Speaker
Description
The report will focus on theorems of the minimum module type and their applications in the theory of the distribution of values.
For entire transcendental functions of finite order having the form $f(z)=\sum\limits_{n}a_{n}z^{p_{n}}$, $p_{n}\in\mathbb{N}$, G. Polia showed that if the density of the sequence $\left\{p_{n}\right\}$ is equal to zero, then for any curve $\gamma$ going to infinity, there is an unlimited sequence $\{\xi_{n}\}\subset\gamma$, such that for $\xi_{n}\rightarrow\infty$ there is a relation:
$$\ln M_{f}(|\xi_{n}|)\sim \ln\left|f(\xi_{n})\right|$$
($M_{f}(r)$ --- the maximum of the module of the function $f$ on a circle of radius $r$). Later, these results were completely transferred by I.D. Latypov to entire Dirichlet series of finite order and finite lower order by Ritt. A further generalization was obtained in the works of N.N. Yusupova--Aitkuzhina to the more general classes $D(\Phi)$ and $\underline{D}(\Phi)$ defined by the convex majorant $\Phi$. In 2022 we obtained necessary and sufficient conditions for the exponents $\lambda_{n}$ under which the logarithm of the modulus of the sum of any Dirichlet series from this classes on the curve $\gamma$ of a bounded $K$--slope was equivalent to the logarithm of the maximum term for $\sigma=Re s\rightarrow +\infty$ over some asymptotic set whose upper density is not less than $\frac{1}{\sqrt{K^{2}+1}}$. It will be shown that this result can be amplified.
The results about iterations of entire transcendental functions with a regular behavior of the modulus minimum will be presented in the report.
