Some notes on asymptotic values of meromorphic functions of smooth growth (Q1062175)

Let f(z) be a nonconstant meromorphic function in \(| z| <\infty\), and let a be a value in the extended complex plane. \textit{W. K. Hayman} [Acta math 141, 115-145 (1978; Zbl 0382.30020)] proved a sufficient condition for a to be an asymptotic value of f, and deduced that (A) if f(z) has perfectly regular growth of order \(\rho\), where \(0<\rho <1/2\), and if \(\delta(a,f)>2\rho\), then a is an asymptotic value. \textit{H.Yoshida} [Hiroshima Math. J. 11, 195-214 (1981; Zbl 0459.30018)] generalized this result: (B) Suppose that f(z) satisfies \(\limsup_{r\to \infty}x^{- \rho}T(r,f)^{-1}T(xr,f)\leq 1\) for any \(x>1\), where \(\rho\) is the order of \(f(0\leq \rho <1/2)\) and that \(\delta (a,f)>2\rho\); then a is an asymptotic value. The author obtains a number of sufficient conditions for a to be an asymptotic value of f and shows that neither Proposition (A) nor (B) is sharp. We state Theorem 1: Suppose that f(z) is meromorphic and nonconstant in \(| z| <\infty\) and that T(r,f) satisfies the condition (C): For some \(\rho\), \(0\leq \rho < 1/2\), \[ \limsup_{r\to \infty}x^{- \rho}T(r,f)^{-1}T(xr,f)\leq 1 \] for any \(x>0\). If \(\delta (a,f)>1- \pi^{1/2}/\Gamma (\rho +1)\Gamma (-\rho)\) then a is asymptotic. A condition for functions f satisfying \[ \limsup_{r\to \infty}T(r,f)/(\log r)^ 2=A<\infty \] is the following: If \[ \lim \inf_{r\to \infty}m(r,a)/\log r>8A \log (1+\sqrt{2}), \] then a is asymptotic.