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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 $$r{d \over dr} r{d \over dr}$$.

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 30D30 Meromorphic functions of one complex variable, general theory
##### Keywords:
deviation; spread; star function of Baernstein; rearrangement
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