# Mean Square Integral Inequalities for Generalized Convex Stochastic Processes via Beta Function

Authors: Putian Yang; Shiqing Zhang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The integral inequalities have become a very popular area of research in recent years. The present paper deals with some important generalizations of convex stochastic processes. Several mean square integral inequalities are derived for this generalization. The involvement of the beta function in the results makes the inequalities more convenient for applied sciences.


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## Body

## 1. Introduction

Just as the probability theory is regarded as the study of mathematical models of random phenomena, the theory of stochastic processes plays an important role in the investigation of random phenomena depending on time. A random phenomenon that arises through a process which is developing in time and controlled by some probability law is called a stochastic process. Thus, stochastic processes can be referred to as the dynamic part of the probability theory. We will now give a formal definition of a stochastic process.Various collections of random variablesXl,·,l∈J, have the property in some sense that Xl is stochastically convex (or −Xl,· is stochastically concave). The stochastic process with convexity properties has a large number of applications. In [1], the authors demonstrated the use of a stochastically convex function in different areas of probability and statistics.In queueing theory, the convexity of steady-state waiting time is used in [2]. More in [1], the authors used the convexity of payoff in the success rate to obtain an imperfect repair.In 1980, Nikodem introduced the study of quadratic and convex stochastic processes (see [3, 4]). In [5, 6], Skowronski explained the properties of the Wright-convex and Jensen-convex stochastic process. Also, Kotrys described results on convex and strongly convex stochastic processes, together with a Hermite-Hadamard-type inequality for convex stochastic processes (see [7–9]).The Hermite-Hadamard inequality for the convex stochastic process is defined as follows:LetX:J×Ω⟶ℝ be a convex and mean square continuous in the interval T×Ω; then, the inequality holds almost everywhere:
(1)Xr+s2,·≤1r−s∫rsXl,·dl≤Xr,·+Xs,·2,for any r,s∈J. For more details on Hermite-Hadamard-type inequalities for the stochastic process, we may refer the reader to [10–12].Definition 1 (see [13]).
A stochastic process is a collection of random variablesXl parameterized by l∈J, where J⊂ℝ. When J=1,2,⋯, then Xl is said to be a stochastic process in discrete time (i.e., a sequence of random variables). When J is an interval in ℝJ=0,∞, then we say that Xl is a stochastic process in continuous time.
For everyω∈Ω, the function
(2)J∋l↦Xl,ωis said to be a path or sample path of Xl.Definition 2 (see [13]).
A familyFl of α-fields on Ω parametrized by l∈J, where J⊂ℝ, is said to be a filtration if
(3)Fs⊂Fl⊂F,for any s,l∈J such that s≤l.Definition 3 (see [13]).
A stochastic processXl parametrized by l∈T is said to be a martingale (supermartingale, submartingale) with respect to a filtration Fl if
(1)
Xl is integrable for each l∈J(2)
Xl is Fl-measurable for each l∈J(3)
Xs=EXl∣Fs (respectively, ≤ or ≥) for every s,l∈J such that s≤l.Definition 4 (see [7]).
LetΩ,A,P be an arbitrary probability space and J⊂ℝ be an interval. A stochastic process X:Ω⟶ℝ is called as follows:
(1)
Stochastically continuous in intervalJ, if ∀l∘∈J(4)P−liml⟶l∘Xl,·=Xl∘,·,where P−lim denotes the limit in probability.(2)
Mean square continuous inJ, if ∀l∘∈J(5)P−liml⟶l∘EXl,·−Xl∘,·=0,where EXl,· denotes the expectation value of the random variable Xl,·.(3)
Increasing (decreasing) if∀μ,ν∈J such that
(6)Xμ,·≤Xν,·,Xμ,·≥Xν,·.(4)
Monotonic if it is increasing or decreasing(5)
If there exists a random variableX′l,·:J×Ω⟶ℝ, then we say that it is differentiable at a point l∈J, such that
(7)X′l,·=P−liml⟶l∘Xl,·−Xl∘,·l−l∘.
A stochastic processX:J×Ω⟶ℝ is continuous (differentiable) if it is continuous (differentiable) at every point of interval J.Definition 5 (see [7, 14]).
Suppose thatΩ,A,P be a probability space and J⊂ℝ be an interval with EXϑ2<∞∀ϑ∈J. If r,s⊂J, r=ϑ0<ϑ1<ϑ2<⋯<ϑn=s is a partition of r,s and Θ∈ϑκ−1,ϑκ for κ=1,2,⋯,n. A random variable Z:Ω⟶ℝ is known as mean square integral of the process Xϑ,· on r,s if
(8)limn⟶∞E∑κ=1∞XΘκ,·ϑκ,ϑκ−1−Z·2=0,
then, we have(9)∫rsXϑ,·dϑ=Z·a.e..
Also, the mean square integral operator is increasing; thus,(10)∫rsXϑ,·dϑ≤∫rsYϑ,·a.e.,where Xϑ,·≤Yϑ,· in r,s.
For more details on stochastic processes, we may refer the reader to [15, 16].Next, we write some basic definitions which will be used in this work:Definition 6 (see [4]).
LetΩ,A,P be a probability space and J⊆R be an interval. A stochastic process X:J×Ω⟶R is called a convex stochastic process; then, the inequality holds almost everywhere:
(11)Xϑr+1−ϑs,·≤ϑXr,·+1−ϑXs,·,
∀r,s∈J and ϑ∈0,1.Definition 7 (see [17]).
A processX:J×Ω⟶ℝ is said to be a p-convex stochastic process, if the following inequality holds:
(12)Xϑrp+1−ϑsp1/p,·≤ϑXr,·+1−ϑXs,·a.e.,for all r,s∈J and ϑ∈0,1.In [18], Barráez et al. defined the definition of the h-convex stochastic process as follows:Definition 8 (see [18]).
Leth:0,1⟶ℝ be a nonnegative function, h≠0. A stochastic process X:J×Ω⟶ℝ is a h-convex stochastic process, if the inequality holds:
(13)Xϑr+1−ϑs,·≤hϑXr,·+h1−ϑXs,·a.e.,for every r,s∈J and ϑ∈0,1.
Obviously, by takinghϑ=ϑ in (13), then the definition of the h-convex stochastic process reduces to the definition of the convex stochastic process [4].Definition 9 (see [9]).
Letc:Ω⟶ℝ be a positive random variable. A stochastic process X:J×Ω⟶ℝ is known as strongly convex with modulus c·>0, if the following inequality holds:
(14)Xϑr+1−ϑs,·≤ϑXr,·+1−ϑXs,·−c·ϑ1−ϑr−s2a.e.,for all r,s∈J and ϑ∈0,1.
For more details on the strongly convex stochastic process, we refer to [9], and for some interesting properties of some special function, see [19, 20]. Obviously, if we omit the term c·ϑ1−ϑr−s2 in (14), then we get the definition of a convex stochastic process (see [4]). On the other hand, if we set c=0, then we get it from (14) in limit case. Also, we use the beta function in this present work which is expressed as
(15)βr,s=∫01ϑr−11−ϑs−1dϑ,Rer>0,Res>0.


## 2. Main Results

Lemma 10 (see [21]).
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. Then, the following equality holds almost everywhere:
(16)∫rsω−rμs−ωνXω,·dω=s−rμ+ν+1∫011−ϑμϑνXϑr+1−ϑs,·dϑ,for some fixed μ,ν>0.Lemma 11.
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. Then, the following equality holds almost everywhere:
(17)∫rpspωp−rpμsp−ωpνXω,·ω1−pdω=sp−rpμ+ν+1p∫011−ϑμϑνXϑrp+1−ϑsp1/p,·dϑ,for some fixed μ,ν>0.Proof.
Letω=ϑrp+1−ϑsp1/p. Then, ϑ=sp−ωp/sp−rp, 1−ϑ=ωp−rp/sp−rp, and dϑ=−p/sp−rpω1−pdω, so
(18)∫011−ϑμϑνXϑrp+1−ϑsp1/p,·dϑ=psp−rpμ+ν+1∫rpspωp−rpμsp−ωpνXω,·ω1−pdωa.e.,which completes the proof.☐Remark 12.
If we takep=1 in Lemma 11, then we obtain Lemma 3.1 of [21].The following results are derived forp-convex stochastic processes.Theorem 13.
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If X is p-convex on r,s, where r,s∈J with r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(19)∫rpspωp−rpμsp−ωpνXω,·ω1−pdω≤sp−rpμ+ν+1pβμ+1,ν+2Xr,·+βμ+2,ν+1Xs,·.Proof.
By using Lemma11, the definition of the p-convexity of X and the beta function yield that
(20)∫rpspωp−rpμsp−ωpνXω,·ω1−pdω≤sp−rpμ+ν+1p∫011−ϑμϑνXϑrp+1−ϑsp1/p,·dϑ≤sp−rpμ+ν+1p∫011−ϑμϑνϑXr,·+1−ϑXs,·dϑ≤sp−rpμ+ν+1pβμ+1,ν+2Xr,·+βμ+2,ν+1Xs,·a.e.,which completes the proof.☐Remark 14.
If we takep=1 in Theorem 13, then we obtain Theorem 3.1 of [21].Theorem 15.
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If Xq is p-convex on r,s for q>1 with 1/κ+1/q=1, where r,s∈J, r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(21)∫rpspωp−rpμsp−ωpνXω,·ω1−pdω≤sp−rpμ+ν+1pβκμ+1,κν+11/κXr,·q+Xs,·q21/q.Proof.
Employing Lemma11 and Hölder’s integral inequality, we have a.e.(22)∫rpspωp−rpμsp−ωpνXω,·ω1−pdω≤sp−rpμ+ν+1p∫011−ϑμϑνXϑrp+1−ϑsp1/p,·dϑ≤sp−rpμ+ν+1p∫011−ϑκμϑκνdϑ1/κ∫01Xϑrp+1−ϑsp1/p,·qdϑ1/q.
SinceXq is a p-convex stochastic process, one can yield that
(23)∫01Xϑrp+1−ϑsp1/p,·qdϑ≤∫01ϑXr,·q+1−ϑXs,·qdϑ=Xr,·q+Xs,·q2a.e.,and by the definition of the beta function, we can write
(24)∫011−ϑκμϑκνdϑ=βκμ+1,κν+1.
Inserting (23) and (24) in (22) yields the required inequality (21).☐Remark 16.
If we takep=1 in Theorem 15, then we get Theorem 3.2 of [21].Theorem 17.
LetX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If Xq is p-convex on r,s for q>1, where r,s∈J with r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(25)∫rpspωp−rpμsp−ωpνXω,·ω1−pdω≤sp−rpμ+ν+1pβκμ+1,κν+11−1/q×βμ+1,ν+2Xr,·q+βμ+2,ν+1Xs,·q1/q.Proof.
Making use of Lemma11 and the power-mean integral inequality for κ≥1 yields that
(26)∫rpspωp−rpμsp−ωpνXω,·ω1−pdω≤sp−rpμ+ν+1p∫011−ϑμϑνXϑrp+1−ϑsp1/p,·dϑa.e.≤sp−rpμ+ν+1p∫011−ϑκμϑκνdϑ1−1/q×∫011−ϑμϑνXϑrp+1−ϑsp1/p,·qdϑ1/qa.e..
By using thep-convexity of the stochastic process Xq and by the definition of the beta function, we have a.e.(27)≤sp−rpμ+ν+1p∫011−ϑκμϑκνdϑ1−1/q×∫011−ϑμϑνϑXr,·q+1−ϑXs,·qdϑ1/q≤sp−rpμ+ν+1pβκμ+1,κν+11−1/q×βμ+1,ν+2Xr,·q+βμ+2,ν+1Xs,·q1/q.which completes the proof. ☐Remark 18.
If we takep=1 in Theorem 17, then we obtain Theorem 3.3 of [21].The following results are derived forh-convex stochastic processes.Theorem 19.
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If X is h-convex on r,s, where r,s∈J with r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(28)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1Xr,·βhϑ+Xs,·βh1−ϑ,where
(29)βhϑμ,ν=∫011−ϑμϑνhϑdϑ,(30)βh1−ϑμ,ν=∫011−ϑμϑνh1−ϑdϑ.Proof.
By Lemma10, the definition of the h-convexity of ∣X∣ and the beta function yield that
(31)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1∫011−ϑμϑνXϑr+1−ϑs,·dϑ≤s−rμ+ν+1∫011−ϑμϑνhϑXr,·+h1−ϑXs,·dϑ≤s−rμ+ν+1Xr,·∫011−ϑμϑνhϑdϑ+Xs,·∫011−ϑμϑνh1−ϑ=s−rμ+ν+1Xr,·βhϑμ,ν+Xs,·βh1−ϑμ,νa.e.,
which completes the proof.☐Remark 20.
If we takehϑ=ϑ in Theorem 19, then we obtain Theorem 3.1 of [21].Theorem 21.
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If Xq is h-convex on r,s for q>1 with 1/κ+1/q=1, where r,s∈J, r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(32)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1βκμ+1,κν+11/κXr,·qβhϑ+Xs,·qβh1−ϑ1/q,where βhϑ=∫01hϑdϑ and βh1−ϑ=∫01h1−ϑdϑ.Proof.
Employing Lemma10 and Hölder’s integral inequality, we have a.e.(33)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1∫011−ϑμϑνXϑr+1−ϑs,·dϑ≤s−rμ+ν+1∫011−ϑκμϑκνdϑ1/κ∫01Xϑr+1−ϑs,·qdϑ1/q.
SinceXq is an h-convex stochastic process, one can yield that
(34)∫01Xϑr+1−ϑs,·qdϑ≤∫01hϑXr,·q+h1−ϑXs,·qdϑ≤Xr,·q∫01hϑdϑ+Xs,·q∫01h1−ϑdϑ≤Xr,·qβhϑ+Xs,·qβh1−ϑa.e.,and by the definition of the beta function, we can write
(35)∫011−ϑκμϑκνdϑ=βκμ+1,κν+1.
Inserting (34) and (35) in (33) yields the desired inequality (32).☐Remark 22.
If we takehϑ=ϑ in Theorem 21, then we obtain Theorem 3.2 of [21].Theorem 23.
LetX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If Xq is h-convex on r,s for q>1, where r,s∈J with r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(36)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1βκμ+1,κν+11−1/qβhϑμ,νXr,·q+βhθμ,νXs,·q1/q,where
(37)βhϑμ,ν=∫011−ϑμϑνhϑdϑ,(38)βh1−ϑμ,ν=∫011−ϑμϑνh1−ϑdϑ.Proof.
By Lemma10 and the power-mean integral inequality for κ≥1, one can yield that
(39)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1∫011−ϑμϑνXϑr+1−ϑs,·dϑa.e.≤s−rμ+ν+1∫011−ϑκμϑκνdϑ1−1/q×∫011−ϑμϑνXϑr+1−ϑs,·qdϑ1/qa.e..
By using theh-convexity of the stochastic process Xq and by the definition of the beta function, we have a.e.(40)≤s−rμ+ν+1∫011−ϑκμϑκνdϑ1−1/q×∫011−ϑμϑνhϑXr,·q+h1−ϑXs,·qdϑ1/q≤s−rμ+ν+1∫011−ϑκμϑκνdϑ1−1/q×Xr,·q∫011−ϑμϑνhϑ+Xs,·q∫01h1−ϑdϑ1/q≤s−rμ+ν+1βκμ+1,κν+11−1/qβhϑμ,νXr,·q+βhθμ,νXs,·q1/q,which completes the proof. ☐Remark 24.
If we takehϑ=ϑ in Theorem 23, then we obtain Theorem 3.3 of [21].The following results are derived for strongly convex stochastic processes.Theorem 25.
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If X is strongly convex on r,s, where r,s∈J with r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(41)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1βμ+1,ν+2Xr,·+βμ+2,ν+1Xs,·−c·r−s2βμ+2,ν+2.Proof.
From Lemma10, the definition of the strong convexity of X and the beta function yield that
(42)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1∫011−ϑμϑνXϑr+1−ϑs,·dϑ≤s−rμ+ν+1∫011−ϑμϑνϑXr,·+1−ϑXs,·−c·ϑ1−ϑr−s2dϑ≤s−rμ+ν+1βμ+1,ν+2Xr,·+βμ+2,ν+1Xs,·−c·r−s2βμ+2,ν+2a.e.,which completes the proof. ☐Remark 26.
If we takec=0 in Theorem 25, then we obtain Theorem 3.1 of [21].Theorem 27.
Suppose thatX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If Xq is strongly convex on r,s for q>1 with 1/κ+1/q=1, where r,s∈J with r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(43)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1βκμ+1,κν+11/κ12Xr,·q+Xs,·q−16c·r−s21/q.Proof.
By Lemma10 and Hölder’s integral inequality, we have a.e.(44)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1∫011−ϑμϑνXϑr+1−ϑs,·dϑ≤s−rμ+ν+1∫011−ϑκμϑκνdϑ1/κ∫01Xϑr+1−ϑs,·qdϑ1/q.
SinceXq is a strongly convex stochastic process, one can yield that
(45)∫01Xϑr+1−ϑs,·qdϑ≤∫01ϑXr,·q+1−ϑXs,·q−c·ϑ1−ϑr−s2dϑ≤12Xr,·q+Xs,·q−16c·r−s2a.e.,and taking the definition of the beta function, we can write
(46)∫011−ϑκμϑκνdϑ=βκμ+1,κν+1.
Replacing (45) and (46) in (44) yields the desired inequality (43). ☐Remark 28.
If we takec=0 in Theorem 27, then we get Theorem 3.2 of [21].Theorem 29.
LetX:J×Ω⟶ℝ be a mean square continuous and mean square integrable stochastic process. If Xq is strongly convex on r,s for q>1, where r,s∈J with r<s, and μ,ν>0 is taken, then the inequality holds almost everywhere:
(47)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1βκμ+1,κν+11−1/q×βμ+1,ν+2Xr,·q+βμ+2,ν+1Xs,·q−c·r−s2βμ+2,ν+21/q.Proof.
By making use of Lemma11 and the power-mean integral inequality for κ≥1, one can yield that
(48)∫rsω−rμs−ωνXω,·dω≤s−rμ+ν+1∫011−ϑμϑνXϑr+1−ϑs,·dϑa.e.≤s−rμ+ν+1∫011−ϑκμϑκνdϑ1−1/q×∫011−ϑμϑνXϑr+1−ϑs,·qdϑ1/qa.e..
By using the strong convexity of the stochastic processXq and taking the definition of the beta function, we have a.e.(49)≤s−rμ+ν+1∫011−ϑκμϑκνdϑ1−1/q×∫011−ϑμϑνϑXr,·q+1−ϑXs,·q−c·ϑ1−ϑr−s2dϑ1/q≤s−rμ+ν+1βκμ+1,κν+11−1/q×βμ+1,ν+2Xr,·q+βμ+2,ν+1Xs,·q−c·r−s2βμ+2,ν+21/q,
which completes the proof. ☐Remark 30.
If we takec=0 in Theorem 29, then we obtain Theorem 3.1 of [21].


## 3. Conclusions

Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography, and telecommunications. In this paper, we studied the generalized convex stochastic processes via a special function “beta function.” We established mean square integral inequalities for these generalized convex stochastic processes.


## 4. Future Directions

It will be interesting for researchers to work on the generalized convex stochastic processes via different fractional internal operators.



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