On the growth of meromorphic functions of order less than \(1/2\) . III (Q762647)

[For part I see the author in ibid. 6, 397-407 (1983; Zbl 0529.30026.] Let \(\rho\) and \(\delta\) be numbers with \(0\leq \delta \leq\), \(1-\cos \pi \rho <\delta \leq 1,\) and let \(m_{\rho,\delta}\) be the set consisting of all meromorphic functions f(z) of order \(\rho\) with the property that there is an \(a\in {\mathbb{C}}\) satisfying \(f(0)\neq a\) and \[ N(r,\infty,f)<(1-\delta)N(r,a,f)+O(1)\quad (r\to \infty). \] Let \(S_ 2\) consist of all functions \(h(r)\) \((r\geq 0)\) which are positive decreasing, continuous, tend to zero as \(r\to \infty\), and for which \(\int^{\infty}_{1}h(t)t^{-1}dt\) is infinite. The author continuing some earlier work proves a number of theorems on growth of meromorphic functions of order less than \(1/2\). A typical theorem follows: Theorem. Let \(h(r)\in S_ 2\) and let \(\rho\) and \(\delta\) be numbers with \(0<\rho <\), \(1-\cos \pi \rho <\delta \leq 1.\) If \(f(z)\in m_{\rho,\delta}\) satisfies the growth restriction \[ T(r,f)=O(r^{\rho}\exp \{(1/(1+\epsilon)C(\rho,\delta))\int^{r}_{1}(h(t)/t)dt\})\quad (r\to \infty) \] with some \(\epsilon >0\), where \[ C(\rho,f)=(\pi (1- \delta)\tan \pi \rho)/(\cos \pi \rho -1+\delta)+(2\pi \rho -\sin 2\pi \rho)/\rho \sin 2\pi \rho, \] then \[ \log \inf_{| z| =r}| f(z)| >(\pi \rho /\sin \pi \rho)(\cos \pi \rho -1+\delta)(1- h(r))T(r,f) \] holds for a sequence of \(r\to \infty\).