Operator (α, m)-convex functions and applications for synchronous and asynchronous functions

In this study, firstly the definition of operator (α, m)-convex function is defined. Secondly, a new lemma is given. Then, new theorems are obtained in terms of this lemma. Finally, they are applied for synchronous and asynchronous functions.

___

[1] Furuta, T., Hot, L. M., P ecari ˘ c´, J. and Seo, Y., Mond-P ecari ˘ c´ Method in Operator Inequalites. Inequalites for Bounded Selfadjoint Operators on a Hilbert space. Element, Zagreb, 2005.

[2] Mihe¸san, V. G., A generalization of the convexity, Seminar on Functionel Equations, Approx Convex, ClujNapoca (Romania), 1993.

[3] Moslehian, M. S. and Najafi, H., Around operator monotone functions. Integr. Equ. Oper. Theory., 2011, 71:575- 582.

[4] Dragomir, S. S., The Hermite-Hadamard type inequalites for operator convex functions. Appl. Math. Comput., 2011, 218(3):766-772.

[5] Erda¸s, Y., Unluyol, E. and Sala¸s, S., The Hermite-Hadamard Type inequalities for operator m-convex functions in Hilbert Space, Journal of New Theory, 5(2015), 80-91.

[6] Sala¸s, S., Unluyol, E. and Erda¸s, Y., The Hermite-Hadamard Type Inequalities for Operator p-Convex Functions in Hilbert Space, Journal of New Theory, 4(2015), 74-79.

[7] Rooin, J., Alikhani, A. and Moslehian, M.S., Operator m-convex functions. Georgian Math. J. 25(2018), no.1, 93-107.

[8] Ghazanfari, A. G., The Hermite-Hadamard type inequalities for operator s-convex functions, JARPM, Vol:6, Issue:3, 2014, 52-61.

[9] Ghazanfari, A. G. and Barani, A., Some Hermite-Hadamard type inequalities for the product of two operator preinvex functions. Banach J. Math. Anal. 9(2015), no.2, 9-20. [10] Dragomir, S. S., Ceby ˘ sev ˘ 0 s type inequalities for functions of selfadjoint operators in Hilbert spaces, Linear and Multilinear Algebra, 58(2010) no. 7-8, 805-814.

[11] Erda¸s, Y., Unluyol, E. and Sala¸s, S., Some new inequalities of operator m-convex functions and applications for Synchronous-Asynchronous functions, accepted and press 2019 in Complex Anal. Oper. Theory.