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 $\varphi:J\to\mathbb{R}$ is called operator $m$-convex‎ ‎if for any $t\in[0,1]$ and any self-adjoint operators‎ ‎$A‎, ‎B\in \mathbb{B}({\mathscr{H}})$‎, ‎whose spectra are contained in $J$‎, ‎$\varphi\big(tA+m(1-t)B\big)\leq t\varphi(A)+m(1-t)\varphi(B)$‎. ‎We first generalize‎ ‎the celebrated Jensen inequality‎ ‎for continuous $m$-convex functions and Hilbert space operators‎ ‎and then use suitable weight functions to give some weighted refinements of it‎. ‎Introducing the notion of operator $m$-convexity‎, ‎we‎ ‎extend the Choi--Davis--Jensen inequality for operator $m$-convex functions‎.‎We also present an operator version of the Jensen--Mercer‎ ‎inequality for $m$-convex functions and generalize this inequality for‎ ‎operator $m$-convex functions involving continuous fields of operators‎ ‎and unital fields of positive linear mappings‎. ‎Employing the Jensen--Mercer‎ ‎operator inequality for operator $m$-convex functions‎, ‎we construct the‎ ‎$m$-Jensen operator functional and obtain an upper bound for it.

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