Let $f$ be a transcendental meromorphic function of order $\rho_f ,$ $g$ a transcendental entire function of lower order $\lambda_g$ $(\lambda_g < +\infty)$, and $a_i(z)$ $(i = 1, 2, \dots , n; n \leq \infty)$ be entire functions satisfying $T(r, a_i(z)) = \circ(T(r, g))$. If $\sum_{i=1}^n\partial(a_i(z),g)=1$ $\partial(a_i(z),g) > 0$ and $a_i(z)\not\equiv\infty$ for each $i$, then
$$\overline{\lim\limits_{r\to\infty}}\log(T(r,f(g)))/T(r,g)=\pi\rho_f.$$