In this article, we obtain the following result: Let f be a transcendental meromorphic function of order $ \lambda _{f}\ (0 \lt \lambda _{f} \lt \infty ) $ , g be a transcendental entire function with $ T(r,g)= O^*(e^{(\log r)^{\alpha }}) $ . Then $$ \overline {\lim _{r\to \infty }}\frac {\log T(r,f(g))}{T(r,g)} = \lambda _f ,\quad (r\notin E), $$ where f (0 < f < 1) is a constant, and E is a set of finite logarithmic measure.
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