Let \(M_n ( \mathbb{C})\) be the set of all \(n\times n\) complex matrices. A function \(f: (0, \infty) \to \mathbb{R}\) is said to be operator monotone if, for any \(n \in \mathbb{N}\) and any pair of selfadjoint matrices \(A, B \in M_n ( \mathbb{C})\) such that \(0 < A \leq B\), the inequality \(f(A) \leq f(B)\) holds. The order \({}\leq{}\) denotes the Löwner partial order, that is \(A \leq B\) if and only if \(B - A\) is a positive-semidefinite matrix. Let \({\mathcal{F}}_{\text{op}}\) denote the class of all functions \(f : (0, \infty) \to (0, \infty)\) such that (i) \(f(1) =1\), (ii) \(x f(x^{-1} ) = f(x), \;x > 0\), (iii) \(f\) is operator monotone. Further, let \[ {\mathcal{F}}_{\text{op}}^{\text{r}} = \{ f \in {\mathcal{F}}_{\text{op}} \mid f(0) \neq 0 \} \qquad \text{and} \qquad {\mathcal{F}}_{\text{op}}^{\text{n}} = \{ f \in {\mathcal{F}}_{\text{op}} \mid f(0) = 0 \}. \] For \(f \in {\mathcal{F}}_{\text{op}}^{\text{r}}\), let \[ \bar{f} (x) = {1 \over 2} \left [ (x+1) - (x-1)^2 {{f(0)} \over {f(x)}} \right ] , \quad x > 0. \] The main result of this paper is the following Theorem. The correspondence \(f \mapsto \bar{f}\) is a bijection between the sets \({\mathcal{F}}_{\text{op}}^{\text{r}}\) and \({\mathcal{F}}_{\text{op}}^{\text{n}}\). As an application, the authors give a new proof of operator monotonicity of certain functions related to the Wigner-Yanse-Dyson skew information.