Abstract
In this study, we identified s-convexity of first and second sense for multidimensional stochastic processes. Concordantly, we verified Hermite-Hadamard type inequalities for these processes. Besides, we exemplified these results on two and three-dimensional stochastic processes. Ultimately, we compared our results with multidimensional harmonically convex stochastic processes in the literature. It must be known that the inequalities in our study are especially necessary to compare the maximum and minimum values of s-convex of first and second sense for multidimensional stochastic process with the expected value of stochastic processes. It is used mean-square integrability for the speciality of stochastic processes to obtain these inequalities in this study
Highlights
Introduction and PreliminariesThe function f : I R ! R, is referred convex [1], if the inequality satis...es for x; y 2 I f ( x + (1 )y) f (x) + (1 )f (y); 2 [0; 1]: It becomes famous the following Hermite-Hadamard inequality for convex functions [2] a+b f f (t) dt : b aaIn terms of probability theory, this inequality gives a lower bound and an upper bound for the expectation value of a random variable X which distributed uniformly on [a; b] [3]
It must be known that the inequalities in our study are especially necessary to compare the maximum and minimum values of s-convex of ...rst and second sense for multidimensional stochastic process with the expected value of stochastic processes
rstly, we investigated primarily s-convex of ...rst sense multidimensional stochastic processes
Summary
The function f : I R ! R, is referred convex [1], if the inequality satis...es for x; y 2 I f ( x + (1 )y) f (x) + (1 )f (y); 2 [0; 1]: It becomes famous the following Hermite-Hadamard inequality for convex functions [2]. Multidimensional stochastic process, s-convexity of ...rst and second sense, mean-square integral, Hermite-Hadamard inequality. From 2010, Kotrys [6] obtained Hermite-Hadamard inequality for some convex stochastic processes. Set et al [12] proved Hermite-Hadamard inequalities for coordinated convex stochastic processes. Okur et al [17]-[18] investigated some convex stochastic processes and obtained the Hermite-Hadamard type inequalities for a few of these processes both on real number set and on the rectangle plane. Viloria et al [20] obtained Hermite-Hadamard type inequalities for harmonically convex functions on n-coordinates. Hermite-Hadamard type inequalities for multidimensional stochastic processes. Let us see the following results related multidimensional harmonically convex stochastic processes [22]: The process X : n Rn+. Our supplemental claim is to obtain Hermite-Hadamard type inequalities for these processes
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