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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 Matemáticas , Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Católica 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 Matemáticas, Decanato de Ciencias y Tecnolog´ıa, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela 4Departamento de Técnicas Cuantitativas, Decanato de Ciencias Económicas y Empresariales, Universidad Centroccidental Lisandro Alvarado, Av. Moran esq. Av. 20, Edf. Los Militares, Ofc.2, ZP: 3001,Matemáticas, 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]

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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 . (11) − tX t + (1 −t)a,· dt (s + 1)(s + 2)(s + 3) b − a 0 2 ! Z 2 a + b = X ′ ,· Proof. Using Lemma 1 is immediate that b − a 2 4 2 a + b 1 a + b 1 b − t X t + (1 −t)a,· X ,· − X(t,·)dt b − a b − a 2 2 b − a a " 0 Z 2 1 2 1 a + b (b − a) 2 ′′ a + b − X t + (1 −t)a,· dt ≤ t X t + (1 −t)a,· dt b − a 2 16 0 2 Z0 Z (b − a)2 1 a + b 2 ′ a + b + (t − 1)2 X ′′ tb + (1 −t) ,· dt. = X .· 16 2 b − a 2 Z0 (12) 8 a + b − X ,· (b − a)2 2 ′′ Now, since |X | is (s,η)−convex on [a,b], it is easy to see 8 1 a + b that + X t + (1 −t)a,· dt. (8) (b − a)2 2 Z0 a + b X ′′ t + (1 −t)a,· 2 Now, using the change of variable x = t ((a + b)/2)+(1− t a for t 0 1 and multiplying the equality (8) by b a + b ) ∈ [ , ] ( − ≤ X ′′(a,·) +tsη X ′′ ,· , X ′′(a,·) (13) a)2/16, it follows that 2

(b − a)2 1 a + b and t2X ′′ t + (1 −t)a,· dt 16 0 2 a + b Z X ′′ tb + (1 −t) ,· b − a a + b 1 a + b 2 = X ′ .· − X ,· (9) 8 2 2 2 a + b a + b ≤ X ′′ ,· +tsη X ′′(b,·) , X ′′ ,· . 1 a+b/2 2 2 X t dt + ( ,·) (14) b − a a Z

c 2020 NSP Natural Sciences Publishing Cor. 496 M.J. Vivas-Cortez et al. : Hermite-Hadamard type mean square integral...

Replacing the inequalities (13)and (14)in(12), we have Theorem 3. Suppose that X : I ×Ω → R bea twice mean b square differentiable stochastic process and mean square a + b 1 ′′ q X ,· − X(t,·)dt integrable on I. If |X | is (s,η)−convex on [a,b], where 2 b − a a a b with a b, and 0 s 1, the following inequality Z , < < ≤ holds almost everywhere

(b − a)2 1 2 ′′ a + b 1 b ≤ t X (a,·) dt X ,· − X(t,·)dt 16 0 2 b − a Z Za 2 1/p 1 (b − a) 1 2+s ′′ a + b ′′ ≤ × + t η X ,· , X (a,·) dt 16 2p + 1 0 2 Z q 1/q η X′′ a+b ,· ,|X′′(a,·)|q q 2 b a 2 1 a b X′′ (a,·) + ( − ) 2 ′′ +  + (t − 1) X ,· dt  s + 1  16 0 2 Z    q 1/q q η X′′ b q X′′ a+b 1 a + b a + b | ( ,·)| , 2 ,· + tsη X ′′(b,·) , X ′′ ,· dt. + X′′ ,· + ,  2 s + 1   0 2 Z     (b − a)2 |X ′′(a,·)| η X ′′ a+b ,· ,|X ′′(a,·)| where q > 1 and 1/p + 1/q = 1. = + 2 16 3 s 3 + ! Proof. Using Lemma 1 and the H¨older inequality, we have 2 ′′ a+b a + b 1 b (b − a) X 2 ,· X ,· − X(t,·)dt + + 2 b − a 16 3 Za 2 (b − a) 1 a + b ≤ t2 X′′ t +(1 −t)a,· dt 16 2 2η |X ′′(b,·)|, X ′′ a+b ,· Z0 2 2 , (b − a) 1 a + b (s + 1)(s + 2)(s + 3) + (t − 1)2 X′′ tb +(1 − t) ,· dt ! 16 2 Z0 2 that is because (b − a) 1 1 ≤ × ts+2dt = 16 s + 3 1/p q 1/q Z0 1 1 a + b t2pdt X′′ t +(1 −t)a,· dt and 2 1 2 "Z0 Z0 (t − 1)2tsdt = . (s + 1)(s + 2)(s + 3) 1 1/p Z0 + (t − 1)2pdt × The proof is complete. Z0 1 a + b q 1/q Remark. If in Theorem 2 we choose η(x,y)= x − y, the X′′ tb +(1 −t) ,· dt (15) 2 inequality (11) can be restated as Z0 #

Using the s η convexity of X ′′ q we observe that a + b 1 b ( , )− | | X ,· − X(t,·)dt 1 a + b q 2 b − a a X′′ t +(1 −t)a,· dt Z 2 b a 2 s X ′′ a 2 X ′′ b Z0 ( − ) | ( ,·)| | ( ,·)| ≤ + q 16(s + 3) 3 (s + 1)(s + 2) q 1 a + b q = X′′ (a,·) + η X′′ ,· , X′′(a,·) (16) s 1 s 2 s 6 6 a b s + 1 2 ( + )( + )( + ) − ′′ + + X ,· , 3(s + 1)(s + 2) 2 and 1 a + b q almost everywhere, which corresponds to s−convex X′′ tb +(1 −t) ,· dt 2 stochastic process in the ﬁrst sense; and if s = 1, we Z0 obtain a + b q = X′′ ,· 2 a b 1 b + q X ,· − X(t,·)dt 1 ′′ q ′′ a + b 2 b − a a + η X (a,·) , X ,· . (17) Z s + 1 2 2 ′′ ′′ (b − a) |X (a,·)| + |X (b,·)| ′′ a + b ≤ + 2 X ,· , Replacing (16)and (17 )in(15 ), we obtain 64 3 2 a + b 1 b X ,· − X(t,·)dt almost everywhere, for a convex stochastic process. 2 b − a Za

c 2020 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 14, No. 3, 493-502 (2020) / www.naturalspublishing.com/Journals.asp 497

(b − a)2 1 1/p ≤ × holds almost everywhere 16 2p + 1 a + b 1 b X ,· − X(t,·)dt q 1/q 2 b − a a ′′ q 1 ′′ a + b ′′ q Z X (a,·) + η X ,· , X (a,·) 2 s + 1 2 (b − a) " ≤ × 16(3)1/p

q 1/q ′′ a+b ′′ q q ′′ q η X ,· ,|X (a,·)| a + b |X (a,·)| 2 + X′′ ,· + 2  3 s + 3     1 q q q q / X′′ a+b ,· 2η |X′′(b,·)| , X′′ a+b ,· q 1/q 2 2 1 q a + b + + , + η X′′(b,·) , X′′ ,·  3 (s + 1)(s + 2)(s + 3)   s + 1 2 #    (18) because where q > 1 and 1/p + 1/q = 1. 1 1/p 1 1/p 1 1/p t2pdt = (t − 1)2pdt = . 0 0 2p + 1 Z Z Proof. Using Lemma 1 and the power mean inequality, it The proof is complete. holds that

a + b 1 b Remark. If in Theorem 3 we choose η(x,y)= x − y, it X ,· − X(t,·)dt 2 b − a follows that Za 2 1 (b − a) 2 a + b ≤ t X′′ t +(1 −t)a,· dt a + b 1 b 16 2 X ,· − X(t,·)dt Z0 2 b − a a 2 1 Z (b − a) 2 ′′ a + b 2 1/p + (t − 1) X tb +(1 − t) ,· dt (b − a) 1 16 0 2 ≤ × Z 1/q 2 16(s + 1) 2p + 1 (b − a) ≤ × q 1/q 16 ′′ q ′′ a + b s X (a,·) + X ,· 1/p q 1/q 2 1 1 a + b " t2dt t2 X′′ t +(1 −t)a,· dt 2 q 1/q "Z0 Z0 ′′ a + b ′′ q + s X ,· + X (b,·) (a.e.) , 1/p 2 1 # + (t − 1)2dt ×

Z0 which corresponds to s−convex stochastic process in the 1 a + b q 1/q (t − 1)2 X′′ tb +(1 −t) ,· dt . (19) ﬁrst sense, and if s = 1, we obtain 2 Z0 #

a + b 1 b X ,· − X(t,·)dt Now, using the s convexity of X ′′ q, it is easy to 2 b − a ( ,η)− | | Za verify that 1/p (b − a)2 1 ≤ × 24+1/q 2p + 1 1 a + b q t2 X ′′ t 1 t a dt q 1/q + ( − ) ,· q a + b 0 2 X ′′ (a,·) + X ′′ ,· Z a+b q q 2 ′′ q η X ′′ ,· ,|X ′′(a,·)| " |X (a,·)| 2 q 1/q ≤ + a + b q 3 s + 3 + X ′′ ,· + X ′′( b,·) 2 # and almost everywhere, for a convex stochastic process. 1 a + b q (t − 1)2 X ′′ tb + (1 −t) ,· dt 2 Theorem 4. Suppose that X : I ×Ω → R be atwice mean Z0 q q square differentiable stochastic process and mean square ′′ a+b q ′′ ′′ a+b q X ,· 2η |X (b,·)| , X 2 ,· integrable on I. If |X ′′| is (s,η)−convex on [a,b], where ≤ 2 + 3 s 1 s 2 s 3 a,b ∈ I with a < b, and 0 < s ≤ 1, the following inequality ( + )( + )(+ )

c 2020 NSP Natural Sciences Publishing Cor. 498 M.J. Vivas-Cortez et al. : Hermite-Hadamard type mean square integral...

Thus, it is attained that where t2, i f t ∈ [0,1/2) a + b 1 b X ,· − X(t,·)dt m(t)= 2 b − a a  Z (1 −t)2, i f t ∈ [1/2,1]. 2  (b − a) ≤ × 16(3)1/p Proof. In a similar way to the Proof of Lemma 1 , easily q 1/q is achieved the desired result . q ′′ a+b ′′ q |X′′(a,·)| η X 2 ,· ,|X (a,·)| + Theorem 5. Suppose that X : I ×Ω → R is a twice mean  3 s + 3  square differentiable stochastic process and mean square  ′′  q  q 1/q integrable on I. If |X | is (s,η)−convex on [a,b], where ′′ a+b ′′ q ′′ a+b X 2 ,· 2η |X (b,·)| , X 2 ,· a,b ∈ I with a < b, and 0 < s ≤ 1, the following inequality + + .  holds almost everywhere  3 (s + 1)(s + 2)(s + 3) 

 a + b 1 b    X ,· − X(t,·)dt The proof is complete. 2 b − a Za 2 ′′ ′′ Remark. If in Theorem 4, we select η(x,y)= x−y. Then (b − a) |X (a,·)| + |X (b,·) | ≤ the inequality (18) can be restated as 4 24 ′′ ′′ ′′ ′′ a + b 1 b +C η X (a,·) , X (b,·) + η X (b,·) , X (a,·) X ,· − X(t,·)dt 2 b − a Za where

(b − a)2 s|X′′(a,·) |q a + b q 1/q 1 ≤ + X′′ ,· C = + B(s + 1,3) − B (s + 1,3). 16(3)1/p(s + 3)1/q 3 2 s+3 1/2 " 2 (s + 3)

q ′′ a+b Proof. Using Lemma 2, we obtain ((s + 1)(s + 2)(s + 3) − 6) X 2 ,· + a + b 1 b  3(s + 1)(s + 2 ) X ,· − X(t,·)dt 2 b − a a  Z 2 ′′ q 1/q (b − a) 2|X (b,·)| ≤ × + , 4 (s + 1)(s + 2) # 1 m(t) X′′ (ta +(1 −t)b,·) + X′′ (tb +(1 −t)a,·) dt which corresponds to s−convex stochastic process in the Z0 ﬁrst sense, and if s = 1, we obtain (b − a )2 ≤ × (21) a + b 1 b 4 X ,· − X(t,·)dt 2 b − a 1 Za m(t) X′′ (ta +(1 −t)b,·) + X′′ (tb +(1 −t)a,·) dt 0 1 q Z (b − a)2 |X′′(a,·)|q a + b q / ≤ + X′′ ,· Now, using the deﬁnition of the function m(t) and the (3)1/p(4)2+1/q 3 2 ′′ " (s,η)−convexity of |X |, one can observe that

q q 1/q 1 a + b |X′′(b,·)| m(t) X′′ (ta +(1 −t)b,·) dt + X′′ ,· + , 2 3 Z0 # which corresponds to convex stochastic process. 1/2 ≤ t2 X′′ (b,·) +tsη X′′ (a,·) , X′′ (b,·) dt The next results are obtained throughout the following Z0 Lemma. 1 2 ′′ s ′′ ′′ Lemma 2. Let X : I × Ω → R be a mean square + (1 −t) X (b,·) +t η X (a,·) , X (b,·) dt Z1/2 differentiable stochastic process where a,b ∈ I anda < b. If X ′′ is mean square integrable, the following equality |X′′ (b,·)| η (|X′′ (a,·)|,|X′′ (b,·)|) holds = + 24 2s+3(s + 3) 1 b a + b X(t,·)dt − X ,· b − a 2 Za ′′ ′′ +(B(s + 1,3) − B1/2(s + 1,3))η X (a,·) , X (b,·) , (22) (b − a)2 = × 4 and similarly 1 ′′ ′′ 1 m(t) X (ta +(1 −t)b,·) + X (tb +(1 −t)a,·) dt, (20) ′′ 0 m(t) X (tb +(1 −t)a,·) dt Z 0 Z

c 2020 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 14, No. 3, 493-502 (2020) / www.naturalspublishing.com/Journals.asp 499

′′ ′′ ′′ 2 1 1/p |X (a,·)| η (|X (b,·)|,|X (a,·)|) (b − a) p ≤ + ≤ |m(t)| dt × 24 2s+3(s + 3) 4 Z0

′′ ′′ 1 1/q +(B(s + 1,3) − B1/2(s + 1,3))η X (b,·) , X (a,·) (23) q X ′′ (ta + (1 −t)b,·) dt Replacing (22) and (23)in(21 ), we get the desired result. 0 "Z The proof is complete. 1 1/q q + X ′′ (tb + (1 −t)a,·) dt (24) Remark. If in Theorem 5 we select η(x,y)= x − y, we Z0 # obtain Now, it is easy to see that a + b 1 b 1 X ,· − X(t,·)dt p 1 2 b − a a |m(t)| dt = , (25) Z 4p(2p + 1) 2 Z0 (b − a) ′′ ′′ ≤ X (a,·) + X (b,·) ′′ q 24 and using the (s,η)−convexity of |X | ,it follows that 1 for convex stochastic process and s−convex stochastic q X ′′ ta 1 t b dt process in the first sense. This makes a coincidence with ( + ( − ) ,·) Z0 the result by Hernández Hernández,J. E. and Gómez, J.F. ′′ q ′′ q q η (|X ( a,·)| ,|X (b,·)| ) in [27]. ≤ X ′′ (b,·) + (26) s + 1

Remark. If the bifunction η has the following property: and η(x,y) = η(y,x), the inequality in Theorem 5 can be 1 written as q X ′′ (tb + (1 −t)a,·) dt b a + b 1 Z0 X ,· − X(t,·)dt ′′ q ′′ q 2 b − a a q η (|X ( b,·)| ,|X (a,·)| ) Z ′′ ≤ X (a,·) + . (27) s + 1 (b − a)2 ≤ × Replacing (25),(26 )and(27)in(24), it follows the desired 4 result. |X′′ (a,·)| + |X′′ (b,·)| + 2C η X′′ (a,·) , X′′ (b,·) 24 Remark. If in Theorem 6 we choose η(x,y)= x − y, it follows that where a + b 1 b 1 X ,· − X(t,·)dt 2 b − a a C = s 3 + B(s + 1,3) − B1/2(s + 1,3). Z 2 + (s + 3) (b − a)2 1 1/p ≤ × Theorem 6. Let q > 1 and 1/p + 1/q = 1. Suppose that 16(s + 1)1/q 2p + 1 R X : I × Ω → is a twice mean square differentiable ′′ q ′′ q 1/q stochastic process and mean square integrable on I. If s X (b,·) + X (a,·) q X ′′ is s η convex on a b , where a b I with a b, h q q 1/q | | ( , )− [ , ] , ∈ < + s X′′ ( b,·) + X′′ (a,·) , and 0 < s ≤ 1, the following inequality holds almost everywhere i which corresponds to s−convex stochastic process in the a + b 1 b ﬁrst sense, and if s = 1, it follows that X ,· − X(t,·)dt 2 b − a b Za a + b 1 X ,· − X(t,·)dt (b − a)2 1 1/p 2 b − a a ≤ × Z 2 1/p 16 2p + 1 (b − a) 1 q q 1/q ≤ X′′ (a, ·) + X′′ (b,·) ′′ q ′′ q 1/q (2)3+1/q 2p + 1 q η |X (a,·)| ,|X (b,·)| X′′ (b,·) + s 1  + ! which corresponds to convex stochastic process.

 ′′ q ′′ q 1/q Theorem 7. Let q > 1 and 1/p + 1/q = 1. Suppose that q η |X (b,·)| ,|X (a,·)| + X′′ (a,·) + . X : I × Ω → R is a twice mean square differentiable s 1 + !  stochastic process and mean square integrable on I. If ′′ q  |X | is (s,η)−convex on [a,b], where a,b ∈ I with a < b, Proof. Using Lemma (2) and the H¨older inequality, it and 0 < s ≤ 1, the following inequality holds almost follows that everywhere a + b 1 b a + b 1 b X ,· − X(t,·)dt X ,· − X(t,·)dt 2 b − a a 2 b − a Z Za

c 2020 NSP Natural Sciences Publishing Cor. 500 M.J. Vivas-Cortez et al. : Hermite-Hadamard type mean square integral...

2 ′′ q 1/q (b − a) |X (b,·)| q q ≤ +Cη X′′ (a,·) , X′′ (b,·) Remark. If in Theorem 7 we choose η(x,y)= x − y, we 4(12)1/p 24 " have a + b 1 b ′′ q 1/q X X t dt |X (a,·)| q q ,· − ( ,·) + +Cη X′′ (b,·) , X′′ (a,·) 2 b − a a 24 Z # 2 1/q (b − a) q 1 q where ≤ C X ′′ (a,·) + −C X ′′ (b,·) 4(12)1/p 24 1 " C = + B(s + 1,3) − B (s + 1,3)). 2s+3 s 3 1/2 ( + ) 1/q q 1 q + C X ′′ (b,·) + −C X ′′ (a,·) Proof. Using the Lemma (2) and the power mean 24 inequality it follows that #

where a + b 1 b X ,· − X(t,·)dt 1 2 b − a a Z C = s 3 + B(s + 1,3) − B1/2(s + 1,3)), 2 + (s + 3)

1/p (b − a)2 1 which corresponds to s−convex stochastic process in the ≤ |m(t)|dt × 4 ﬁrst sense, if s = 1, then it follows that Z0 1 ∼ −2 1/q C = + B(2,3) − B1/2(2,3)) = 4.1666 × 10 1 q 26 |m(t)| X′′ (ta +(1 −t)b,·) dt + "Z0 and

a + b 1 b 1 1/q X ,· − X(t,·)dt q 2 b − a |m(t)| X′′ (tb +(1 −t)a,·) dt . (28) Za

Z0 # (b − a)2 ≤ × Observing that 22+2/p(3)1/p 1 |m(t)|dt = 1/12 (29) q q 1/q 4.1666 × 10−2 X′′ (a,·) + 6.6667 × 10−7 X′′ (b,·) Z0 and 1/q 1 2 q 7 q q + 4.1666 × 10− X′′ (b,·) + 6.6667 × 10− X′′ (a,·) m(t) X′′ (ta +(1 −t)b,·) dt 0 Z which corresponds to convex stochastic process. 1 2 / q q q ≤ t2 X′′ (b,·) +tsη X′′ (a,·) , X′′ (b,·) dt Z0 4 Conclusion 1 q q q + (1 −t)2 X′′ (b,·) +tsη X′′ (a,·) , X′′ (b,·) dt In the present paper, we established some new Hermite - Z1/2 Hadamard type mean square integral inequalities for q q stochastic processes whose mean square derivative is |X′′ (b,·)|q η |X′′ (a,·)| ,|X′′ (b,·)| = + generalized η−convex. From these results, some other 24 s+3 s 2 ( + 3) inequalities are deduced for convex stochastic processes and s−convex stochastic processes in the first sense. ′′ q ′′ q +B(3,s + 1)(1 − I1/2(3,s + 1))η X (a,·) , X (b,·) , (30) and similarly Acknowledgement 1 ′′ m(t) X (tb +(1 −t)a,·) dt Miguel Vivas-Cortez is strongly grateful to Dirección de Z0 Investigación from Pontificia Universidad Católica del q q |X′′ (a,·)|q η |X′′ (b,·)| ,|X′′ (a,·)| Ecuador for the technical support offered to the research ≤ + 24 2s+3(s + 3) project entitled: ”Algunas desigualdades integrales para funciones convexas generalizadas y aplicaciones”. Jorge q q E. Hernandez´ Hernandez´ would like thank to the +B(s + 1,3) − B (s + 1,3))η X′′ (b,·) , X′′ (a,·) . (31) 1/2 Consejo de Desarrollo Cient´ıfico, Human´ıstico y Replacing (29),(30)and(31)in( 28), we obtain the desired Tecnológico (CDCHT) from Universidad Centroccidental result. Lisandro Alvarado (Venezuela) also for the technical The proof is complete. support that developed of this article.

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References [20] A. Skowronski, On some properties of J-convex stochastic processes,Aequat. Math., 44(1), 249–258 (1922). [1] J. Hadamard , Etude sur les propriétés des fonctions entiéres [21] A. Skowronski, On wright-convex stochastic et en particulier d’une fonction considéré par Riemann,J. processes,Annal. Math. Silec., 9(1), 29–32 (1995). Math. Pures et Appliquées ,58, 171-216 (1893). [22] M.J. Vivas-Cortez , J.E. Hernández Hernández , On [2] Ch. Hermite, Sur deux limites dune integrale definie, (m,h1,h2)−-GA-Convex stochastic processes, Appl. Math. & Mathesis 3, 1-82 (1883). Inf. Sci., 11(3) 649-657 (2017). [3] K. Nikodem , On convex stochastic processes,Aequat. Math., [23] M.J. Vivas-Cortez , Y.C. Rangel Oliveros, On some 20(2), 184-197 (1980). Hermite-Hadamard type inequalities for functions whose [4] D. Kotrys, Hermite-Hadamard inequality for convex second derivative are convex generalized, Rev. Matua stochastic processes, Aequat. Math., 83(1) , 143-151 (2012) . (Universidad del Atlántico, Colombia), 5(2), 21-31 (2018). [5] M. Alomari, M. Darus , On the hadamard’s inequality for log- [24] T. Mikosh, , Elementary Stochastic Calculus with Finance J. Ineq. Appl. 1 convex functions on the coordinates. , , 1-13 in View,Series: Advan. Ser. Stat. Sci. Appl. Prob., World (2009). Scientific Publishing , Singapore,55-60, (1998). [6] M. Alomari , M. Darus , S.S. Dragomir , P. Cerone , [25] K. Sobczyk , Stochastic Differential Equations with Ostrowski type inequalities for functions whose derivatives Applications to Physics and Engineering. Kluwer Academic are s-convex in the second sense, Appl. Math. Lett. 23(9) , Publishers, Dordrecht , 125-140,(1991). 1071-1076 (2010) [26] S. Maden , M. Tomar, E. Set, Hermite-Hadamard type [7] A. Barani , S. Barani , S.S. Dragomir, Refinements of inequalities for s-convex stochastic processes in first Hermite-Hadamard inequalities for functions when a power sense.Pure Appl. Math. Lett. , 2015(1) , 1-7 (2015). of the absolute value of the second derivative is p-convex, J. [27] J.E. Hernández Hernández , J.F. and Gómez, Hermite Appl. Math. 2012, 1-10 (2012). [8] M. Eshagy Gordji , S.S. Dragomir, M.R. Delavar, An hadamard type inequalities for stochastic processes whose second derivatives are (m,h ,h )−-convex using Riemann- inequality related to η−convex functions (II), Intern. J. 1 2 Liouville fractional integral, Rev. Matua, Universidad del Nonlin. Anal. Appl. 6(2), 27-33 (2015). [9] H. Greenberg, W. Pierskalla , A review of quasi convex Atlantico ,Colombia, 5(1), 13-26 (2018) . functions, Repr. Operat. Research, 19(7), 1553-1569(1971) . [10] W. Liu, W. Wen , J. Park , Hermite-Hadamard type Miguel J. Vivas inequalities for MT-convex functions via classical integrals C. earned his Ph.D. degree and fractional integrals,J. Nonlinear Sci. Appl. ,9(3), 766-777 (2016). from Universidad Central [11] S. Varo˘sanec , On h-convexity, J. Math. Anal. Appl., 326(1), de Venezuela, Caracas, 303-311 (2007). Distrito Capital (2014) [12] J. Medina Viloria , M. Vivas-Cortez, Hermite-Hadamard in the field Pure Mathematics type inequalities for harmonically convex functions on n- (Nonlinear Analysis), coordinates, Appl. Math. and Inf. Sci Lett., 5(2), 53-58 (2018). and earned his Master Degree [13] M. Vivas-Cortez, Relative strongly h-convex functions and in Pure Mathematics in the integral inequalities, Appl. Math. and Inf. Sci. lett., 4(2), 39- area of Differential Equations 45 (2016). (Ecological Models). He has vast experience of teaching [14] M. Vivas, J. Hernández, N. Merentes, New Hermite- and research at university levels. It covers many areas of Hadamard and Jensen type inequalities for h-convex Mathematical such as Inequalities, Bounded Variation functions on fractal sets, Rev. Col. Mat , 50(2), 145-164 Functions and Ordinary Differential Equations. He has (2016). written and published several research articles in reputed [15] M. Bracamonte , J. Gimenez , N. Merentes, M. Vivas, Fejer international journals of mathematical and textbooks. He type inequalities for m-convex functions, Publ. en Cs. y Tec. was Titular Professor in Decanato de Ciencias y 10(1), 10-15 (2016). [16] J.E. Hernádez Hernández, Some fractional integral Tecnolog´ıa of Universidad Centroccidental Lisandro inequalities for stochastic processes whose first and second Alvarado (UCLA), Barquisimeto, Lara state, Venezuela, derivatives are quasi-convex, Rev. Matua, Universidad del and invited Professor in Facultad de Ciencias Naturales y Atlantico, Colombia 5(2), 1-13 (2018). Matemáticas from Escuela Superior Politécnica del [17] J.E. Hernández Hernández , J.F. Gómez J.F, Hermite- Litoral (ESPOL), Guayaquil, Ecuador, actually is Hadamard type inequalities, convex stochastic processes and Principal Professor and Researcher in Pontificia Katugampola fractional integral.Rev. Integración: temas de Universidad Católica del Ecuador. 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Artion Kashuri Jorge E. Hernandez´ earned his Ph.D. degree H. earned his M.Sc. from University Ismail degree from Universidad Qemail of Vlora (2016) in the Centroccidental Lisandro area of Analysis and Algebra Alvarado, Barquisimeto, with Doctoral Thesis entitled: Estado Lara (2001) in A new integral transform the ﬁeld Pure Mathematics and its applications, and (Harmonic Analysis). He has being his resaerch areas vast experience of teaching Differential Equations, at university levels. It covers Numerical Analysis , Integral transforms, Mathematical many areas of Mathematical such as Mathematics applied Inequalities, Mathematical Analysis. He has vast to Economy, Functional Analysis, Harmonical Analysis experience of teaching asignatures such as Differential (Wavelets). He is currently Associated Professor in Equations, Numerical Analysis, Calculus, Linear Decanato de Ciencias Econ´omicas y Empresariales of Algebra, Real Analysis, Complex Analysis Topology, Universidad Centroccidental Lisandro Alvarado (UCLA), Real and Complex Analysis. His current position in the Barquisimeto, Lara state, Venezuela. aforementioned University is Lecturer in the Department of Mathematics.

Carlos Eduardo Garc´ıa. obtained his Master of the University Centroccidental Lisandro Alvarado, State Lara of Barquisimeto, in the ﬁeld of the mathematical pure (ergodic theory ). A student of PhD from the Universidad Centroccidental Lisandro Alvarado. Currently is Professor in the Decanato de Ciencias y Tecnologia de la University Centroccidental Lisandro Alvarado (UCLA), Barquisimeto, State Lara, Venezuela. His ﬁeld of interest is the ergodic theory and the nonlinear analysis (convexity generalized).

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