On the growth of gap power series of homogeneous polynomials

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 $\mathbb{C}^p$.