On close-to-pseudoconvex Dirichlet series

Abstract

For a Dirichlet series of form $F(s)=\exp\{s\lambda_1\}+\sum\nolimits_{k=2}^{+\infty}f_k\exp\{s\lambda_k\}$ absolutely convergent in the half-plane $\Pi_0=\{s\colon \mathop{\rm Re}s<0\}$ new sufficient conditions
for the close-to-pseudoconvexity are found and the obtained result is applied to studying of solutions linear differential equations of second order with exponential coefficients. In particular, are proved the following statements:

1) Let $\lambda_k=\lambda_{k-1}+\lambda_1$ and $f_k>0$ for all $k\ge 2$. If $1\le\lambda_2f_2/\lambda_1\le 2$ and $\lambda_kf_k-\lambda_{k+1}f_{k+1}\searrow q\ge 0$ as $k\to+\infty$ then function of form {\bf(1)} is close-to-pseudoconvex in $\Pi_0$ (Theorem 3). This theorem complements Alexander's criterion obtained for power series.
2) If either $-h^2\le\gamma\le0$ or $\gamma=h^2$ then differential equation $(1-e^{hs})^2w''-h(1-e^{2hs})w'+\gamma e^{2hs}=0$ $(h>0, \gamma\in{\mathbb R})$ has a solution $w=F$ of form {\bf(1)} with the exponents $\lambda_k=kh$ and the the abscissa of absolute convergence $\sigma_a=0$ that is close-to-pseudoconvex in $\Pi_0$ (Theorem 4).