The aim of this paper is to present a comprehensive study of operator $m$-convex functions‎.

‎Let $m\\in[0,1]$ and $J=[0,b]$ for some $b\\in\\mathbb{R}$ or‎ ‎$J=[0,\\infty)$‎. ‎A continuous‎

‎function ...

We present a Hermite--Hadamard--Mercer type inequality

and then generalize it for Hilbert space operators. We also prove f left(M+m- sum_{i=1}^{n}x_iA_i right) leq f(M)+f(m)- sum_{i=1}^{n}f(x_i)A_i , where f is ...