On the growth of gap power series of homogeneous polynomials

Abstract

Let $$$f$$$ be an entire functions $$$f\colon \mathbb{C}^{p}\to\mathbb{C}$$$, represented by power series of the form
$$f(z)=\sum\limits_{k=0}^{+\infty} P_k(z), z\in\mathbb{C}^p$$
where $$$P_0(z)\equiv a_{0}\in\mathbb{C}$$$, $$$P_k(z)=\sum\limits_{\|n\|=\lambda_k} a_{n}z^{n}$$$ is a homogeneous polynomial of degree $$$\lambda_k\in\mathbb{N}$$$ and $$$ 0=\lambda_0< \lambda_k\uparrow +\infty$$$ ($$$1\leq k\uparrow +\infty$$$).
In this paper, we present conditions such that the equality
$$\rho_f:=\varlimsup\limits_{r\to +\infty}\frac{\ln\ln M_f(r)}{\ln r}=\rho(f,\mathbf{K}):=\varlimsup\limits_{r\to +\infty}\frac{\ln\ln M_f(r,\mathbf{K})}{\ln r}$$
holds, where
$$M_f(r):=\sup\{|f(z)|\colon |z|\leq r\},\quad M_f(r,\mathbf{K}):=\sup\{|f(z)|\colon z\in \mathbf{K},|z|\leq r\}$$
and $$$\mathbf{K}$$$ is a real cone in $$$\mathbb{C}^p$$$ such that the set $$$\mathbb{C}\overline{\mathbf{K}}$$$ is non-pluripolar in $$$\mathbb{C}^p$$$.