Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients

Abstract

Dirichlet series $F(s)=e^{s}+\sum_{k=1}^{\infty}f_ke^{s\lambda_k}$ with the exponents $1<\lambda_k\uparrow+\infty$ and the abscissa of absolute convergence $\sigma_a[F]\ge 0$ is said to be pseudostarlike of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if

$\left|\dfrac{F'(s)}{F(s)}-1\right|<\beta\left|\dfrac{F'(s)}{F(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0=\{s\colon \,\text{Re}\,s<0\}$.

 Similarly, the function $F$ is said to be pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if

$\left|\dfrac{F''(s)}{F'(s)}-1\right|<\beta\left|\dfrac{F''(s)}{F'(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0$.

 Some conditions are found on the parameters $b_0,\,b_1,\,c_0,\,c_1,\,\,c_2$ and the coefficients $a_n$, under which the differential equation

 $\dfrac{d^2w}{ds^2}+(b_0e^{s}+b_1)\dfrac{dw}{ds}+(c_0e^{2s}+c_1e^{s}+c_2)w=\sum\limits_{n=1}^{\infty}a_ne^{ns}$
has an entire solution which is pseudostarlike or pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$. It is proved that by some conditions for such solution the asymptotic equality holds
 $\ln\,\max\{|F(\sigma+it)|\colon t\in {\mathbb R}\}=\dfrac{1+o(1)}{2}\left(|b_0|+\sqrt{|b_0|^2+4|c_0|}\right)$ as $\sigma \to+\infty$.