder’s inequality, and the fact that f″q is strongly s-convex functions of the order σ>0, we obtain
(28)1b−a2xα+k+kafx−kafaka−x+aα+k+kxfx−kxfbkx−b+α+kΓkα+kkb−a2xx−aα/k+1Jkx−αfa+ab−xα/k+1Jkx+αfb≤x−ab−a2∫01tα/k+1x+taf″tx+1−tadt+b−xb−a2∫011−tα/k+1a+1−txf″tb+1−txdt≤x−ab−a2∫01tα/k+1x+tadt1/p×∫01tα/k+1x+taf″tx+1−taqdt1/q+b−xb−a2∫011−tα/k+1a+1−txdt1/p×∫011−tα/k+1a+1−txf″tb+1−txqdt1/q≤x−ab−a2∫01tα/k+1x+tadt1/p×∫01tα/k+1x+tatsf″xq+1−tsf″aq−μtσ1−t+t1−tσx−aσdt1/q+b−xb−a2∫011−tα/k+1a+1−txdt1/p×∫011−tα/k+1a+1−txtsf″bq+1−tsf″xq−μtσ1−t+t1−tσb−xσdt1/q=x−ab−a2xkα+2k+a21/pxkα+ks+2k+as+2f″xq+xkBkα+2k,ks+k+akBk2k,ks+kf″aq−μxk2α+kσ+2kα+kσ+3k+xkBkα+3k,kσ+k+aσ+2σ+3+akBk3k,kσ+kx−aσ1/q+b−xb−a2akα+2k+x21/pakBkks+k,α+2k+xs+1s+2f″bq+kaα+ks+2k+xs+2f″xq−μkaBkkσ+k,α+3k+kaBk2k,α+kσ+2k+xkBkkσ+k,3k+xkBk2k,kσ+2kb−xσ1/q.
The proof of Theorem12 is complete.


## 4. Conclusion

In this paper, we establish two identities involving differentiable functions and thek-Riemann-Liouville fractional integrals. Utilizing the identities, we obtain the estimates of the upper bound for functions whose first or second derivatives absolute values are higher order strongly s-convex functions. It is worth mentioning that our results contain, as a special case k,s,σ=1,1,2, the estimates of the upper bound of functions for the classical Riemann-Liouville fractional integrals and strongly convex functions.



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*Source: 5091857-2020-07-03.xml*