VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA117 10.15559/18-VMSTA117 Research Article On generalized stochastic fractional integrals and related inequalities BudakHüseyinhsyn.budak@gmail.com∗ SarikayaMehmet Zekisarikayamz@gmail.com Department of Mathematics, Faculty of Science and Arts, Düzce University, Konuralp Campus, Düzce-Turkey Corresponding author. 2018 249201854471481 2952018 1392018 1392018 © 2018 The Author(s). Published by VTeX2018 Open access article under the CC BY license.

The generalized mean-square fractional integrals Jρ,λ,u+;ωσ and Jρ,λ,v−;ωσ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals.

 Hermite–Hadamard inequality stochastic fractional integrals convex stochastic process 26D15 26A51 60G99 Introduction

In 1980, Nikodem [11] introduced convex stochastic processes and investigated their regularity properties. In 1992, Skwronski [17] obtained some further results on convex stochastic processes.

Let (Ω,A,P) be an arbitrary probability space. A function X:Ω→R is called a random variable if it is A -measurable. A function X:I×Ω→R , where I⊂R is an interval, is called a stochastic process if for every t∈I the function X(t,.) is a random variable.

Recall that the stochastic process X:I×Ω→R is called

(i) continuous in probability in interval I, if for all t0∈I we have P-limt→t0X(t,.)=X(t0,.), where P-lim denotes the limit in probability.

(ii) mean-square continuous in the interval I, if for all t0∈I limt→t0E[(X(t)−X(t0))2]=0, where E[X(t)] denotes the expectation value of the random variable X(t,.) .

Obviously, mean-square continuity implies continuity in probability, but the converse implication is not true.

Suppose we are given a sequence {Δm} of partitions, Δm={am,0,…,am,nm} . We say that the sequence {Δm} is a normal sequence of partitions if the length of the greatest interval in the n-th partition tends to zero, i.e., limm→∞sup1≤i≤nm|am,i−am,i−1|=0.

Now we would like to recall the concept of the mean-square integral. For the definition and basic properties see [18].

Let X:I×Ω→R be a stochastic process with E[X(t)2]<∞ for all t∈I . Let [a,b]⊂I , a=t0<t1<t2<⋯<tn=b be a partition of [a,b] and Θk∈[tk−1,tk] for all k=1,…,n . A random variable Y:Ω→R is called the mean-square integral of the process X on [a,b] , if we have limn→∞E[( ∑k=1nX(Θk)(tk−tk−1)−Y)2]=0 for all normal sequences of partitions of the interval [a,b] and for all Θk∈[tk−1,tk] , k=1,…,n . Then, we write Y(·)= ∫abX(s,·)ds(a.e.). For the existence of the mean-square integral it is enough to assume the mean-square continuity of the stochastic process X.

Throughout the paper we will frequently use the monotonicity of the mean-square integral. If X(t,·)≤Y(t,·) (a.e.) in some interval [a,b] , then ∫abX(t,·)dt≤ ∫abY(t,·)dt(a.e.). Of course, this inequality is an immediate consequence of the definition of the mean-square integral.

We say that a stochastic processes X:I×Ω→R is convex, if for all λ∈[0,1] and u,v∈I the inequality X(λu+(1−λ)v,·)≤λX(u,·)+(1−λ)X(v,·)(a.e.) is satisfied. If the above inequality is assumed only for λ=12 , then the process X is Jensen-convex or 12 -convex. A stochastic process X is concave if (−X) is convex. Some interesting properties of convex and Jensen-convex processes are presented in [11, 18].

Now, we present some results proved by Kotrys [6] about Hermite–Hadamard inequality for convex stochastic processes.

If X:I×Ω→R is a stochastic process of the form X(t,·)=A(·)t+B(·) , where A,B:Ω→R are random variables, such that E[A2]<∞,E[B2]<∞ and [a,b]⊂I , then ∫abX(t,·)dt=A(·)b2−a22+B(·)(b−a)(a.e.).

Let X:I×Ω→R be a convex stochastic process and t0∈intI . Then there exists a random variable A:Ω→R such that X is supported at t0 by the process A(·)(t−t0)+X(t0,·) . That is X(t,·)≥A(·)(t−t0)+X(t0,·)(a.e.). for all t∈I .

Let X:I×Ω→R be a Jensen-convex, mean-square continuous in the interval I stochastic process. Then for any u,v∈I we have X(u+v2,·)≤1v−u ∫uvX(t,·)dt≤X(u,·)+X(v,·)2(a.e.)

In [7], Kotrys introduced the concept of strongly convex stochastic processes and investigated their properties.

Let C:Ω→R denote a positive random variable. The stochastic process X:I×Ω→R is called strongly convex with modulus C(·)>0 0$]]>, if for all λ∈[0,1] and u,v∈I the inequality X(λu+(1−λ)v,·)≤λX(u,·)+(1−λ)X(v,·)−C(·)λ(1−λ)(u−v)2a.e. is satisfied. If the above inequality is assumed only for λ=12 , then the process X is strongly Jensen-convex with modulus C(·) .

In [5], Hafiz gave the following definition of stochastic mean-square fractional integrals.

For the stochastic proces X:I×Ω→R , the concept of stochastic mean-square fractional integrals Iu+α and Iv+α of X of order α>0 0$]]> is defined by Iu+α[X](t)=1Γ(α) ∫ut(t−s)α−1X(x,s)ds(a.e.),t>u, u,\]]]> and Iv−α[X](t)=1Γ(α) ∫tv(s−t)α−1X(x,s)ds(a.e.),t<v.

Using this concept of stochastic mean-square fractional integrals Ia+α and Ib+α , Agahi and Babakhani proved the following Hermite–Hadamard type inequality for convex stochastic processes:

Let X:I×Ω→R be a Jensen-convex stochastic process that is mean-square continuous in the interval I. Then for any u,v∈I , the following Hermite–Hadamard inequality X(u+v2,·)≤Γ(α+1)2(v−u)α[Iu+α[X](v)+Iv−α[X](u)]≤X(u,·)+X(v,·)2(a.e.) holds, where α>0 0$]]>.

For more information and recent developments on Hermite–Hadamard type inequalities for stochastic process, please refer to [2–4, 9–11, 14, 16, 15, 20, 19].

 Main results

In tis section, we introduce the concept of the generalized mean-square fractional integrals Jρ,λ,u+;ωσ and Jρ,λ,v−;ωσ of the stochastic process X.

In [13], Raina studied a class of functions defined formally by Fρ,λσ(x)=Fρ,λσ(0),σ(1),…(x)= ∑k=0∞σ(k)Γ(ρk+λ)xk(ρ,λ>0;|x|<R), 0;|x|<\mathcal{R}\big),\]]]> where the cofficents σ(k) (k∈N0=N∪{0}) make a bounded sequence of positive real numbers and R  is the set of real numbers. For more information on the function (4), please refer to [8, 12]. With the help of (4), we give the following definition.

Let X:I×Ω→R be a stochastic process. The generalized mean-square fractional integrals Jρ,λ,a+;ωα and Jρ,λ,b−;ωα of X are defined by Jρ,λ,u+;ωσ[X](x)=∫ux(x−t)λ−1Fρ,λσ[ω(x−t)ρ]X(t,·)dt,(a.e.)x>u, u,\]]]> and Jρ,λ,v−;ωσ[X](x)=∫xv(t−x)λ−1Fρ,λσ[ω(t−x)ρ]X(s,·)dt,(a.e.)x<v, where λ,ρ>0,ω∈R 0,\omega \in \mathbb{R}$]]>.

Many useful generalized mean-square fractional integrals can be obtained by specializing the coefficients σ(k) . Here, we just point out that the stochastic mean-square fractional integrals Ia+α and Ib+α can be established by coosing λ=α , σ(0)=1 and w=0 .

Now we present Hermite–Hadamard inequality for generalized mean-square fractional integrals Jρ,λ,a+;ωσ and Jρ,λ,b−;ωσ of X.

Let X:I×Ω→R be a Jensen-convex stochastic process that is mean-square continuous in the interval I. For every u,v∈I,u<v , we have the following Hermite–Hadamard inequality X(u+v2,·)≤12(v−u)λFρ,λ+1σ[ω(v−u)ρ][Jρ,λ,u+;ωσ[X](t)+Jρ,λ,v−;ωσ[X](t)]≤X(u,·)+X(v,·)2.a.e.

Since the process X is mean-square continuous, it is continuous in probability. Nikodem [11] proved that every Jensen-convex and continuous in probability stochastic process is convex. Since X is convex, then by Proposition 1, it has a supporting process at any point t0∈intI . Let us take a support at t0=u+v2 , then we have X(t,·)≥A(·)(t−u+v2)+X(u+v2,·).a.e. Multiplying both sides of (8) by [(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]] , then integrating the resulting inequality with respect to t over [u,v] , we obtain ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]X(t,·)dt≥A(·) ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]](t−u+v2)dt+X(u+v2,·) ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]dt=A(·) ∫uv[t(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+t(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]dt−A(·)u+v2 ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]dt+X(u+v2,·) ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]dt.

Calculating the integrals, we have ∫uvt(v−t)λ−1Fρ,λσ[ω(v−t)ρ]dt=− ∫uv(v−t)λFρ,λσ[ω(v−t)ρ]dt+v ∫uv(v−t)λ−1Fρ,λσ[ω(v−t)ρ]dt=−(v−u)λ+1Fρ,λσ1[ω(v−u)ρ]+v(v−u)λFρ,λ+1σ[ω(v−u)ρ] and similarly, ∫uvt(t−u)λ−1Fρ,λσ[ω(t−u)ρ]dt= ∫uv(t−u)λFρ,λσ[ω(t−u)ρ]dt+u ∫uv(t−u)λ−1Fρ,λσ[ω(t−u)ρ]dt=(v−u)λ+1Fρ,λσ1[ω(v−u)ρ]+u(v−u)λFρ,λ+1σ[ω(v−u)ρ] where σ1(k)=σ(k)ρk+λ+1 , k=0,1,2,… . Using the identities (10) and (11) in (9), we obtain Jρ,λ,u+;ωσ[X](t)+Jρ,λ,v−;ωσ[X](t)≥A(·)(u+v)(v−u)λFρ,λ+1σ[ω(v−u)ρ]−A(·)u+v22(v−u)λFρ,λ+1σ[ω(v−u)ρ]+X(u+v2,·)2(v−u)λFρ,λ+1σ[ω(v−u)ρ]=X(u+v2,·)2(v−u)λFρ,λ+1σ[ω(v−u)ρ]. That is, X(u+v2,·)≤12(v−u)λFρ,λ+1σ[ω(v−u)ρ][Jρ,λ,u+;ωσ[X](t)+Jρ,λ,v−;ωσ[X](t)]a.e., which completes the proof of the first inequality in (7).

By using the convexity of X, we get X(t,·)=X(v−tv−uu+t−uv−uv,·)≤v−tv−uX(u,·)+t−uv−uX(v,·)=X(v,·)−X(u,·)v−ut+X(u,·)v−X(v,·)uv−ua.e. for t∈[u,v] . Using the identities (10) and (11), it follows that ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]X(t,·)dt≤X(v,·)−X(u,·)v−u× ∫uv[t(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+t(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]dt+X(u,·)v−X(v,·)uv−u× ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]dt=X(v,·)−X(u,·)v−u(u+v)(v−u)λFρ,λ+1σ[ω(v−u)ρ]+X(u,·)v−X(v,·)uv−u2(v−u)λFρ,λ+1σ[ω(v−u)ρ]=[X(u,·)+X(v,·)](v−u)λFρ,λ+1σ[ω(v−u)ρ]. That is, 12(v−u)λFρ,λ+1σ[ω(v−u)ρ][Jρ,λ,u+;ωσ[X](t)+Jρ,λ,v−;ωσ[X](t)]≤X(u,·)+X(v,·)2a.e., which completes the proof.  □

i) Choosing λ=α , σ(0)=1 and w=0 in Theorem 3, the inequality (7) reduces to the inequality (3).

ii) Choosing λ=1 , σ(0)=1 and w=0 in Theorem 3, the inequality (7) reduces to the inequality (2).

Let X:I×Ω→R be a stochastic process, which is strongly Jensen-convex with modulus C(·) and mean-square continuous in the interval I so that E[C2]<∞ . Then for any u,v∈I , we have X(u+v2,·)−C(·){2(v−u)λ+2Fρ,λσ2[ω(v−u)ρ]−2(v−u)λFρ,λσ1[ω(v−u)ρ]+(u2+v2)(v−u)λFρ,λ+1σ[ω(v−u)ρ]−(u+v2)2}≤12(v−u)λFρ,λ+1σ[ω(v−u)ρ][Jρ,λ,u+;ωσ[X](t)+Jρ,λ,v−;ωσ[X](t)]≤X(u,·)+X(v,·)2−C(·){u2+v22+2(v−u)λ+2Fρ,λσ2[ω(v−u)ρ]−2(v−u)λFρ,λσ1[ω(v−u)ρ]+(u2+v2)(v−u)λFρ,λ+1σ[ω(v−u)ρ]}a.e.

It is known that if X is strongly convex process with the modulus C(·) , then the process Y(t,·)=X(t,·)−C(·)t2 is convex [7, Lemma 2]. Appying the inequality (7) for the process Y(t,.) , we have Y(u+v2,·)≤12(v−u)λFρ,λ+1σ[ω(v−u)ρ] ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]Y(t,·)dt≤Y(u,·)+Y(v,·)2a.e. That is X(u+v2,·)−C(·)(u+v2)2≤12(v−u)λFρ,λ+1σ[ω(v−u)ρ]{ ∫uv[(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]X(t,.)dt−C(·) ∫uv[t2(v−t)λ−1Fρ,λσ[ω(v−t)ρ]+t2(t−u)λ−1Fρ,λσ[ω(t−u)ρ]]dt}≤X(u,·)−C(·)u2+X(v,·)−C(·)v22a.e.

Calculating the integrals, we obtain ∫uvt2(v−t)λ−1Fρ,λσ[ω(v−t)ρ]dt= ∫uvt2(v−t)λ−1Fρ,λσ[ω(v−t)ρ]dt+ ∫uvt2(v−t)λ−1Fρ,λσ[ω(v−t)ρ]dt+ ∫uvt2(v−t)λ−1Fρ,λσ[ω(v−t)ρ]dt=(v−u)λ+2Fρ,λσ2[ω(v−u)ρ]−2v(v−u)λ+1Fρ,λσ1[ω(v−u)ρ]+v2(v−u)λFρ,λ+1σ[ω(v−u)ρ] and similarly, ∫uvt2(t−u)λ−1Fρ,λσ[ω(t−u)ρ]dt=(v−u)λ+2Fρ,λσ2[ω(v−u)ρ]+2u(v−u)λ+1Fρ,λσ1[ω(v−u)ρ]+u2(v−u)λFρ,λ+1σ[ω(v−u)ρ], where σ2(k)=σ(k)ρk+λ+2 , k=0,1,2,… . Then it follows that X(u+v2,·)−C(·)(u+v2)2≤12(v−u)λFρ,λ+1σ[ω(v−u)ρ][Jρ,λ,a+;ωσ[X](t)+Jρ,λ,b−;ωσ[X](t)]−C(·)[2(v−u)λ+2Fρ,λσ2[ω(v−u)ρ]−2(v−u)λFρ,λσ1[ω(v−u)ρ]+(u2+v2)(v−u)λFρ,λ+1σ[ω(v−u)ρ]]≤X(u,·)+X(v,·)2−C(·)u2+v22a.e. Then X(u+v2,·)−C(·){2(v−u)λ+2Fρ,λσ2[ω(v−u)ρ]−2(v−u)λFρ,λσ1[ω(v−u)ρ]+(u2+v2)(v−u)λFρ,λ+1σ[ω(v−u)ρ]−(u+v2)2}≤12(v−u)λFρ,λ+1σ[ω(v−u)ρ][Jρ,λ,u+;ωσ[X](t)+Jρ,λ,v−;ωσ[X](t)]≤X(u,·)+X(v,·)2−C(·){u2+v22+2(v−u)λ+2Fρ,λσ2[ω(v−u)ρ]−2(v−u)λFρ,λσ1[ω(v−u)ρ]+(u2+v2)(v−u)λFρ,λ+1σ[ω(v−u)ρ]}a.e. This completes the proof.  □

Choosing λ=α , σ(0)=1 and w=0 in Theorem 4, it reduces to Theorem 7 in [1].

 Acknowledgments

Authors thank the reviewer for his/her thorough review and highly appreciate the comments and suggestions.

 References Agahi, H., Babakhani, A.: On fractional stochastic inequalities related to Hermite–Hadamard and Jensen types for convex stochastic processes. Aequationes mathematicae 90(5), 1035–1043 (2016) MR3547706. https://doi.org/10.1007/s00010-016-0425-z Barráez, D., González, L., Merentes, N., Moros, A.: On h-convex stochastic processes. Mathematica Aeterna 5(4), 571–581 (2015) Budak, H., Sarikaya, M.Z.: A new Hermite-Hadamard inequality for h-convex stochastic processes. RGMIA Research Report Collection 19, 30 (2016) González, L., Merentes, N., Valera-López, M.: Some estimates on the Hermite-Hadamard inequality through convex and quasi-convex stochastic processes. Mathematica Aeterna 5(5), 745–767 (2015) Hafiz, F.M.: The fractional calculus for some stochastic processes. Stochastic analysis and applications 22(2), 507–523 (2004) MR2038026. https://doi.org/10.1081/SAP-120028609 Kotrys, D.: Hermite–Hadamard inequality for convex stochastic processes. Aequationes mathematicae 83(1), 143–151 (2012) MR2885506. https://doi.org/10.1007/s00010-011-0090-1 Kotrys, D.: Remarks on strongly convex stochastic processes. Aequationes mathematicae 86(1–2), 91–98 (2013) MR3094634. https://doi.org/10.1007/s00010-012-0163-9 Luo, M.-J., Raina, R.K.: A note on a class of convolution integral equations. Honam Math. J 37(4), 397–409 (2015) MR3445221. https://doi.org/10.5831/HMJ.2015.37.4.397 Maden, S., Tomar, M., Set, E.: Hermite-Hadamard type inequalities for s-convex stochastic processes in first sense. Pure and Applied Mathematics Letters 2015, 1 (2015) Materano, J., N., M., Valera-Lopez, M.: Some estimates on the Simpson’s type inequalities through Nikodem, K.: On convex stochastic processes. Aequationes mathematicae 20(1), 184–197 (1980) MR0577487. https://doi.org/10.1007/BF02190513 Parmar, R.K., Luo, M., Raina, R.K.: On a multivariable class of Mittag-Leffler type functions. Journal of Applied Analysis and Computation 6(4), 981–999 (2016) MR3502352 Raina, R.K.: On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian mathematical journal 21(2), 191–203 (2005) Sarikaya, M.Z., Yaldiz, H., Budak, H.: Some integral inequalities for convex stochastic processes. Acta Mathematica Universitatis Comenianae 85(1), 155–164 (2016) MR3456531 Set, E., Sarıkaya, M.Z., Tomar, M.: Hermite-Hadamard type inequalities for coordinates convex stochastic processes. Mathematica Aeterna 5(2), 363–382 (2015) Set, E., Tomar, M., Maden, S.: Hermite Hadamard type inequalities for s-convex stochastic processes in the second sense. Turkish Journal of Analysis and Number Theory 2(6), 202–207 (2014) Skowroński, A.: On some properties ofj-convex stochastic processes. Aequationes Mathematicae 44(2), 249–258 (1992) MR1181272. https://doi.org/10.1007/BF01830983 Sobczyk, K.: Stochastic Differential Equations: with Applications to Physics and Engineering vol. 40. Springer (2013) Tomar, M., Set, E., Bekar, N.O.: Hermite-Hadamard type inequalities for strongly-log-convex stochastic processes. Journal of Global Engineering Studies 1, 53–61 (2014) Tomar, M., Set, E., Maden, S.: Hermite-Hadamard type inequalities for log-convex stochastic processes. Journal of New Theory 2, 23–32 (2015)