zbMATH — the first resource for mathematics

On the magnitudes of deviations and spreads of meromorphic functions of finite lower order. (English. Russian original) Zbl 0854.30023
Sb. Math. 186, No. 3, 391-408 (1995); translation from Mat. Sb. 186, No. 3, 85-102 (1995).
Let \(f(z)\) be a meromorphic function, \(F_1 = \{e^{i \varphi} : |f(re^{i \varphi})|> 1\}\), \(p(r, \infty, f)\) be a number of open connected arcs forming \(F_1\) which include a point of the global maximum of the function \(|f (re^{i \varphi})|\) on the circle \(|z| = r\), and let \[ p (\infty,f) = \varliminf_{r \to \infty} p(r,\infty,f). \] Let \(\lambda\) be a lower order of \(f\). The author proves that some significant theorems of Nevanlinna’s theory of the distribution values and Petrenko’s theory of the growth of meromorphic functions are remained valid after changing \(\lambda\) to \(\lambda/p (\infty,f)\) in their formulations. He shows this for five theorems. For example, the estimates \[ \beta (a,f) \leq \begin{cases} {\pi \lambda \over \sin \pi \lambda}, & \text{ if } \lambda \leq 0.5 \\ \pi \lambda, & \text{ if } \lambda > 0.5, \end{cases} \] \[ \beta (a,f) \leq \begin{cases} {\pi \lambda \over p (\infty, f)}, & \text{ if } {\pi \lambda \over p (\infty, f)} \geq 0.5 \\ {\pi \lambda \over \sin \pi \lambda}, & \text{ if } p (\infty, f) = 1 \text{ and } \lambda < 0.5 \\ {\pi \lambda \over p (\infty, f)} \sin {\pi \lambda \over p (\infty, f)}, & \text{ if } p (\infty, f) > 1 \text{ and } {\pi \lambda \over p (\infty, f)} < 0.5 \end{cases} \] where \(\beta (a,f)\) is magnitude of the Petrenko’s deviation of the meromorphic function at the point \(a\) are the results of Petrenko and the author. Furthermore the inequalities \[ \sigma (\infty, f) \geq \min \left( 2 \pi, {4 \over \lambda} \arcsin \sqrt {{\delta (\infty, f) \over 2}} \right), \] \[ \sigma (\infty, f) \geq \min \left( 2 \pi, {4p (\infty, f) \over \lambda} \arcsin \sqrt {{\delta (\infty, f) \over 2}} \right) \] where \(\sigma (\infty, f) = \varlimsup_{r \to \infty} \text{mes} \{\theta : f(re^{i \theta}) > 1\} \) are results of A. Baernstein and the author. The main tool of the author is his inequality \[ LT^* (r, \theta, f) \geq - {p^2 (r, \infty, f) \over \pi} {\partial u^* (r, \theta) \over \partial \theta} \] where \(T^*\) is the star function of Baernstein for \(\ln^+ |f (re^{i \theta}) |\), \(u^* (r, \theta)\) is the rearrangement of \(\ln^+ |f(re^{i \theta}) |\) which is even decreasing on \([0,\pi]\) and \(L\) is the correctly defined operator \(r{d \over dr} r{d \over dr}\).

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable, general theory
Full Text: DOI