Some integral inequalities of the Hermite-Hadamard type for s-convex stochastic processes on n-coordinates

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

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  • Hadamard, J., Étude sur les propriétés des fonctions entières et en particulier d.une function considerée par Riemann, J. Math Pures Appl., 58,(1893), 171--215.
  • Dragomir, S.S., Pearce, C.E., Sandor, M.J., A note on the Jensen-Hadamard inequality, Anal. Num. Theo. Approx., 19, (1990), 29--34.
  • De la Cal, J., Carcamo, J., Multidimensional Hermite-Hadamard inequalities and the convex order, J. Math. Anal. Appl., 324, (2006), 248--261.
  • Kumar, P., Inequalities involving moments of a continuous random variable defined over a finite interval, Computers and Mathematics with Appl., 48, (2004), 257--273.
  • Gavrea, B., A Hermite-Hadamard type inequality with applications to the estimation of moments of continuous random variables, Appl. Math. and Comp., 254, (2015), 92--98.
  • Kotrys, D., Hermite-Hadamard inequality for convex stochastic processes, Aequat. Math., 83, (2012), 143--151.
  • Shaked, M., Shanthikumar, J.G., Stochastic convexity and its applications, Advances in Applied Probability, 20, 1988, 427--446.
  • Nikodem, K., On convex stochastic processes, Aequat. Math., 20, (1980), 184--197.
  • Skowronski, A., On some properties of J-convex stochastic processes, Aequat. Math., 44, (1992), 249--258.
  • Maden S., Tomar M., Set E., Hermite-Hadamard type inequalities for s-convex stochastic processes in the first sense, Pure and Appl. Math. Letters, (2014),1--7.
  • Set E., Tomar M., Maden S., Hermite-Hadamard type inequalities for s-convex stochastic processes in the second sense, Turkish Journal of Analysis and Number Theory, 2(6), (2014), 202--207.
  • Set E., Sarıkaya, M.Z. Tomar M., Hermite-Hadamard inequalities for coordinates convex stochastic processes, Math. Aeterna, 5(2), (2015), 363--382.
  • Sarikaya, M.Z., On Hermite-Hadamard type inequalities for ϕ_{h}-convex functions, Kochi J. of Math., 9, (2014), 83--90.
  • Materano, J., Merentes, N., Valera-Lopez, M., On inequalities of Hermite-Hadamard type for stochastic processes whose third derivative absolute values are quasi-convex, Tamkang J. of Math., 48(2), (2017), 203--208.
  • Barraez, D., Gonzalez, L., Merentes, N., Moros, A.M., On h-convex stochastic process, Math. Æterna, 5, (2015), 571-581.
  • Agahi, H., Yadollahzadeh, M., Comonotonic stochastic processes and generalized mean-square stochastic integral with applications, Aequ. Math., 91, (2017), 153--159.
  • Okur, N., Iscan, I., Yuksek Dizdar, E., Hermite-Hadamard inequalities for harmonically convex stochastic processes, International Journal of Economic and Administrative Studies, 11, (2018), 281--292.
  • Okur, N., Iscan, I., Usta, Y., Some Integral inequalities for harmonically convex stochastic processes on the coordinates, Adv. Math. Models and App., 3(1), (2018), 63--75.
  • Ellahi, H., Farid, G. and Rehman, A. U., Hadamard's Inequality for s-convex function on n-coordinates, Proceedings of 1st ICAM Attock, Pakistan, (2015).
  • Viloria, J.M., Cortez, M.V., Hermite-Hadamard type inequalities for harmonically convex functions on n-coordinates, Appl. Math. Inf. Sci. Lett., 6(2), (2018), 1--6.
  • Karahan, V., Okur, N., Hermite-Hadamard Type Inequalities for Convex Stochastic Processes On n-Coordinates, Turk. J. Math. Comp. Sci., 10, (2018), 256--262.
  • Karahan, V., Okur, N., New Integral Inequalities of Hadamard's for Harmonically Convex Stochastic Processes on n-coordinates, ICAAMM 2019, Istanbul, Abstract Book, 10-13 March 2019, (2019), 201.
  • Karahan, V., Okur, N., Some Generalized Hermite-Hadamard Type Integral Inequalities for I†-Convex Stochastic Processes on n-coordinates, ICAAMM 2019, Istanbul, Abstract Book, 10-13 March 2019, (2019), 202.
  • Okur, N., Karahan, V., Hermite-Hadamard Inequalities for Multidimensional Preinvex Convex Stochastic Processes, Karadeniz 1. Uluslararası Multidisipliner Bilimsel Araştırmalar Kongresi, Giresun, 15-17 Mart 2019, (2019)