Connections between points of convexity of functions and centrality of elements in $C^*$-algebras

In this paper, we give a characterization of central elements in a $C^*$-algebra $\mathcal{A}$ in terms of points of convexity of scalar functions. We prove that if $D$ is an open interval and $f\in\mathcal{C}^2(D)$ is a convex function satisfying a certain inequality, then a self-adjoint element $a\in\mathcal{A}$ with spectrum in $D$ is central if and only if it is a point of convexity of $f$. The class of functions with these properties contains the nontrivial real exponential ones and the power ones with exponent outside $[-1,2]$.