Wiman's type inequality in the unit ball and the diagonal maximal term

Abstract

Let $\mathbb{C}^{p}$ be the $p$-dimensional complex vector space  $(p\geq 1)$, $\|n\|=n_1+\cdots +n_p$, $|z|=\sqrt{|z_1|^2+\ldots+|z_n|^2}$ for $n=(n_1,\ldots,n_p)\in\mathbb{Z}_{+}^{p}$ and $z=(z_1,\ldots , z_p)\in\mathbb{C}^{p}$,\ $\mathbb{R}_{+}= [0, +\infty)$. In the paper we consider the class of analytic functions $f$, represented in the unit ball $\mathbb{B}_p=\{z\in\mathbb{C}^p\colon |z|<1\}$ by power series of the form $f(z)=\sum\limits_{k=0}^{+\infty} P_k(z)$; here $P_0(z)\equiv a_{0}\in\mathbb{C},$ $P_k(z)$ is a homogeneous polynomial of degree $k\in\mathbb{Z}_+$.  We denote $M_f(r)=\max\{|f(z)|\colon |z|=r\}$ and $m(r,f)=\max\{|P_k(z)|\colon k\geq 0\}$, the maximum  modulus and maximal term of series, respectively; $r\in [0, 1)$. In particular, the following statements are proved: $1^0.$\ For every analytic function $f\in\mathcal{A}^{p}$, $p\geq 2$ and for any $\varepsilon>0$ there exists a set $E=E(\varepsilon, f)\subset (0,1)$ of finite logarithmic measure such that  the inequality $M(r, f)\leq \frac{m(r, f)}{(1-r)^{1+\varepsilon}}\Big(\ln\Big(\frac{m(r, f)}{1-r}\Big)\Big)^{1/2+\varepsilon}$ holds for all $ r\in (0, 1)\setminus E$.   $2^0.$  Let $h$ be a continuous positive increasing to $+\infty$ on $[0;1)$ functions such that   $\int^1_{0} h(r) d r =+\infty.$  If the function $f\in\mathcal{A}^p$ is unbounded, then  there exists a set $E_1:=E(f,h)\subset(0,1)$ such that $\ln M(r,f)\leq (1+o(1))\ln\big(h(r)m(r,f)\big)$\ $(r\to 1-0,\ r\in (0,1)\setminus E_1)$, and $h$-${\rm meas\ }E_1=\int_{E_1\cap(0,1)}h(r)dr < +\infty$.