On a conjecture of C. C. Yang for the class \(F\) of meromorphic functions (Q1316523)

The author gives a positive answer to C. C. Yang's conjecture. More generally, they arrive at the following two results: Theorem 1. Let \(f,g,\mu\) and \(\lambda\) be nonconstant meromorphic functions, satisfying \[ T(r,\mu)=o \{T(r,f)\},\;T(r, \lambda)=0 \{T(r,g)\}. \] If \(E(\infty,f)=E(\infty,g)\), \(E(\mu,f)=E(\lambda,g)\), and \[ \delta (0,f)+\Theta(\infty,f)>{3 \over 2},\;\delta(0,g)+\Theta (\infty,g)>{3 \over 2}, \] then \[ {f \over \mu}={g \over \lambda} \text{ or } fg=\mu\lambda. \] Theorem 2. Let \(f,g, \varphi_ 1, \varphi_ 2\), \(h_ 1\) and \(h_ 2\) be nonconstant meromorphic functions, satisfying \[ T(r,\varphi_ i)=o \{T(r,f)\},\;T(r,h_ 1)=o \{T(r,g)\},\;(i=1,2). \] If \(E(\infty,f)=E(\infty,g)\), \(E(\varphi_ if)=E(h_ i,g)\), \((i=1,2)\) and \[ \delta(0,f)+\Theta (\infty,f)>{3 \over 2},\;\delta (0,g) +\Theta (\infty,g)>{3 \over 2}, \] then \({f \over \varphi_ 1}={g \over h_ 1}\) and \({\varphi_ 1 \over h_ 1}={\varphi_ 2 \over h_ 2}\), or \(fg=\varphi_ 1 h_ 1\) and \(\varphi_ 1 h_ 1=\varphi_ 2 h_ 2\). Corollary. The conjecture of C. C. Yang is true.