Lévy constants of transcendental numbers

We prove that every $\gamma \ge \log \frac{1+\sqrt 5}{2}$ is the Lévy constant of a transcendental number; i.e., there exists a transcendental number $\alpha$ such that $\gamma =$ ${\lim \limits _{m\to \infty }}\frac{1}{m}\log q_{m}(\alpha )$, where $q_{m}(\alpha )$ denotes the denominator of the $m$th convergent of $\alpha$.