In this paper, we obtain the following results:
Let $f_1,f_2$ and $g_1,g_2$ be four transcendental entire functions with $T(r,f_1)=O^{\ast }((\log r{)}^{\nu }{e}^{(\log r{)}^{\alpha }})$ and $T(r,g_1)=O^{\ast }((\log r{)}^{\beta })$ (i.e., there exist four positive constants $K_1,K_2,K_3$ and $K_4$ such that $K_1\le \frac{T(r,f_1)}{(\log r{)}^{\nu }{e}^{(\log r{)}^{\alpha }}}\le K_2$ and $K_3\le \frac{T(r,g_1)}{(\log r{)}^{\beta }}\le K_4$).
If $T(r,f_1)\sim T(r,f_2),$ $T(r,g_1)\sim T(r,g_2)$ $(r\to \mathrm{\infty })$, then
$$T(r,f_1(g_1))\sim T(r,f_2(g_2))(r\to \mathrm{\infty },r\notin E)$$
where $\nu >0,0<\alpha <1,\beta >1$ and $\alpha \beta <1$ and $E$ is a set of finite logarithmic measure.
We solved a problem due to C. C. Yang concerning the characteristic functions of the composite functions.