On the growth of the maximum modulus of an entire function depending on the growth of its central index

Let $h$ be a positive function continuous on $(0,+\infty)$, $f(z)=\sum_{n=0}^\infty a_nz^n$ be an entire function, and $M_f(r)=\max\{|f(z)|\colon|z|=r\}$, $\mu_f(r)=\max\{|a_n|r^n\colon n\ge0\}$, and $\nu_f(r)=\max\{n\ge0\colon|a_n|r^n=\mu_f(r)\}$ be the maximum modulus, the maximal term, and the central index of the function $f$, respectively. We establish necessary and sufficient conditions for the growth of $\nu_f(r)$ under which $M_f(r)=O(\mu_f(r)h(\ln\mu_f(r)))$, $r\to+\infty$.