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.
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∈Ilimt→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.
IfX:I×Ω→Ris a stochastic process of the formX(t,·)=A(·)t+B(·), whereA,B:Ω→Rare random variables, such thatE[A2]<∞,E[B2]<∞and[a,b]⊂I, then∫abX(t,·)dt=A(·)b2−a22+B(·)(b−a)(a.e.).
LetX:I×Ω→Rbe a convex stochastic process andt0∈intI. Then there exists a random variableA:Ω→Rsuch that X is supported att0by the processA(·)(t−t0)+X(t0,·). That isX(t,·)≥A(·)(t−t0)+X(t0,·)(a.e.).for allt∈I.
LetX:I×Ω→Rbe a Jensen-convex, mean-square continuous in the interval I stochastic process. Then for anyu,v∈Iwe haveX(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(·)>00$]]>, if for all λ∈[0,1] and u,v∈I the inequality
X(λu+(1−λ)v,·)≤λX(u,·)+(1−λ)X(v,·)−C(·)λ(1−λ)(u−v)2a.e.
is satisfied. If the above inequality is assumed only for λ=12, then the process X is strongly Jensen-convex with modulus C(·).
In [5], Hafiz gave the following definition of stochastic mean-square fractional integrals.
For the stochastic proces X:I×Ω→R, the concept of stochastic mean-square fractional integrals Iu+α and Iv+α of X of order α>00$]]> 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:
LetX:I×Ω→Rbe a Jensen-convex stochastic process that is mean-square continuous in the interval I. Then for anyu,v∈I, the following Hermite–Hadamard inequalityX(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, whereα>00$]]>.
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∈R0,\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.
LetX:I×Ω→Rbe a Jensen-convex stochastic process that is mean-square continuous in the interval I. For everyu,v∈I,u<v, we have the following Hermite–Hadamard inequalityX(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) Choosingλ=α,σ(0)=1andw=0in Theorem3, the inequality (7) reduces to the inequality (3).
ii) Choosingλ=1,σ(0)=1andw=0in Theorem3, the inequality (7) reduces to the inequality (2).
LetX:I×Ω→Rbe a stochastic process, which is strongly Jensen-convex with modulusC(·)and mean-square continuous in the interval I so thatE[C2]<∞. Then for anyu,v∈I, we haveX(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. □
Choosingλ=α,σ(0)=1andw=0in Theorem4, it reduces to Theorem 7 in [1].
Acknowledgments
Authors thank the reviewer for his/her thorough review and highly appreciate the comments and suggestions.
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