On certain subclass of Dirichlet series absolutely convergent in half-plane

Abstract

Denote by $\mathfrak{D}_0$ a class of absolutely convergent in half-plane $\Pi_0=\{s\colon \text{Re}\,s<0\}$ Dirichlet series$F(s)=e^{sh}-\sum_{k=1}^{\infty}f_k\exp\{s(\lambda_k+h)\},\, s=\sigma+it$, where $h> 0$, $h<\lambda_k\uparrow+\infty$ and $f_k>0$.For $0\le\alpha\frac{\alpha}{h}$,and belongs to the class $\mathfrak{DG}_h(l,\alpha)$ if and only if$\text{Re}\{e^{-hs}((1-l)F'(s)+\frac{l}{h}F''(s))\}>\alpha$ for all $s\in \Pi_0$. It is provedthat $F\in \mathfrak{DF}_h(l,\alpha)$ if and only if $ \sum_{k=1}^{\infty}(h+l\lambda_k)f_k\le h-\alpha$, and$F\in \mathfrak{DG}_h(l,\alpha)$ if and only if $\sum_{k=1}^{\infty}(h+l\lambda_k)(\lambda_k+h)f_k\le h(h-\alpha)$. If $F_j\in \mathfrak{DF}_h(l_j,\alpha_j)$, $j=1, 2$, where $l_j\ge0$ and $0\le \alpha_j0$ and $\delta>0$ the neighborhood of the function $F\in \mathfrak{D}_0$ is defined as follows$O_{j,\delta}(F)=\{G(s)=e^{s}-\sum_{k=1}^{\infty}g_k\exp\{s\lambda_k\}\in \mathfrak{D}_0\colon \sum_{k=1}^{\infty}\lambda^j_k|g_k-f_k|\le\delta\}$. It is described the neighborhoods of functions from classes $\mathfrak{DF}_h(l,\alpha)$ and $\mathfrak{DG}_h(l,\alpha)$. Conditions on real parameters $\gamma_0,\,\gamma_1,\,\gamma_2,\,a_1$ and $a_2$ of the differential equation $w''+(\gamma_0e^{2hs}+\gamma_1e^{hs}+\gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solutioneither in $\mathfrak{DF}_h(l,\alpha)$ or in $\mathfrak{DG}_h(l,\alpha)$.