Some Mean Square Integral Inequalities For Preinvexity Involving The Beta Function

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1 Int. J. Nonlinear Anal. Appl. Volume 2, Special Issue, Winter and Spring 22, ISSN: electronic Some Mean Suare Integral Ineualities For Preinvexity Involving The Beta Function Muhammad Shoaib Saleem a, Miguel Vivas-Cortez b,, Sana Sajid a a Department of Mathematics, University of Okara b Pontificia Universidad Católica del Ecuador, Facultad de Ciencias Naturales y Exactas, Escuela de Ciencias Físicas y Matemáticas, Sede Quito, Ecuador. Communicated by Madjid Eshaghi Gordji Abstract In the present research, we will deal with mean suare integral ineualities for preinvex stochastic process and η-convex stochastic process in setting of beta function. Further, we will present some novel results for improved Hölder integral ineuality. The results given in this present paper are generalizations of already existing results in the literature. Keywords: Mean suare integral ineualities, Convex stochastic process, η-convex stochastic process, Preinvex stochastic process, Beta function 2 MSC: Primary 9C33; Secondary 26B25.. Introduction In the last few years, a lot of attention has been given to the study of stochastic processes due to a wide range of applications in finance, probabilistic model, economics, information theory, noise in the physical system, operation research, weather forecasting, astronomy, and so on. The study of stochastic processes makes use of mathematical expertise and strategies for probability, calculus, set theory, functional analysis, linear algebra, real analysis, topology, and Fourier analysis. Stochastic processes is a branch of mathematics, beginning with the axioms of probability and containing a wealthy and captivating set of results following from one s axioms [2]. Yet the results are useful to several areas [], they re best understood at the start in respect of their mathematical structure and correlation [8]. Corresponding author addresses: shoaib83455@gmail.com Muhammad Shoaib Saleem, mjvivas@puce.edu.ec Miguel Vivas-Cortez, sanasajid22@gmail.com Sana Sajid Received: August 22 Accepted: August 22

2 68 Shoaib Saleem, Vivas-Cortez, Sajid The study on convex stochastic processes was originated by B. Nagy in 974 see [2]. In [2, 22], K. Nikodem et al. investigated some more properties of convex stochastic stochastic processes. Later on, at the end of the twentieth century, Skowronski established some more results on the convex stochastic process which generalized several well-known characterizations [27, 28]. In [24], Set et al. considered the Hermite-Hadamard integral type ineualities in the second sense for convex stochastic processes within the year 24. Recently, in [7, 3] numerous Hermite-Hadamard type ineualities connected with fractional integrals have been considered. D. Kotrys described results on convex and strongly convex stochastic processes, together with, a Hermite-Hadamard type ineuality for convex stochastic processes see [4, 5, 6]. For convex stochastic process, Hermite-Hadamard integral ineuality is expressed as follows: Assume ξ : I Ω R be a convex and mean suare continuous in the interval T Ω, then + ν ξ, 2 ν ν ξα, ξ, + ξν, 2 a.e.. for all, ν I. For more on these ineualities, we allude the reader see [, 2?, 29, 3]. Fractional calculus attained uick improvement in both applied and pure mathematics because of its giant use in system learning, physics, image processing, networking, and other branches of science. For more on these, we allude the reader to see [9, 23, 5]. The fractional derivative acuires speedy interest together with experts from disparate branches of science. The vast majority of applied issues can t be modeled by classical derivations. The difficulty in real-world issues is addressed via way of means fractional differential euations. The present paper is organized as follows. In second section we will give some preliminary material. However, in third section we will present our main results and at last we give concluding remarks to our present research. 2. Preliminaries Definition 2.. [6] A stochastic process is a collection of random variables ξ parametrized by I, where I R. When I = {, 2,...}, then ξ is known as a stochastic process in discrete time a seuence of random variables. When I is an interval in R I = [,, then ξ be a stochastic process in continuous time. For every ω Ω the function I ξ, ω is called a path or sample path of ξ. Definition 2.2. [6] A family F of α-fields on Ω parametrized by I, where I R, is known as a filtration if F ν F F for any ν, I such that ν. Definition 2.3. [6] A sochastic process ξ parametrized by T is known as a martingale supermartingale, submartingale with respect to a filtration F if ξ is integrable for each I; 2 ξ is F -measurable for each I;

3 Some Mean Suare Integral Ineualities... ; Volume 2, Special Issue, Winter and Spring 22, ξν = EX F ν respectively, or for every ν, I such that ν. Definition 2.4. [4] Let Ω, A, P be an arbitrary probability space and I R. A stochastic process ξ : Ω R is termed as Stochastically continuous in I, if I P lim ξ, = ξ,, where P lim denotes the limit in probability. 2 Mean-suare continuous in I, if I P lim Eξ, ξ, =, where Eξ, represents the expectation value of the random variable ξ,. 3 If there exist a random variable ξ, : I Ω R then it is differentiable at a point I, such that ξ, = P lim ξ, ξ,. A stochastic process ξ : I Ω R is termed to be continuous differentiable if it is continuous differentiable at every point of I. Definition 2.5. [4, 3] Assume Ω, A, P be a probability space and I R be an interval with Eξ 2 < α I. If [a, b] I, a = < < 2 <... < n = b be a partition of [a, b] and Θ [ κ, κ ] for κ =, 2,..., n. A random variable Z : Ω R is termed as mean-suare integral of the process ξ on [a, b] if [ lim E n then we have b a κ= ξθ κ, κ, κ Z ] 2 =, ξ, d = Z a.e.. Also, mean suare integral operator is increasing, then, b a ξ, d b where ξ, Y, in [a, b]. a Y, a.e. For more details on stochastic processes see [4, 7, 8, 25, 26]. Now, we give some definitions related to our present work: Definition 2.6. [22] Let Ω, A, P be a probability space and I R be an interval. A stochastic process ξ : I Ω R is termed as a convex stochastic process, then ξ α + αν, αξ, + αξ ν, a.e. 2., ν I and α [, ].

4 62 Shoaib Saleem, Vivas-Cortez, Sajid Definition 2.7. [3] Let ξ : I Ω R not necessarily mean-suare differentiable on η-invex index set I. ξ, is said to be preinvex with respect to η if ξ + αην,, αξ, + αξν, a.e. 2.2 holds, ν I and α [, ]. Definition 2.8. [3] Let Ω, A, P be a probability space and I R be an interval, then ξ : IΩ R is called η-convex stochastic process, if ξα + αν, ξν, + αηξ,, ξν, a.e. 2.3 holds, ν I and α [, ]. Also, we will use the Beta function in this present paper which is defined as follows: β, ν = α α ν, Re >, Reν >. Theorem 2.9. Hölder-İşcan integral ineuality [9] Let ξ, ξ 2 : [, ν] Ω R be a real stochastic processes and ξ, ξ 2 be a mean suare integrable on [, ν]. If p > and + =, then we have almost everywhere p { ν ν ξ z, ξ 2 z, dz ν + z ξ z, p dz ν z ξ z, p p ν dz ν z ξ 2 z, dz p Theorem 2.. Improved power-mean integral ineuality [9] z ξ 2 z, dz }. be a mean suare inte- Let ξ, ξ 2 : [, ν] Ω R be a real stochastic processes and ξ, ξ ξ 2 grable on [, ν]. If, then we have almost everywhere { ξ z, ξ 2 z, dz ν ν + z ξ z, dz ν ν z ξ z, dz ν z ξ z, ξ 2 z dz z ξ z, ξ 2 z, dz }. 3. Main Results In this section, we will establish some new lemmas and results for η-convex stochastic process and preinvex stochastic process under beta function.

5 Some Mean Suare Integral Ineualities... ; Volume 2, Special Issue, Winter and Spring 22, Lemma 3.. [] Assume that ξ : I Ω R be a mean suare continuous and mean suare integrable stochastic process. Then the euality holds almost everywhere for some fixed θ, ϑ >. z θ ν z ϑ ξz, dz = ν θ+ϑ+ α θ α ϑ ξα + αν,, Lemma 3.2. Let ξ : I = [, + ην, ] Ω R be a mean suare integrable and mean suare continuous stochastic process with < + ην,. Then the euality holds almost everywhere +ην, for some fixed θ, ϑ >. z θ + ην, z ϑ ξz, dz = ην, θ+ϑ+ α θ α ϑ ξ + αην,, Proof. Note that +ην, = z θ + ην, z ϑ ξz, dz + αην, θ + ην, αην, ϑ ξ + αην,, = ην, θ+ϑ+ α θ α ϑ ξ + αην,,. This completes the proof. We will derive the next results for η-convex stochastic process. Theorem 3.3. Suppose ξ : I Ω R be a mean suare continuous and mean suare integrable stochastic process. Take θ, ϑ >, if ξ is a η-convex on [, ν], where, ν I and < ν, then the ineuality holds almost everywhere z θ ν z ϑ ξz, dz ν βθ θ+ϑ+ +, ϑ + ξν, + βθ +, ϑ + 2η ξ,, ξν,.

6 622 Shoaib Saleem, Vivas-Cortez, Sajid Proof. From Lemma 3. and take the definition of η-convex stochastic process of ξ yields that z θ ν z ϑ ξz, dz ν θ+ϑ+ α θ α ϑ ξα + αν, ν θ+ϑ+ α θ α ϑ ξν, + αη ξ,, ξν, a.e. and by the definition of beta function, we have ν βθ θ+ϑ+ +, ϑ + ξν, + βθ +, ϑ + 2η ξ,, ξν, a.e. This completes the proof. Theorem 3.4. Suppose ξ : I Ω R be a mean suare continuous and mean suare integrable stochastic process. Take θ, ϑ >, if ξ is a η-convex on [, ν] for > with + =, where p, ν I and < ν, then the ineuality holds almost everywhere z θ ν z ϑ ξz, dz ν βpθ θ+ϑ+ p +, pϑ + ξν, + 2 η ξ,, ξν,. 3. Proof. By using Lemma 3. and from Hölder integral ineuality, we have a.e. z θ ν z ϑ ξz, dz ν θ+ϑ+ α θ α ϑ ξα + αν, ν θ+ϑ+ Since ξ is a η-convex stochastic process, we have a.e. ξα + αν, α pθ α pϑ p ξα + αν, Now, by the definition of beta function, we can deduce ξν, + αη ξ,, ξν, 3.2 ξν, + 2 η ξ,, ξν,, 3.3 α pθ α pϑ = βpθ +, pϑ So, by inserting 3.3 and 3.4 in 3.2 yields the reuired ineuality 3..

7 Some Mean Suare Integral Ineualities... ; Volume 2, Special Issue, Winter and Spring 22, Theorem 3.5. Let ξ : I Ω R be a mean suare continuous and mean suare integrable stochastic process. Take θ, ϑ >, if ξ is a η-convex on [, ν] for >, where, ν I and < ν, then the ineuality holds almost everywhere z θ ν z ϑ ξz, dz ν θ+ϑ+ β θ +, ϑ + βθ +, ϑ + ξν, + βθ +, ϑ + 2η ξ,, ξν,. Proof. By using Lemma 3. and power-mean integral ineuality for, we can write z θ ν z ϑ ξz, dz ν θ+ϑ+ α θ α ϑ ξα + αν, ν θ+ϑ+ α θ α ϑ α θ α ϑ ξα + αν, a.e.. Take the definition of η-convex stochastic process of ξ and empolying the beta function yields that z θ ν z ϑ ξz, dz ν θ+ϑ+ α θ α ϑ α θ α ξν, ϑ + αη ξ,, ξν, ν θ+ϑ+ βθ +, ϑ + βθ +, ϑ + ξν, + βθ +, ϑ + 2η ξ,, ξν, a.e. which completes the proof. Theorem 3.6. Let ξ : I Ω R be a mean suare continuous and mean suare integrable stochastic process. Take θ, ϑ >, if ξ is a η-convex on [, ν] for > with + =, where, ν I and p < ν, then the ineuality holds almost everywhere

8 624 Shoaib Saleem, Vivas-Cortez, Sajid z θ ν z ϑ ξz, dz [ ν θ+ϑ+ β pθ + 2, pϑ + p 2 ξν, + 6 η ξ,, ξν, + β pθ +, pϑ + 2 p 2 ξν, + ] 3 η ξ,, ξν,. 3.6 Proof. By Lemma 3. and using Hölder- Iscan integral ineuality, we have a.e. z θ ν z ϑ ξz, dz ν θ+ϑ+ α θ α ϑ ξα + αν, ν θ+ϑ+ [ α α pθ α pϑ p α ξα + αν, + α α pθ α pϑ p ] α ξα + αν,. Since ξ is a η-convex stochastic process and by using the definition of beta function, we have a.e. z θ ν z ϑ ξz, dz ν θ+ϑ+ [ α pθ+ α pϑ p α ξν, + αη ξ,, ξν, + α pθ α pϑ+ p ] α ξν, + αη ξ,, ξν, ] [ ν θ+ϑ+ β pθ + 2, pϑ + p 2 ξν, + 6 η ξ,, ξν, + β pθ +, pϑ + 2 p 2 ξν, + ] 3 η ξ,, ξν, which completes the proof.

9 Some Mean Suare Integral Ineualities... ; Volume 2, Special Issue, Winter and Spring 22, Theorem 3.7. Suppose that ξ : I Ω R be a mean suare continuous and mean suare integrable stochastic process. Take θ, ϑ >, if ξ is a η-convex on [, ν] for >, where, ν I and < ν, then the ineuality holds almost everywhere z θ ν z ϑ ξz, dz ν βθ θ+ϑ+ + 2, ϑ + + βθ + 2, ϑ + ξν, + βθ + 2, ϑ + 2η ξ,, ξν, βθ +, ϑ + 2 βθ +, ϑ + 2 ξν, + βθ +, ϑ + 3η ξ,, ξν,. Proof. By using Lemma 3. and improved power-mean integral ineuality for yields that z θ ν z ϑ ξz, dz ν θ+ϑ+ α θ α ϑ ξα + αν, ν θ+ϑ+ α α θ α ϑ α α θ α ϑ ξα + αν, + α α θ α ϑ α α θ α ϑ ξα + αν, a.e..

10 626 Shoaib Saleem, Vivas-Cortez, Sajid Take the definition of η-convex stochastic process of ξ and beta function yields that z θ ν z ϑ ξz, dz ν θ+ϑ+ α θ+ α ϑ α θ+ α ξν, ϑ + αη ξ,, ξν, + α θ α ϑ+ α θ α ξν, ϑ+ + αη ξ,, ξν, ν θ+ϑ+ βθ + 2, ϑ + + which completes the proof. βθ + 2, ϑ + ξν, + βθ + 2, ϑ + 2η ξ,, ξν, βθ +, ϑ + 2 βθ +, ϑ + 2 ξν, + βθ +, ϑ + 3η ξ,, ξν, We will derive the following results for preinvex stochastic process. Theorem 3.8. Suppose ξ : I Ω R be a mean suare continuous and mean suare integrable stochastic process. If ξ is preinvex on [, + ην, ], where, + ην, I with < + ην, and take θ, ϑ >, then the ineuality holds almost everywhere +ην, z θ + ην, z ϑ ξz, dz ην, βθ θ+ϑ+ +, ϑ + 2 ξ, + βθ + 2, ϑ + ξν, 3.7 for some fixed θ, ϑ >. Proof. From Lemma 3.2 and take the preinvex stochastic process of ξ yields that +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ α θ α ϑ ξ + αην,, ην, θ+ϑ+ α θ α ϑ α ξ, + α ξn, a.e.

11 Some Mean Suare Integral Ineualities... ; Volume 2, Special Issue, Winter and Spring 22, and by the definition of beta function, we get ην, βθ θ+ϑ+ +, ϑ + 2 ξ, + βθ + 2, ϑ + ξν, a.e. which completes the proof. Remark 3.9. For ην, = ν in Theorem 3.8, then we obtain Theorem 3. of []. Theorem 3.. Assume that ξ : I = [, +ην, ]Ω R be a mean suare continuous and mean suare integrable stochastic process. If ξ is a preinvex on [, + ην, ], where < + ην, for > with + = and take θ, ϑ >, then the ineuality holds almost everywhere p +ην, for some fixed θ, ϑ >. z θ + ην, z ϑ ξz, dz ην, βpθ θ+ϑ+ +, pϑ + p 2 ξ, + ξν, 3.8 Proof. By using Lemma 3.2 and from Hölder integral ineuality, we have a.e. +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ α θ α ϑ ξ + αην,, ην, θ+ϑ+ Since ξ is a preinvex stochastic process, we have a.e. ξm + αην,, α pθ α pϑ p ξm + αην,, 2 α ξ, + α ξν, ξ, + ξν, Now, by the definition of beta function, we can deduce α pθ α pϑ = βpθ +, pϑ So, by inserting 3. and 3. in 3.9 yields the reuired ineuality 3.8. Remark 3.. For ην, = ν in Theorem 3., then we obtain Theorem 3.2 of [].

12 628 Shoaib Saleem, Vivas-Cortez, Sajid Theorem 3.2. Assume that ξ : I = [, + ην, ] Ω R be a mean suare continuous and mean suare integrable stochastic process. If ξ is preinvex on [, + ην, ] for >, where < + ην, and take θ, ϑ >, then the ineuality holds almost everywhere +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ βθ +, ϑ + βθ +, ϑ + 2 ξ, + βθ + 2, ϑ + ξν,. 3.2 for some fixed θ, ϑ >. Proof. By using Lemma 3.2 and power-mean integral ineuality for yields that +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ α θ α ϑ ξ + αην,, ην, θ+ϑ+ α θ α ϑ α θ α ϑ ξ + αην,, a.e.. Take the definition of preinvex stochastic process of ξ and the beta function yields that +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ α θ α ϑ α θ α ϑ α ξ, + α ξν, ην, θ+ϑ+ βθ +, ϑ + βθ +, ϑ + 2 ξ, + βθ + 2, ϑ + ξν, a.e. which completes the proof. Remark 3.3. For ην, = ν in Theorem 3.2, then we obtain Theorem 3.3 of []. Theorem 3.4. Let ξ : I = [, + ην, ] Ω R be a mean suare continuous and mean suare integrable stochastic process. If ξ is a preinvex on [, + ην, ], where < + ην, for >

13 Some Mean Suare Integral Ineualities... ; Volume 2, Special Issue, Winter and Spring 22, with p + = and take θ, ϑ >, then the ineuality holds almost everywhere +ην, z θ + ην, z ϑ ξz, dz [ ην, θ+ϑ+ p βpθ +, pϑ + 2 for some fixed θ, ϑ >. 3 ξ, + ξν, 6 + βpθ + 2, pϑ + p 6 ξ, + ξν, 3 ], 3.3 Proof. By using Lemma 3.2 and Hölder- Iscan integral ineuality, we have a.e. +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ α θ α ϑ ξ + αην,, ην, θ+ϑ+ [ αα pθ α pϑ p α ξm + αην,, + αα pθ α pϑ p ] α ξm + αην,,. Take the definition of preinvex stochastic process of ξ and the beta function yields that +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ α θ α ϑ ξ + αην,, ην, θ+ϑ+ [ α pθ α pϑ+ p α α ξ, + α ξν,

14 63 Shoaib Saleem, Vivas-Cortez, Sajid + α pθ+ α pϑ p ] α α ξ, + α ξν, ην, θ+ϑ+ [ βpθ +, pϑ + 2 p 3 ξ, + ξν, 6 + βpθ + 2, pϑ + p 6 ξ, + ξν, 3 ] a.e.. which completes the proof. Theorem 3.5. Let ξ : I = [, + ην, ] Ω R be a mean suare continuous and mean suare integrable stochastic process. If ξ is preinvex on [, + ην, ] for >, where < + ην, and take θ, ϑ >, then the ineuality holds almost everywhere +ην, z θ + ην, z ϑ ξz, dz ην, βθ θ+ϑ+ +, ϑ for some fixed θ, ϑ >. βθ +, ϑ + 3 ξ, + βθ + 2, ϑ + 2 ξν, βθ + 2, ϑ + βθ + 2, ϑ + 2 ξ, + βθ + 3, ϑ + ξν,. 3.4 Proof. By using Lemma 3.2 and improved power-mean integral ineuality for yields that +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ α θ α ϑ ξ + αην,, ην, θ+ϑ+ αα θ α ϑ αα θ α ϑ ξ + αην,, + αα θ α ϑ αα θ α ϑ ξ + αην,, a.e..

15 Some Mean Suare Integral Ineualities... ; Volume 2, Special Issue, Winter and Spring 22, Take the definition of preinvex stochastic process of ξ and the beta function yields that +ην, z θ + ην, z ϑ ξz, dz ην, θ+ϑ+ + α θ α ϑ+ α θ α ϑ+ α ξ, + α ξν, α θ+ α ϑ α θ+ α ϑ α ξ, + α ξν, ην, θ+ϑ+ βθ +, ϑ + 2 a.e. + βθ +, ϑ + 3 ξ, + βθ + 2, ϑ + 2 ξν, βθ + 2, ϑ + βθ + 2, ϑ + 2 ξ, + βθ + 3, ϑ + ξν, a.e. which completes the proof. 4. Conclusion The study of convex stochastic processes has significant applications in optimization theory and linear programming. The new ineualities in stochastic processes is always appreciable as it provides various interesting bounds to study ualitative properties. The mean suare integral ineualities for preinvex and η-convex stochastic processes are derived in this research. Further, results are extended to Hölder improved ineualities. The results are new and interesting for this particular area of research. References [] H. Agahi, Refinements of mean-suare stochastic integral ineualities on convex stochastic processes, Aeuat. Math [2] H. Agahi and M. Yadollahzadeh, On stochastic pseudo-integrals with applications, Stat. Probab. Lett [3] H. G. Akdemir, N. O. Bekar and I. Iscan, On preinvexity for stochastic processes. Istatistik Journal of The Turkish Statistical Association [4] A. Bain and D. Crisan, Fundamentals of stochastic filtering, Springer-Verlag, New York, 29. [5] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, 2nd edn. World Scientific, Singapore, 26. [6] Z. Brzezniak and T. Zastawniak, Basic stochastic processes: a course through exercises, Springer Science and Business Media, London, 2.

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