On $(m, h1, h2)$-G-Convex Dominated Stochastic Processes

Download PDF

Authors: J. E. H. HERNáNDEZ

DOI: 10.46793/KgJMat2202.215H


In this paper is introduced the concept of (m,h1,h2)-convexity for stochastic processes dominated by other stochastic processes with the same property, some mean square integral Hermite-Hadamard type inequalities for this kind of generalized convexity are established and from the founded results, other mean square integral inequalities for the classical convex, s-convex in the first and second sense, P-convex and MT-convex stochastic processes are deduced.


(m,h1,h2)-convexity, dominated convexity, mean square integral inequalities, stochastic processes.


[1]   M. Alomari, M. Darus, S. S. Dragomir and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett. 23 (2010), 1071–1076.

[2]   A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability 60, Springer, New York, 2009.

[3]   P. Devolder, J. Janssen and R. Manca, Basic Stochastic Processes, Mathematics and Statistics Series, ISTE, John Wiley and Sons, Inc., London, 2015.

[4]   S. S. Dragomir, C. E. Pearce and J. Pećarić, Means, g convex dominated functions and Hadamard-type inequalities, Tamsui Oxf. J. Inf. Math. Sci. 18 (2002), 161–173.

[5]   M. E. Gordji, S. S. Dragomir and M. R. Delavar, An inequality related to η-convex functions (II), Int. J. Nonlinear Anal. Appl. 6 (2015), 27–33.

[6]   M. Grinalatt and J. T. Linnainmaa, Jensen’s inequality, parameter uncertainty and multiperiod investment, Review of Asset Pricing Studies 1 (2011), 1–34.

[7]   J. E. Hernádez-Hernández and J. Gómez, Hermite hadamard type inequalities for stochastic processes whose second derivatives are (m,h1,h2)-convex using Riemann-Liouville fractional integral, Revista Matua, Universidad del Atlántico 5 (2018), 13–28.

[8]   D. Kotrys, Hermite-Hadamard inequality for convex stochastic processes, Aequationes Math. 83 (2012), 143–151.

[9]   D. Kotrys, Remarks on strongly convex stochastic processes, Aequationes Math. 86 (2013), 91–98.

[10]   D. Kotrys, On strongly wright-convex stochastic process, Tatra Mt. Math. Publ. 66 (2016), 67–72.

[11]   M. A. Latif, S. S. Dragomir and M. A. Momoniat, Some estimates on the Hermite-Hadamard inequality through geometrically quasi-convex functions, Miskolc Math. Notes 18 (2017), 933–946.

[12]   S. Maden, M. Tomar and E. Set, Hermite-Hadamard type inequalities for s-convex stochastic processes in first sense, Pure and Applied Mathematics Letters 2015 (2015), 1–7.

[13]   T. Mikosch, Elementary Stochastic Calculus with Finance in View, Advanced Series on Statistical Science and Applied Probability, World Scientific Publishing Co., Inc., Singapore, London, Hong Kong, 2010.

[14]   P. O. Mohammed, Some new Hermite-Hadamard type inequalities for mt-convex functions on differentiable coordinates, Aequationes Math. 28 (1985), 229–232.

[15]   B. Nagy, On a generalization of the Cauchy equation, Aequationes Math. 11 (1974), 165–171.

[16]   E. Newman, Inequalities involving a logarithmically convex functions and their applications to special functions, Journal of Inequalities in Pure and Applied Mathematics 7 (2006), 1–3.

[17]   K. Nikodem, On convex stochastic processes, Aequationes Math. 20 (1980), 184–197.

[18]   M. A. Noor, K. I. Noor, S. Ifthikar and K. Al-Ban, Inequalities for mt-harmonic convex functions, J. Adv. Math. Stud. 9 (2016), 194–207.

[19]   N. Okur, I. Işcan and Y. Dizdar, Hermite-Hadamard type inequalities for harmonically convex stochastic processes, International Journal of Economic and Administrative Studies 8 (2018), 281–291.

[20]   N. Okur, I. Işcan and Y. Dizdar, Hermite-Hadamard type inequalities for p-convex stochastic processes, Int. J. Optim. Control. Theor. Appl. IJOCTA 9 (2019), 148–153.

[21]   J. J. Ruel and M. P. Ayres, Jensen’s inequality predicts effects of environmental variations, Trends in Ecology and Evolution 14 (1999), 361–366.

[22]   E. Set and M. A. Ardiç, Inequalities for log-convex and p functions, Miskolc Math. Notes 18 (2017), 1033–1041.

[23]   E. Set, M. Tomar and S. Maden, Hermite Hadamard type inequalities for s-convex stochastic processes in the second sense, Turkish Journal of Analysis and Number Theory 2 (2014), 202–207.

[24]   M. Shaked and J. Shantikumar, Stochastic Convexity and its Applications, Arizona University, Tucson, 1985.

[25]   J. J. Shynk, Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications, Wiley, New Jersey, 2013.

[26]   A. Skowronski, On some properties of j-convex stochastic processes, Aequationes Math. 44 (1992), 249–258.

[27]   A. Skowronski, On wright-convex stochastic processes, Ann. Math. Sil. 9 (1995), 29–32.

[28]   K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991.

[29]   S. Varo˘s   anec, On h-convexity, J. Math. Anal. Appl. 326 (2007), 303–311.

[30]   M. J. Vivas-Cortez and J. E. Hernández-Hernández, On (m,h1,h2)-ga-convex stochastic processes, Appl. Math. Inf. Sci. 11 (2017), 649–657.