Connections between centrality and local monotonicity of certain functions on $C^*$-algebras

We introduce a quite large class of functions (including the exponential function and the power functions with exponent greater than one), and show that for any element $f$ of this function class, a self-adjoint element $a$ of a $C^*$-algebra is central if and only if $a \leq b$ implies $f(a) \leq f(b).$ That is, we characterize centrality by local monotonicity of certain functions on $C^*$-algebras. Numerous former results (including works of Ogasawara, Pedersen, Wu, and Moln\'ar) are apparent consequences of our result.