[1] 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.
[2] Alomari M.W., A generalization of Hermite-Hadamard’s inequality, Krag. J. Math., 41(2) (2017) 313–328.
[3] Dragomir S.S., On Hadamard’s inequality for convex functions on the coordinates in a rectangle from the plane, Taiwanese J. Math., 4 (2001) 775–788.
[4] Nwaeze E.R., Generalized Hermite-Hadamard’s inequality for functions convex on the coordinates, Applied Mathematics E-Notes, 18 (2018) 275-283.
[5] Hanson M.A., On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications, 80 (1981) 545-550.
[6] Ben-Isreal A., Mond B., What is invexity? Journal of Australian Mathematical Society, Series B., 28(1) (1986) 1-9.
[7] Noor M.A., Hermite-Hadamard integral inequalities for log-preinvex functions, Journal of Mathematical Analysis and Approximation Theory, 2 (2007) 126-131.
[8] Mohan S.R., Neogy S.K., On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications, 189 (1995) 901-908.
[9] Pini R., Invexity and generalized Convexity. Optimization, 22 (1991) 513-525.
[10] Weir T., Mond B., Preinvex functions in multiple bijective optimization, Journal of Mathematical Analysis and Applications, 136 (1998) 29-38.
[11] Yang X.M., Li D., On properties of preinvex functions, J. Math. Anal. Appl, 256 (2001) 229-241.
[13] Mishra S.K., Giorgi G., Invexity and optimization, Nonconvex optimization and Its Applications, Vol.88, Berlin: Springer-Verlag, 2008.
[14] Kumar P., Inequalities involving moments of a continuous random variable defined over a finite interval, Computers and Math. with Appl., 48 (2004) 257-273.
[15] 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.
[16] Feller W., An introduction to Probability Theory and its Applications, Vol.2, New York: J John Wiley, 1971.
[17] Ross S.M., Stochastic Processes, 2rd ed.; J.Wiley&Sons, 1996.
[18] Nikodem K., On convex stochastic processes, Aequat. Math., 20 (1980) 184-197.
[19] Kotrys D., Hermite-Hadamard inequality for convex stochastic processes, Aequat. Math., 83 (2012) 143-151.
[20] Okur N., Multidimensional general convexity for stochastic processes and associated witH Hermite-Hadamard type integral inequalities, Thermal Sci., Suppl. 6(23) (2019), 1971-1979.
[21] Okur N., Aliyev R., Some Hermite-Hadamard type integral inequalities for multidimensional general preinvex stochastic processes, Comm. Statist. Theory Methods, 49 (2020), in press.
[22] Set E., Sarıkaya M.Z., Tomar M., Hermite-Hadamard inequalities for coordinates convex stochastic processes, Mathematica Aeterna, 5 (2) (2015) 363-382.
[23] Akdemir G.H., Okur B. N., Iscan,I., On Preinvexity for Stochastic Processes, Statistics, Journal of the Turkish Statistical Association, 7(1) (2014) 15-22.
[24] Okur B. N., Gunay Akdemir H., Iscan I., Some extensions of preinvexity for stochastic processes, Comp. Analysis, Springer Proceedings in Mathematics & Statistics,, New York: Springer, 2016; 155, 259-270.
[25] Usta Y., Stokastik Süreçler için Koordinatlarda Bazı Konvekslik Çeşitleri ve Hermite-Hadamard Eşitsizliği, Yüksek Lisans Tezi, Giresun Üniversitesi, Fen Bilimleri Enstitüsü, 2018.
Some generalised integral inequalities for bidimensional preinvex stochastic processes
Year 2020,
Volume: 41 Issue: 4, 845 - 853, 29.12.2020
In this study, we generalized some integral inequalities
for bidimensional preinvex stochastic processes. For this reason, we used
mean-square integrable preinvex stochastic processes on the real line and on the
coordinates, respectively. Therefore, we obtained some generalized integral
inequalitİies for preinvex stochastic processes on the real line.
[1] 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.
[2] Alomari M.W., A generalization of Hermite-Hadamard’s inequality, Krag. J. Math., 41(2) (2017) 313–328.
[3] Dragomir S.S., On Hadamard’s inequality for convex functions on the coordinates in a rectangle from the plane, Taiwanese J. Math., 4 (2001) 775–788.
[4] Nwaeze E.R., Generalized Hermite-Hadamard’s inequality for functions convex on the coordinates, Applied Mathematics E-Notes, 18 (2018) 275-283.
[5] Hanson M.A., On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications, 80 (1981) 545-550.
[6] Ben-Isreal A., Mond B., What is invexity? Journal of Australian Mathematical Society, Series B., 28(1) (1986) 1-9.
[7] Noor M.A., Hermite-Hadamard integral inequalities for log-preinvex functions, Journal of Mathematical Analysis and Approximation Theory, 2 (2007) 126-131.
[8] Mohan S.R., Neogy S.K., On invex sets and preinvex functions, Journal of Mathematical Analysis and Applications, 189 (1995) 901-908.
[9] Pini R., Invexity and generalized Convexity. Optimization, 22 (1991) 513-525.
[10] Weir T., Mond B., Preinvex functions in multiple bijective optimization, Journal of Mathematical Analysis and Applications, 136 (1998) 29-38.
[11] Yang X.M., Li D., On properties of preinvex functions, J. Math. Anal. Appl, 256 (2001) 229-241.
[13] Mishra S.K., Giorgi G., Invexity and optimization, Nonconvex optimization and Its Applications, Vol.88, Berlin: Springer-Verlag, 2008.
[14] Kumar P., Inequalities involving moments of a continuous random variable defined over a finite interval, Computers and Math. with Appl., 48 (2004) 257-273.
[15] 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.
[16] Feller W., An introduction to Probability Theory and its Applications, Vol.2, New York: J John Wiley, 1971.
[17] Ross S.M., Stochastic Processes, 2rd ed.; J.Wiley&Sons, 1996.
[18] Nikodem K., On convex stochastic processes, Aequat. Math., 20 (1980) 184-197.
[19] Kotrys D., Hermite-Hadamard inequality for convex stochastic processes, Aequat. Math., 83 (2012) 143-151.
[20] Okur N., Multidimensional general convexity for stochastic processes and associated witH Hermite-Hadamard type integral inequalities, Thermal Sci., Suppl. 6(23) (2019), 1971-1979.
[21] Okur N., Aliyev R., Some Hermite-Hadamard type integral inequalities for multidimensional general preinvex stochastic processes, Comm. Statist. Theory Methods, 49 (2020), in press.
[22] Set E., Sarıkaya M.Z., Tomar M., Hermite-Hadamard inequalities for coordinates convex stochastic processes, Mathematica Aeterna, 5 (2) (2015) 363-382.
[23] Akdemir G.H., Okur B. N., Iscan,I., On Preinvexity for Stochastic Processes, Statistics, Journal of the Turkish Statistical Association, 7(1) (2014) 15-22.
[24] Okur B. N., Gunay Akdemir H., Iscan I., Some extensions of preinvexity for stochastic processes, Comp. Analysis, Springer Proceedings in Mathematics & Statistics,, New York: Springer, 2016; 155, 259-270.
[25] Usta Y., Stokastik Süreçler için Koordinatlarda Bazı Konvekslik Çeşitleri ve Hermite-Hadamard Eşitsizliği, Yüksek Lisans Tezi, Giresun Üniversitesi, Fen Bilimleri Enstitüsü, 2018.
Okur, N. (2020). Some generalised integral inequalities for bidimensional preinvex stochastic processes. Cumhuriyet Science Journal, 41(4), 845-853. https://doi.org/10.17776/csj.634250