On the decay rate of $(p,A)$-lacunary series

A power series $\sum\limits^\infty_{k=0} a_k x^{n_k}$ with radius of convergence equal to 1 is said to be $(p,A)$-lacunary if $n_k\ge Ak^p$, $A>0$, $1<p<\infty$. It is proved that if a $(p,A)$-lacunary series $f$ satisfies the condition
$$ |f(x)|\exp\biggl(B(1-x)^{-\frac1{p-1}}+\varepsilon(1-x)^{-\frac1{p-1}}\bigg/(|\log(1-x)|+1)\biggr)\underset{x\to1-0}{\longrightarrow}0, $$
for $1<p<2$, where
$$ B=(p-1)\biggl(\frac\pi p\biggr)^{\frac p{p-1}}\cdot\frac1{A^{1/(p-1)}}\cdot\frac1{|\cos\frac{\pi p}2|^{1/(p-1)}}, $$
and $\varepsilon>0$, then $f\equiv0$.
We also construct a $(p,A)$-lacunary series $f_0$ such that
$$ |f_0(x)|\exp\biggl(B(1-x)^{-\frac1{p-1}}+C_0(1-x)^{-\frac1{p-1}}\bigg/(|\log(1-x)|^2+1)\biggr)\underset{x\to1-0}{\longrightarrow}0. $$
for a constant $C_0=C_0(p,A)>0$.