On a generalization of some Shah equation

Abstract

A Dirichlet series $F(s)=e^{hs}+\sum_{k=2}^{\infty}f_ke^{s\lambda_k}$ with the exponents $0<h<\lambda_k\uparrow+\infty$ and the abscissa of absolute convergence $\sigma_a[F]\ge 0$ is said to be pseudostarlike of order $\alpha\in [0,\,h)$ and type $\beta\in(0,\,1]$ in $\Pi_0=\{s:\,\text{Re}\,s<0\}$ if \[\left|\frac{F'(s)}{F(s)}-h\right|<\beta\left|\frac{F'(s)}{F(s)}-(2\alpha-h)\right|\] for all $s\in \Pi_0$. Similarly, the function $F$ is said to be pseudoconvex of order $\alpha\in [0,\,h)$ and type $\beta\in(0,\,1]$ if \[\left|\frac{F''(s)}{F'(s)}-h\right|<\beta\left|\frac{F''(s)}{F'(s)}-(2\alpha-h)\right|\] for all $s\in \Pi_0$, and $F$ is said to be close-to-pseudoconvex if there exists a pseudoconvex (with $\alpha=0$ and $\beta=1$) function $\Psi$ such that $\text{Re}\{F'(s)/\Psi'(s)\}>0$ in $\Pi_0$. Conditions on parameters $a_1,\,a_2,\,b_1,\,b_2,\,c_1,\,\,c_2$, under which the differential equation \[\dfrac{d^n w}{ds^n}+(a_1 e^{hs}+a_2)\dfrac{dw}{ds}+(b_1e^{hs}+b_2) w=c_1e^{hs}+c_2, \quad n\ge 2,\] has an entire solution pseudostarlike or pseudoconvex of order $\alpha\in [0,\,h)$ and type $\beta\in(0,\,1]$, or close-to-pseudoconvex in $\Pi_0$ are found. It is proved that for such solution\[\ln\,M(\sigma,F)=(1+o(1))\dfrac{n\root{n}\of{|b_1|}}{h}e^{h\sigma/n}\quad \text{as}\quad \sigma \to+\infty,\] where $M(\sigma,F)=\sup\{|F(\sigma+it)|:\, t\in {\mathbb R}\}$.