On the slow growth of power series convergent in the unit disk

We construct two power series $f(z)=\sum_{n=0}^\infty a_nz^n$ convergent in the unit disk with the maximum modulus $M_f(r)=\max\{|f(z)|:|z|=r\}$ and the maximum term $\mu_f(r)=\max\{|a_n|r^n:n\ge 0\}$ such that
1) $a_n\not= O(1)$ $(n\to\infty)$, $\ln\mu_f(r)$ is a slowly growing function and $\ln M_f(r)$ is not slowly growing;
2) $a_n= O(1)$ $(n\to\infty)$, and $\ln M_f(r)$ is not a slowly growing function are constructed.