Grows estimated of analytic functions in D×ℂ having bounded L-index i joint variables

Анотація

We prove some properties of power series expansion of analytic functions in $\mathbb{D}\times \mathrm{ℂ}$ having bounded $L$-index in joint variables, where $L\left(\mathrm{z},\mathrm{w}\right)=\left(l_1(z,w),l_2(z,w)\right)$ with $l_j:\mathbb{D}\times \mathrm{ℂ}\to {\mathrm{ℝ}}_{+}$ $\left(j\in \left\{1,2\right\}\right)$ are positive continuous functions and $l_1(z,w)>\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$\left(1-\left|z\right|\right)$}\right.$ for all $\left(z,w\right)\in \mathbb{D}\times \mathrm{ℂ}$ and some $\beta >1$. Moreover, we provide growth estimates of these function class. They describe the behavior of logarithm of maximum modulus of analytic function on a skeleton in a bidisc by behavior of the function $\mathit{L}$. These estimates are sharp in a general case. The presented results are based on bidisc exhaustion of the Cartesian product of the unit disc and complex plane.