# Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for GeneralizedP-Convex Stochastic Processes

Authors: Fangfang Ma; Waqas Nazeer; Mamoona Ghafoor
Journal: Journal of Mathematics
 (2021)

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## Abstract

The stochastic process is one of the important branches of probability theory which deals with probabilistic models that evolve over time. It starts with probability postulates and includes a captivating arrangement of conclusions from those postulates. In probability theory, a convex function applied on the expected value of a random variable is always bounded above by the expected value of the convex function of that random variable. The purpose of this note is to introduce the class of generalizedp-convex stochastic processes. Some well-known results of generalized p-convex functions such as Hermite-Hadamard, Jensen, and fractional integral inequalities are extended for generalized p-stochastic convexity.


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## Body

## 1. Introduction

A stochastic process is a mathematical tool commonly defined as a set of random variables in various fields of probability. Verifiably, random variables were related to or listed by a lot of numbers, normally as focuses in time, giving the translation of a stochastic process, speaking to numerical estimations, some systems randomly changing over time, such as the growth of bacterial populations, fluctuations in electrical flow due to thermal noise, or the production of gas molecules. Stochastic systems are commonly used as scientific models of systems that tend to alter in an arbitrary manner. They have applications in various fields, especially in sciences, for instance, chemistry, physics, biology, neuroscience, and ecology, in addition to technology and engineering fields, for example, picture preparing, cryptography, signal processing, telecommunications, PC science, and data theory. Furthermore, apparently, arbitrary changes in money-related markets have inspired the broad utilization of stochastic processes in fund.Convex stochastic processes and their applications have a fundamental significance in mathematics and in probability. Nikodem [1] in 1980 proposed the idea of convex stochastic processes in his article. In 1992, λ-convex and Jensen-convex stochastic processes were initiated by Skowronski [2]. More recently, Kotrys presented in [3] the results on convex stochastic processes.For more details, refer to [4]. Many studies in the literature have been performed on some extensions of convex stochastic processes and on Hermite-Hadamard type inequalities for these extensions [5].In the present note, we purpose to investigate the idea of generalizedp-convex stochastic processes. The notion of inequality as convexity has a significant place in literature [6], as it yields a broader setting in order to investigate the mathematical programming and optimization problems. Therefore, Schur type, Hermite-Hadamard, Jensen, and fractional integral inequalities and some important results for the above said processes will be obtained in this study.We start by definition of the stochastic process [7].Definition 1 (see [3]).
Assume a probability spaceΩ,A,P. A random variable is the function ξ:Ω⟶ℝ if ξ is A-measurable; whereas, a stochastic process is the function ξ:I×Ω⟶ℝ if ξt,. is a random variable for every t∈I.
Let us review some basic notions about stochastic processes.Definition 2 (see [3]).
The stochastic processξ:I×Ω⟶ℝ is as follows:(1)
Continuous onI, if for all u∘∈I,(1)P−limu⟶u∘ξu,.=ξu∘,.,whereP−lim represents the limit in the probability;(2)
Mean-square continuous onI, if for every u∘∈I,(2)limu⟶u∘Eξu,.−ξu∘,.2=0,WhereEξu,. represents an expectation of the random variable ξ.It is obvious in probability that if a stochastic process is mean-square continuous, then it is also continuous, but the converse is not true;(3)
Mean square is differentiable att∈I, if there is a random variable ξ′v,.:I×Ω⟶ℝ, such that for every u∘∈I,(3)ξ′u∘,.=P−limu⟶u∘ξu,.−ξu∘,.u−u∘.Definition 3 (see [3]).
Considerξ:I×Ω⟶ℝ, a stochastic process with Eξt,.2<∞. We say a random variable ν:Ω⟶ℝ to be the mean-square integral of the process ξ on a1,a2 if for each normal sequence of partitions of a1,a2, a1=u∘<u1<⋯<un=a2, and for all Θk∈uk−1,uk, we have(4)limn⟶∞E∑k=1nξΘk,.uk−uk−1−ν.2=0.
Then,ν. is written as(5)ν.=∫a1a2ξs,.ds,a.e.
For more on mean-square integrable stochastic processes, refer [8].Theorem 1 (see [3]).
Let us consider the Jensen-convex stochastic processξ:I×Ω⟶ℝ that is mean-square continuous on I; then, we have(6)ξa1+a22,.≤1a2−a1∫a1a2ξu,.du≤ξa1,.+ξa2,.2,a.e,for all a1,a2∈I, a1<a2. The above inequality is Hermite-Hadamard inequality for stochastic convexity.Let us present some important generalizations of convex stochastic processes.Definition 4 (see [9]).
A stochastic processξ:I×Ω⟶ℝ is said to be generalized convex if for θ∈0,1 and u,v∈I,(7)ξθu+1−θv,.≤ξv,.+θηξu,.,ξv,.,a.e.Definition 5 (see [10]).
An intervalI is a p-convex set if θup+1−θvp1/p∈I for all u,v∈I, θ∈0,1, and p = 2m + 1 or p=r/n, r = 2s + 1, n = 2t + 1, and m,s,t∈N.Definition 6 (see [10]).
A functionf:I⟶ℝ is p-convex, if for θ∈0,1 and u,v∈I, we have(8)fθup+1−θvp1/p≤θfu+1−θfv,where I is a p-convex set.Remark 1 (see [11]).
IfI⊂0,∞ be a real interval and p∈ℝ0, then(9)txp+1−typ1/p∈I,for allx,y∈Iandt∈0,1.
According to Remark1, we can give a different version of the definition of p-convex function as follows:Definition 7 (see [11]).
IfI⊂0,∞ be a real interval and p∈ℝ0. A function f:I⟶ℝ is said to be a p-convex function if(10)ftxp+1−typ1/p≤tfx+1−tfy,for all x,y∈I and t∈0,1. If the inequality (10) is reversed, then f is said to be p-concave.Definition 8 (see [12]).
A processξ:I×Ω⟶ℝ, where I is a p-convex set, is said to be a p-convex stochastic process if for θ∈0,1 and u,v∈I, we have(11)ξθup+1−θvp1/p,.≤θξu,.+1−θξv,.,a.e.
In [13], the following functions are defined.Definition 9.
(12)βu,v=∫01θu−11−θv−1dθ,u,v>0.Definition 10.
Forw>v>0, z<1,(13)F21u,v;w;z=1βv,w−v∫01θv−11−θw−v−11−zθ−udθ.
Now, we are in position to define the main notion of this article.Definition 11.
A stochastic processξ:I×Ω⟶ℝ is said to be generalized p-convex, if for θ∈0,1 and , we have(14)ξθup+1−θvp1/p,.≤ξv,.+θηξu,.,ξv,.,a.e.
If the inequality in (14) is reversed, then ξ is the generalized p-concave.Remark 2.
It is obvious that the inequality (14) reduces to the convex stochastic process for p=1 and ηx,y=x−y.Example 1.
Consider a stochastic processξ:0,∞×Ω⟶ℝ defined by ξu,.=up, p≠0, and ηx,y=x−y; then, ξ is the generalized p-convex.
We organize our study as follows. First we derive some basic properties for this generalization. In next section, Schur type inequality is obtained. The third, fourth, and fifth sections are devoted to Hermite-Hadamard, Jensen, and fractional integral inequalities for generalizedp-convex stochastic processes.Proposition 1.
Letξ1,ξ2:I×Ω⟶ℝ be two generalized p-convex stochastic processes:(1)
Ifη is additive, then ξ1+ξ2:I×Ω⟶ℝ is also a generalized p-convex stochastic process(2)
Ifη is nonnegatively homogeneous, then λξ1:I×Ω⟶ℝ, for any λ≥0, is the generalized p-convex stochastic processThe proof Proposition1 is straightforward.Theorem 2.
Assume a nonempty collectionξj:I×Ω⟶ℝ,j∈J of generalized p-convex stochastic processes, such that(1)
There existα∈0,∞ and β∈−1,∞, such that ηu,v=αu+βv for all u,v∈ℝ(2)
For eachu∈I,maxj∈Jξju,. exists in ℝ; then, the stochastic process defined by ξu,.=maxj∈Jξju,. for all u∈I is the generalized p-convex.Proof.
For anyu,v∈I and θ∈0,1, we have(15)ξθup+1−θvp1/p,.=maxj∈Jξjθup+1−θvp1/p,.≤maxj∈Jξjv,.+θηξju,.,ξjv,.=maxj∈Jξjv,.+θαξju,.+βξjv,.=maxj∈J1+βθξjv,.+αθξju,.≤1+βθmaxj∈Jξjv,.+αθmaxj∈Jξju,.=1+βθξv,.+αθξu,.=ξv,.+θαξu,.+βξv,.=ξv,.+θηξu,.,ξv,.,which is as required.


## 2. Schur Type Inequality

Theorem 3.
ForI⊂0,∞ and p>0, let ξ:I×Ω⟶ℝ is the generalized p-convex stochastic process. Then, ∀u1,u2,u3∈I, such that u1<u2<u3 and u3p−u1p,u3p−u2p,u2p−u1p∈0,1, and we have(16)ξu3,.u3p−u1p−ξu2,.u3p−u1p+u3p−u2pηξu1,.,ξu3,.≥0,a.e.Proof.
Letu1,u2,u3∈I be given. Then, we can easily see that(17)u3p−u2pu3p−u1p,u2p−u1pu3p−u1p∈0,1,u3p−u2pu3p−u1p+u2p−u1pu3p−u1p=1.
Settingθ=u3p−u2p/u3p−u1p, u2p=θu1p+1−θu3p. As ξ is generalized p-convex, so(18)ξu2,.≤ξu3,.+u3p−u2pu3p−u1pηξu1,.,ξu3,..
By assumingu3p−u1p>0 and multiplying the above inequality by u3p−u1p, we get inequality (16).


## 3. Hermite-Hadamard Type Inequality

Theorem 4.
ForI⊂0,∞ and p>0, let a mean-square generalized p-convex stochastic process ξ:u1,u2×Ω⟶ℝ, which is integrable. Then, for any u1,u2∈I, u1<u2, the following inequality holds almost everywhere:(19)ξu1p+u2p21/p,.−p2u2p−u1p∫u1u2xp−1ηξu1p+u2p−xp1/p,.,ξx,.dx≤pu2p−u1p∫u1u2xp−1ξx,.dx≤ξu1,.+ξu2,.2+14ηξu1,.,ξu2,.+ηξu2,.,ξu1,..Proof.
Takexp=θu1p+1−θu2p and yp=1−θu1p+θu2p; so,(20)ξu1p+u2p21/p,.=ξxp+yp21/p,..
Sinceξ is the generalized p-convex, so we have(21)ξu1p+u2p21/p,.=ξ12θu1p+1−θu2p1/p,.p+121−θu1p+θu1p1/pp≤ξ1−θu1p+θu2p1/p,.+12ηξθu1p+1−θu2p1/p,.,ξ1−θu1p+θu2p1/p,..
Integrating w.r.t “θ,” the above inequality on [0, 1],(22)ξu1p+u2p21/p,.≤∫01ξ1−θu1p+θu2p1/p,.dθ+12∫01ηξθu1p+1−θu2p1/p,.,ξθu1p+1−θu2p1/p,.dθ,which implies(23)ξu1p+u2p21/p,.−p2u2p−u1p×∫u1u2xp−1ηξu1p+u2p−xp1/p,.,ξx,.dx≤pu2p−u1p∫u1u2xp−1ξx,.dx.
Now,(24)∫u1u2xp−1ξx,.dx=u2p−u1pp∫01ξθu1p+1−θu2p1/p,.dθ≤u2p−u1ppξu2,.+∫01θηξu1,.,ξu2,.dθ,which implies(25)pu2p−u1p∫u1u2xp−1ξx,.dx≤ξu2,.+∫01θηξu1,.,ξu2,.dθ.
Similarly,(26)pu2p−u1p∫u1u2xp−1ξx,.dx≤ξu1,.+∫01θηξu2,.,ξu1,.dθ.
Adding (25) and (26),(27)pu2p−u1p∫u1u2xp−1ξx,.dx≤ξu1,.+ξu2,.2+14ηξu1,.,ξu2,.+ηξu1,.,ξu2,..
Combining (23) and (27), we obtain the inequality (22).Remark 3.
Forp=1 and ηu,v=u−v in (22), we get Hermite-Hadamard inequality (6) for the convex stochastic process.


## 4. Jensen Type Inequality

The following result will be helpful in the derivation of Jensen’s type inequality for the generalizedp-convex stochastic process.Lemma 1.
Letw1,…,wn be the positive real numbers n≥2. Assume ξ:I×Ω⟶ℝ be a generalized p-convex stochastic process and u1,u2,…,un∈I; then, we have almost everywhere(28)ξ1Wn∑i=1nwiuip1/p,.=ξWn−1Wn∑i=1n−1wiWn−1uip+wnWnunp1/p,.≤ξun,.+Wn−1Wnηξ∑i=1n−1wiWn−1uip1/p,.,ξun,.,where Wn=∑i=1nwi.Theorem 5 (Jensen type inequality).
Letξ:I×Ω⟶ℝ be a generalized p-convex stochastic process and η:A×B⟶ℝ be nondecreasing, nonnegatively sublinear in the first variable; then, we have almost everywhere(29)ξ1Wn∑i=1nwiuip1/p,.≤ξun,.+∑i=1nWiWnηξui,ui+1,…,un,.,where Wn=∑i=1nwi and ηξui,ui+1,…,un,.=ηηξui,ui+1,…,un−1,.,ξun,. and ηξu,.=ξu,.∀u∈I.Proof.
Sinceη is nondecreasing, nonnegatively sublinear in the first variable, so using Lemma 1, we get(30)ξ1Wn∑i=1nwixip1/p,.=ξwnWnunp+∑i=1n−1wiWnuip1/p,.=ξWn−1Wn∑i=1n−1wiWn−1uip+wnWnunp1/p,.≤ξun,.+Wn−1Wnηξ∑i=1n−1wiWn−1uip1/p,.,ξun,.=ξun,.+Wn−1WnηξWn−2Wn−1∑i=1n−2wiWn−2uip+wn−1Wn−1un−1p1/p,.,ξun,.≤ξun,.+Wn−1Wnηξun−1,.+Wn−2Wn−1η×ξ∑i=1n−2wiWn−2uip1/p,.,ξun−1,.,ξun,.≤ξun,.+Wn−1Wnηξun−1,.,ξun+Wn−2Wnη×ηξ∑i=1n−2wiWn−2uip1/p,.,ξun−1,.,ξun,.≤⋯≤ξun,.+Wn−1Wnηξun−1,.,ξun,.+Wn−2Wn×ηηξun−2,.,ξun,.,ξun,.+⋯+W1Wnηη…ηξu1,.,ξu2,.,ξu3,.…,ξun−1,.,ξun,.=ξun,.+Wn−1Wnηξun−1,un,.+Wn−2Wnηξun−2,un−1,un,.+⋯+W1Wnηξu1,u2,…,un−1,un,.=ξun,.+∑i=1n−1WiWnηξui,ui+1,…,un,..


## 5. Fractional Integral Inequalities

Lemma 2 (see [12]).
Assume a stochastic processξ:I⊂0,∞×Ω⟶ℝ which is mean-square differentiable on Io and u1,u2∈Io with u1<u2. If ξ′∈Lu1,u2, then we have almost everywhere(31)ξu1,.+ξu2,.2−pu2p−u1p∫u1u2ξx,.x1−pdx=u2p−u1p2p∫011−2θθu1p+1−θdp1−1/p×ξ′θu1p+1−θu2p1/p,.dθ.Theorem 6.
ForI⊂0,∞ and p>0 and under the assumptions of Lemma 2 with ξ′q, a generalized p-convex stochastic process on u1,u2 for q≥1, then we have almost everywhere(32)ξu1,.+ξu2,.2−pu2p−u1p∫u1u2ξx,.x1−pdx≤u2p−cp2pL11−1/qL1ξ′u2,.q+L2ηξ′u1,.q,ξ′u2,.q1/q,where(33)L1u1,u2;p=14u1p−u2p21/p−1F211−1p,2,3:u1p−u2pu1p+u2p+F211−1p,2,3:u2p−u1pcp+u2p,L2u1,u2;p=124u1p+u2p21/p−1F211−1p,2,4:u1p−u2pu1p+u2p+2F11−1p,2,4:u2p−u1pu1p+vp.Proof.
By making use of Lemma2 and power mean-integral inequality, we have(34)ξu1,.+ξu2,.2−1u2p−u1p∫u1u2ξx,.x1−pdx≤u2p−u1p2p∫011−2θθu1p+1−θu2p1−1/p×ξ′θu1p+1−θu2p1/p,.dθ≤u2p−u1p2p∫011−2θθu1p+1−θu2p1−1/pdt1−1/q×∫011−2θθu1p+1−θu2p1−1/pξ′θu1p+1−θu2p1/p,.qdθ1/q.
Hence, by generalizedp-convexity of ξ′q on u1,u2, we have(35)ξu1,.+ξu2,.2−1u2p−u1p∫abξx,.x1−pdx≤u2p−u1p2p∫011−2θθu1p+1−θu2p1−1/pdt1−1/q∫011−2θξ′u2,.q+θηξ′u1,.q,ξ′u2,.qθu1p+1−θu2p1−1/pdt1/q≤u2p−u1p2pL11−1/qL1ξ′u2,.q+L2ηξ′u1,.q,ξ′u2,.q1/q.
It is easy to check that(36)∫011−2θθu1p+1−θu2p1−1/pdθ=L1u1,u2;p,∫011−2θθθu1p+1−θu2p1−1/pdθ=L2u1,u2;p.Remark 4.
By settingηu1,u2=u1−u2 in (32), we get Theorem 4 of [12].
We will get the following Corollary by takingq=1 in (32).Corollary 1.
Ifξ′ is generalized p-convex on u1,u2, then we have almost everywhere(37)ξu1,.+ξu2,.2−pu2p−u1p∫u1u2ξx,.x1−pdx≤u2p−u1p2pL1ξ′u2,.+L2ηξ′u1,.,ξ′u2,.,where L1 and L2 are defined in Theorem 6.Remark 5.
If we takeηu1,u2=u1−u2 in (37), then we have Corollary 4 of [12].Theorem 7.
ForI⊂0,∞ and p>0 and under the assumptions of Lemma 2 with ξ′q, a generalized p-convex stochastic process on u1,u2 for 1<q, 1/r+1/q=1, then we have almost everywhere(38)ξu1,.+ξu2,.2−pu2p−u1p∫u1u2ξx,.x1−pdx≤u2p−u1p2p1r+11/rL4ξ′u2,.q+L5ηξ′u1,.q,ξ′u2,.q1/q,where(39)L3=L3u1,u2;p;q=1u1qp−q⋅F21q−qp,1;2;1−u2u1p,ifp<0,1u2qp−q⋅F21q−qp,1;2;1−u1dp,ifp>0,L4=L4u1,u2;p;q12u1qp−q⋅F21q−qp,1;3;1−u2u1p,ifp<0,12u2qp−q⋅F21q−qp,2;3;1−u1u2p,ifp>0.Proof.
By making use of Lemma2, Holder’s inequality, and generalized p-convexity of ξ′q on u1,u2, we have(40)ξu1,.+ξu2,.2−pu2p−u1p∫u1u2ξx,.x1−pdx≤u2p−u1p2p∫011−2θrdt1/r×∫011θu1p+1−θu2pq−q/pξ′θu1p+1−θu2p1/p,.qdt1/q≤dp−u1p2p1r+11/r×∫01ξ′u2,.q+θηξ′u1,.q,ξ′u2,.qθu1p+1−θu2pq−q/pdθ1/q,where an easy calculation gives(41)∫011θu1p+1−θu2pq−q/pdθ=L3u1,u2;p;q,(42)∫01θθu1p+1−θu2pq−q/pdθ=L4u1,d;p;q.
Substituting equations (41) and (42) into (38), the proof is completed.Remark 6.
By takingηu1,u2=u1−u2 in Theorem 7, then we obtain Theorem 6 of [12].Theorem 8.
ForI⊂0,∞ and p>0 and under the assumptions of Lemma 2 with ξ′q, a generalized p-convex on u1,u2 for 1<q, 1/r+1/q=1, then we have almost everywhere(43)ξu1,.+ξu2,.2−pu2p−u1p∫u1u2ξx,.x1−pdx≤u2p−u1p2pL51/r1q+11/qξ′u2,.q+12ηξ′u1,.q,ξ′u2,.q1/q,where(44)L5=L5u1,u2;p;r=1u1pr−r⋅F21r−rp,1;2;1−u2u1p,ifp<0,1u2pr−r⋅F21r−rp,1;2;1−u1u2p,ifp>0.Proof.
From Lemma2, Holder’s inequality, and generalized p-convexity of ξ′q on u1,u2, we have(45)ξu1,.+ξu2,.2−pu2p−u1p∫u1u2ξx,.x1−pdx≤u2p−u1p2p∫011θu1p+1−θu2pq−q/pdθ1/r×∫011−2θqξ′θu1p+1−θu2p1/p,.qdθ1/q≤u2p−u1p2pL61/r1q+11/qξ′u2,.q+12ηξ′u1,.q,ξ′u2,.q1/q,where an easy calculation gives(46)∫011θu1p+1−θu2pq−r/pdθ=L5c,u2;p;r,(47)∫011−2θqdθ=1q+1.
Substituting (46) and (47) into (43), we obtain the required result.


## 6. Conclusion

There are many applications of stochastic processes, for instance, the Kolmogorov–Smirnoff test on equality of distributions. The other application includes sequential analysis and quickest detection. In this study, we have presented a new class of convex stochastic processes which are generalizedp-convex and established Jensen, Hermite-Hadamard, and fractional integral inequalities for this class. Our conclusions are applicable, since the expected value of a random variable is consistently bounded above by the expected value of the convex function of that random variable. It will be interesting to find parallel results by using the proposed definition in this study in the setting of other fractional integrals [14, 15].



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