On an estimate for a function represented by a Dirichlet series

The article considers the series
$$ f(z)=\sum_{k=1}^\infty a_ke^{\lambda_kz},\qquad0<\lambda_k\uparrow\infty,\quad\sum_{k=1}^\infty\lambda_k^{-1}<\infty, $$
convergent in the whole plane.
Theorem 1. {\it Let $|f(x)|<H(x),$ $-\infty<x<\infty,$ where $0<H(x)\uparrow\infty$. For given $\varepsilon>0$ and $h>0$ there exists a constant $A$, not depending on $f(z)$ and $H(x)$ such that $|f(z)|<AH(x+\varepsilon),$ $x=\operatorname{Re}z,$ $|y|<h$.}
Theorem 2. {\it If in addition
$$ \delta=\varlimsup_{k\to\infty}\frac1{\lambda_k}\ln\biggl|\frac1{L'(\lambda_k)}\biggr|<\infty,\qquad L(\lambda)=\prod_{k=1}^\infty\biggl(1-\frac\lambda{\lambda_k}\biggr), $$
then for arbitrary $z$ we have $|f(z)|<AH(x+\delta+\varepsilon),$ $x=\operatorname{Re}z$.}
The quantity $\delta$ cannot be replaced by a smaller one. These results strengthen corresponding results due to Gaier (RZhMat., 1967, 10B155) and Anderson and Binmore (RZhMat., 1972, 7B1115).
Bibliography: 7 titles.