Growth of a class of composite entire functions (Q1433485)

The author considered the problem due to Chuang and Yang: For two pairs of entire functions \(f_{1},f_{2}\) and \(g_{1},g_{2}\), do we have \(T(r,f_{1}(g_{1}))\sim T(r,f_{2}(g_{2}))\) as \(r\to\infty\), provided that \(T(r,f_{1})\sim T(r,f_{2})\) and \(T(r,g_{1})\sim T(r,g_{2})\) as \(r\to\infty\)? The author gave an example to show that the answer is negative in general, and gave some conditions to assure that the answer is positive.