Bitlyan-Gol'dberg type inequality for entire functions and diagonal maximal term

Abstract

In the article is obtained an analogue of Wiman-Bitlyan-Gol'dberg type inequality for entire $f\colon\mathbb{C}^p\to \mathbb{C}$ from the class $\mathcal{E}^{p}(\lambda)$ of functions represented by gap power series of the form$$f(z)=\sum\limits_{k=0}^{+\infty} P_k(z),\quadz\in\mathbb{C}^p.$$Here $P_0(z)\equiv a_{0}\in\mathbb{C},$ $P_k(z)=\sum_{\|n\|=\lambda_k} a_{n}z^{n}$ is homogeneouspolynomial of degree $\lambda_k\in\mathbb{Z}_+,$ ànd $ 0=\lambda_0<\lambda_k\uparrow +\infty$\ $(1\leq k\uparrow +\infty ),$$\lambda=(\lambda_k)$.\ We consider the exhaustion of thespace\ $\mathbb{C}^{p}$\by the system $(\mathbf{G}_{r})_{r\geq 0}$ of a bounded complete multiple-circular domains $\mathbf{G}_{r}$with the center at the point $\mathbf{0}=(0,\ldots,0)\in \mathbb{C}^{p}$. Define $M(r,f)=\max\{|f(z)|\colon z\in\overline{G}_r\}$, $\mu(r,f)=\max\{|P_k(z))|\colon z\in\overline{G}_r\}$.Let $\mathcal{L}$ be the class of positive continuous functions $\psi\colon \mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $\int_{0}^{+\infty}\frac{dx}{\psi(x)}<+\infty$, $n(t)=\sum_{\lambda_k\leq t}1$ counting function of the sequence $(\lambda_k)$ for $t\geq 0$. The following statement is proved:{\it If a sequence $\lambda=(\lambda_{k})$ satisfy the condition\begin{equation*}(\exists p_1\in (0,+\infty))(\exists t_0>0)(\forall t\geq t_0)\colon\quad n(t+\sqrt{\psi(t)})-n(t-\sqrt{\psi(t)})\leq t^{p_1}\end{equation*}for some function $\psi\in \mathcal{L}$,then for every entire function $f\in\mathcal{E}^{p}(\lambda)$, $p\geq 2$ and for any$\varepsilon>0$ there exist a constant $C=C(\varepsilon, f)>0$ and a set $E=E(\varepsilon, f)\subset [1,+\infty)$ of finite logarithmic measure such that the inequality\begin{equation*}M(r, f)\leq C m(r,f)(\ln m(r, f))^{p_1}(\ln\ln m(r, f))^{p_1+\varepsilon}\end{equation*}holds for all $ r\in[1,+\infty]\setminus E$.}The obtained inequality is sharp in general.At $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and the Bitlyan-Gol'dberg inequality (1959) it follows. In the case $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and from obtained statement we get the assertion on the Bitlyan-Gol'dberg inequality (1959), and at $p=1$ about the classical Wiman inequality it follows.

On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

“…By Lemma 1, we get from ( 14) that either Ψ 1 (z)e P (z)+Q(z+c) ≡ 1 or Ω 1 (z)e −P (z)+Q(z+c) ≡ 1 where Ψ 1 (z) and Ω 1 (z) are given after (14). Similarly, by using Lemma 1, we deduce from (15) that either Ψ 2 (z)e Q(z)+P (z+c) ≡ 1 or Ω 2 (z)e −Q(z)+P (z+c) ≡ 1, where Ψ 2 (z) and Ω 2 (z) are given after (15). Now we will discuss the following cases.…”

“…But they allow to consider functions of infinite order. The first approach is the multidimensional Wiman-Valiron theory which examines the properties of the maximal term and the central index of the power series [13][14][15][16]. This theory is applicable for any entire solution of differential equations.…”

Wiman’s type inequality for entire multiple Dirichlet series with arbitrary complex exponents

“…Without a loss of generality we can assume that F ∈ D 0 , λ 0 = 0, 0 = µ 0 ≤ µ m ↗ +∞ (1 ≤ m → +∞). To prove Theorem 4, it is enough to use Theorem 3 and arguments according to the scheme of proving Theorem 1 in [13]. On the one hand, for a given R > 0 and every fixed z ∈ γ R we will obtain that ln µ(tz, F ) ≥ (n 1 (3µ ν )) α c 1 (ν), ν = ν(tz, F ), holds for all t > 0 such that tz ̸ ∈ E 1 , where the set E 1 ⊂ γ + (F ), by Theorem 3, such that τ 2p (E ∩ γ + (F )) ≤ C p /2, and c 1 (ν) → +∞ as tz → ∞ uniformly in K (by Proposition 3).…”

“…Let D 1 be the class of absolutely convergent for all Dirichlet series in C of form (1) with sequence of the exponents (λ n ) such that λ n ≥ 0 (n ≥ 0) and sup{λ n : n ≥ 0} = +∞. It should be noted that some asymptotic properties of functions F ∈ D 1 were investigated in the papers [8][9][10][11][12][13].…”

Wiman’s Type Inequality in Multiple-Circular Domain

“…Some analogues of Wiman's inequality for entire functions of several complex variables can be found in [16][17][18][19][20][21][22], and for analytic functions in the polydisc D p , p ≥ 2, in [23,24].…”