Appl. Math. Inf. Sci. 14, No. 3, 493-502 (2020) 493 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/140317 Hermite-Hadamard Type Mean Square Integral Inequalities for Stochastic Processes whose Twice Mean Square Derivative are Generalized η−convex. Miguel Vivas-Cortez1,∗, Artion Kashuri2, Carlos Garc´ıa3 and Jorge E. Hernandez´ Hernandez´ 4 1Escuela de Ciencias F´ısicas y Matem´aticas , Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Cat´olica del Ecuador, Av. 12 de Octubre 1076. Apartado, Quito 17-01-2184, Ecuador 2Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, L. Pavaresia, Vlora 1001, Vlore, Albania 3Departamento de Matem´aticas, Decanato de Ciencias y Tecnolog´ıa, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela 4Departamento de T´ecnicas Cuantitativas, Decanato de Ciencias Econ´omicas y Empresariales, Universidad Centroccidental Lisandro Alvarado, Av. Moran esq. Av. 20, Edf. Los Militares, Ofc.2, ZP: 3001,Matem´aticas, Decanato de Ciencias y Tecnolog´ıa, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela Received: 2 Dec. 2019, Revised: 3 jAN. 2020, Accepted: 14 Frb. 2020 Published online: 1 May 2020 Abstract: In the present work, a new concept of generalized convexity (i.e. generalized η−convexity ) is established and applied to stochastic process.Using the aforementioned concept, some new Hermite - Hadamard type inequalities for stochastic processes are found. From these results, some other inequalities for convex stochastic processes and s-convex stochastic processes in the first sense are deduced. Some Lemmas are introduced and the classical H¨older and power mean inequalities are used as tools for the development of the main results. Keywords: Hermite-Hadamard inequality, generalized η-convex stochastic processes, mean square integral inequalities 1 Introduction everywhere Let f : I ⊂ R → R a convex function defined on the interval X(tu + (1 −t)v,·) ≤ tX(u,·) + (1 −t)X(v,·) (2) I of real numbersand a,b ∈ I with a < b, then the following for all u,v ∈ I and t ∈ [0,1]. inequality With this definition, in 2012, David Kotrys established in [4] the Hermite-Hadamard inequality version for convex a + b 1 1 f (a)+ f (b) f ≤ f (t)dt ≤ (1) stochastic processes as follows: 2 b − a 0 2 Z Theorem 1. If X : I × Ω → R is Jensen-convex, mean holds for all t ∈ [0,1]. This double inequality (1) is known square continuous in the interval I, then for any u,v ∈ I in the literature as Hermite-Hadamard integral inequality we have for convex functions from the works of Jacques Hadamard u + v 1 v X(u,·)+ X(v,·) and Charles Hermite [1,2]. Both inequalities hold in the X ,· ≤ X(t,·)dt ≤ . reversed direction if f is concave. 2 v − u u 2 Z (3) Kazimierz Nikodem in [3] makes an analogy of this almost everywhere. inequality by defining the convex stochastic processes as follows. Let (Ω,(A), µ) be a probability space and I ⊂ R The evolution of the concept of convexity has had a great be an interval. It is said that a stochastic process X : I × impact on the community of investigators. Recently, Ω → R is convex if the following inequality holds almost generalized concepts, such as log-convexity, s-convexity, ∗ Corresponding author e-mail: [email protected] c 2020 NSP Natural Sciences Publishing Cor. 494 M.J. Vivas-Cortez et al. : Hermite-Hadamard type mean square integral... P-convexity, η-convexity, quasi convexity, MT-convexity, then h-convexity, as well as combinations of these new b X(t,·)dt = Y (·) (a.e.). concepts have been introduced. The following references Za comprises much relevant information [5,6,7,8,9,10,11, The book of K. Sobczyk [25] involves substantial 12,13,14,15]. properties of mean-square integral. In a direct relation, the results found for generalized In [26], S. Maden et al. established the following convex functions have had a counterpart with stochastic definition: processes, as illustrated in [16,17,18,19,20,21,22]. Motivated by the works of M.J. Vivas and Y. Rangel Definition 1. Let 0 < s ≤ 1. A stochastic process in [23], we have established some Hermite-Hadamard type X : I × Ω → R is said to be s−convex in the first sense if mean square integral inequalities using generalized convex the inequality stochastic processes. X(tu + (1 −t)v,·) ≤ tsX(u,·) + (1 −ts)X(v,·) (4) holds almost everywhere for all u,v ∈ I andallt ∈ [0,1]. 2 Preliminaries In this work, the authors introduce the following The followingnotionscan be found in some text booksand definitions: articles. The reader can review then in [4,24,21]. Definition 2. Let η : R R R be a bifunction. A A × → Let (Ω, , µ) be an arbitrary probability space. A stochastic process X : I Ω R is called convex with R × → function X : Ω → is called a random variable if it is respect to η, or briefly η−convex, if the inequality A -measurable. Let I ⊂ R be time. A function X : I × Ω → R is called stochastic process if for all u ∈ I X(tu + (1 −t)v,·) ≤ X(v,·)+tη(X(u,·),X(v,·) (5) the function X(u,·) is a random variable. A stochastic process X : I × Ω → R is called holds almost everywhere for all u,v ∈ I andallt ∈ [0,1]. continuous in probability in the interval I if for all t I. 0 ∈ R R R Then Definition 3. Let η : × → be a bifunction and 0 < s ≤ 1. A stochastic process X : I × Ω → R is called µ − lim X(t,·)= X(t0,·), t→t0 s−convex in the first sense with respect to η, or briefly (s,η)−convex in the first sense, if the inequality where µ − lim denotes the limit in probability, and it is called mean-square continuous in the interval I if for all X(tu + (1 −t)v,·) ≤ X(v,·)+tsη(X(u,·),X(v,·) (6) t0 ∈ I µ − lim E(X(t,·) − X(t0,·)) = 0, holds almost everywhere for all u,v ∈ I andallt ∈ [0,1]. t→t0 E Example 1. Let X(t,·) be an stochastic process defined where (X(t,·)) denotes the expectation value of the 2 random variable X(t,·). by X(t,·)= t A(·), where A(·) is a random variable, then In addition, the monotony property is attained. A X(t,·) is a convex stochastic process and (1/2,η)− stochastic process is called increasing (decreasing) if for convex in the first sense with respect to the bifunction all u,v ∈ I such that t < s, η(u,v)= 2u + v. Example 2. Let X(t,·) be an stochastic process defined X u X v X u X v a e ( ,·) ≤ ( ,·), ( ( ,·) ≥ ( ,·)) ( . .) by X(t,·)= tnA(·), where A(·) is a random variable, then X(t,·) is a convex stochastic process and (s,η)− convex in respectively, and, it called monotonic if it is increasing or the first sense for 0 < s ≤ 1, with respect to the bifunction decreasing. n In terms of differentiability, stochastic processes are n k n η u v v1− n u1/n v1/n differentiable at a point t ∈ I if there is a random variable ( , )= ∑ k − . ′ k 1 X (t,·) such that = In some cases, the functions B(x,y) and B (x,y) defined X(t,·) − X(t ,·) α X ′(t,·)= µ − lim 0 . as the Beta and incomplete Beta functions are used, t→t0 t −t0 respectively.They are defined as follows: Let [a,b] ⊂ I,a = t0 < t1 < ... < tn = b be a partition 1 x−1 y−1 of [a,b] and θk ∈ [tk−1,tk] for k = 1,2,...,n. Let X be a B(x,y)= t (1 −t) dt stochastic process such that E(X(u,·)2) < ∞. A random Z0 variable Y : Ω → R is called mean-square integral of the and α process X(t,·) on [a,b] if the following identity holds: x−1 y−1 Bα (x,y)= t (1 −t) dt 0 2 Z lim E[X(θk,·)(tk −tk−1) −Y(·)] = 0 n→∞ for x,y > 0 and 0 < α < 1. c 2020 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 14, No. 3, 493-502 (2020) / www.naturalspublishing.com/Journals.asp 495 3 Main Results Similarly, we have 2 1 (b − a) 2 ′′ a + b First we establish the following Lemma as an auxiliary (t − 1) X tb + (1 −t) ,· dt 16 0 2 result. Z b − a a + b 1 a + b = − X ′ .· − X ,· (10) 8 2 2 2 Lemma 1. Let X : I × Ω → R be a mean square 1 b differentiable stochastic process where a,b ∈ I anda < b. + X(t,·)dt If X ′′ is mean square integrable. Then, the following b − a Za+b/2 equality holds Adding the equations (9) and (10), then the desired result 1 b a + b (7) follows. X(t,·)dt − X ,· b − a 2 Za Theorem 2. Suppose that X : I ×Ω → R be atwice mean 2 1 (b − a) 2 ′′ a + b square differentiable stochastic process and mean square = t X t + (1 −t)a,· dt ′′ 16 0 2 integrable on I. If |X | is (s,η)−convex on [a,b], where Z a,b ∈ I with a < b, and 0 < s ≤ 1 , the following inequality 1 a + b + (t − 1)2X ′′ tb + (1 −t) ,· dt . (7) holds almost everywhere 2 Z0 a + b 1 b X ,· − X(t,·)dt Proof. Using integration by parts, we found that 2 b − a Za (b − a)2 |X ′′(a,·)| + X ′′ a+b ,· 1 a + b ≤ 2 t2X ′′ t + (1 −t)a,· dt 16 3 2 Z0 1 η X ′′ a+b ,· ,|X ′′(a,·)| 2 2 ′ a + b + 2 = t X t + (1 −t)a,· s + 3 b − a 2 0 ′′ ′′ a+b 1 2η |X (b,·)|, X ,· 4 ′ a + b + 2 .