Maximum modulus of entire functions of two variables and arguments of coefficients of double power series

Let $mathcal{L}$ be the class of positive continuous functions on $(-infty,+infty)$ and let $mathcal{L}_+^2$ be the class of positive continuous increasing with respect to each variable functions $gamma$ in $mathbb{R}^2$ such that $ gamma(r_1,r_2)to +infty $ as $r_1+r_2to+infty.$ We prove the following} statement: for all entire functions of the form $ f(z_1,z_2)=sum_{n+m=0}^{+infty}a_{nm}z_1^nz_2^m$ such that $ |a_{nm}|leqexp{-(n+m)psi(n,m)} mbox{ for } n+mgeq k_0(f)$ and functions $f(z_1,1), f(1,z_2)$ are transcendent, $ psiinmathcal{L}_+^2, $ the inequality $$ mathfrak{M}_f(r_1,r_2)=O(M_f(r_1,r_2) h(ln M_f(r_1,r_2))), hinmathcal{L}, r^{vee}=min{r_1,r_2}to+infty, $$ holds where $M_f(r_1,r_2)=max{|f(z_1,z_2)|colon |z_1|=r_1,|z_2|=r_2},$ $mathfrak{M}_f(r_1,r_2)=sum_{n+m=0}^{+infty}|a_{nm}|times$ $times r_1^nr_2^m,$ if and only if begin{equation*} (forallgammainmathcal{L}_+^2)colon sqrt{r_1r_2}=Obig(h(gamma(r_1,r_2)psi(r_1,r_2))big), r^{vee}to+infty. end{equation*}