An example of a fast decaying $(p,A)$-lacunary function

A function $f(z)=\sum^\infty_{k=0}a_kz^{n_k}$ analytic in the unit disc is said be $(p,A)$-lacunary if $n_k\ge Ak^p$, $1<p<\infty$, $A>0$ for all $k\ge0$. For $1<p<2$, $A>0$, a $(p,A)$-lacunary function $f_{1,p,A}(z)$ is constructed that decays as $x\to1-0$ almost optimally.