In this paper we prove some analogues of Wiman's inequality for analytic f (z) and random analytic functionsis a multiplicative system of random variables on the Steinhaus probability space, uniformly bounded by the number 1. In particular, we prove the following statements: For every δ > 0 there exist sets.hold for all r ∈ T \ E 1 and for all r ∈ T \ E 2 a.s. in t, respectively. Also sharpness of the obtained inequalities is proved.1. Introduction and the main result. Let f be an analytic function in the disc D R = {z : |z| < R}, 0 < R ≤ +∞, represented by the power seriesPut [6,7,8]) that for each nonconstant entire function f (z) and for every ε > 0 there exists a set E(ε, f ) ⊂ [1, +∞) such that Wiman's inequality M f (r) ≤ µ f (r)(ln µ f (r))2010 Mathematics Subject Classification: 30B20, 30D20.
Let $\mathcal{A}^2$ be a class of analytic functions $f$ represented by power series of the from $$ f(z)=f(z_1,z_2)=\sum^{+\infty}_{n+m=0}a_{nm}z_1^nz^m_2$$ with the domain of convergence $\mathbb{T}=\{ z\in \mathbb{C}^2 \colon |z_1|<1, |z_2|<+\infty \} $ such that $\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}$ and there exists $r_0=(r^0_1, r^0_2)\in [0,1)\times[0,+\infty)$ such that for all $r\in(r^0_1,1)\times(r^0_2,+\infty)$ we have $ r_1\frac{\partial}{\partial r_1}\ln M_f(r)+\ln r_1>1, \ $ where $M_f(r)=\sum_{n+m=0}^{+\infty}|a_{nm}|r_1^nr_2^m.$ Let $K(f,\theta)=\{f(z,t)=\sum_{n+m=0}^{+\infty}a_{nm}e^{2\pi it(\theta_n+\theta_m)}:t\in \mathbb{R}\}$ be class of analytic functions, where $(\theta_{nm})$ is a sequence of positive integer such that its arrangement $(\theta^*_k)$ by increasing satisfies the condition $$ \theta^*_{k+1}/\theta^*_{k}\geq q>1, k>0. $$ For analytic functions from the class $\mathcal{K}(f,\theta)$ Wiman's inequality is improved.
We prove some properties of power series expansion of analytic functions in D × C having bounded L-index in joint variables, where L(z, w) = (l1(z, w), l2(z, w)) with lj : D × C → R+ (j ∈ {1, 2}) are positive continuous functions and l1(z, w) > β/(1 − |z|) for all (z, w) ∈ D × C and some β > 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 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.
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