On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients

Abstract

UDC 517.537

In his study of the geometric properties of functions analytic in a disk ${\mathbb D} = \{z\colon |z|<1\},$ G. S. Sălăgean introduced a class $S_j(\alpha)$ of functions $f(z) = z + \sum _{k = 2}^{\infty}f_kz^k$ such that $\operatorname{Re} \frac{D^{j + 1}f(z)}{D^{j}f(z)} > \alpha\in [0, 1)$ for each $ z\in{\mathbb D},$ where $D^jf$ is the Sălăgean derivative. For Dirichlet series $F(s) = e^{s}-\sum _{k = 1 }^{\infty}f_k\exp\{s\lambda_k\}$ with $f_k\ge0$ absolutely convergent in the half plane $\Pi_0 = \{s\colon \operatorname{Re} s<0\},$ an analog of the Sălăgean class is the class$D_{j}(\alpha)$ defined by the condition $\operatorname{Re} \frac{F^{(j + 1)}(s)}{F^{(j)}(s)} > \alpha$ for each $s\in \Pi_0.$ By analogy with the neighborhood of an analytic function in ${\mathbb D}$ defined by A. V. Goodman, for $F\in D_{j}(\alpha),$ we introduce the concept of a neighborhood $O_{j,\delta}(F)$ and establish the conditions under which all functions from $O_{j,\delta}(F)$ belong to $D_{j}(\alpha_1),$ $0\le \alpha_1<\alpha<1,$ and vice versa. The problem of belonging of solutions of the differential equation  $\frac{d^2w}{ds^2} + (\gamma_0e^{2s} + \gamma_1 e^s + \gamma_2)w = 0$ with real parameters to the class $D_{j}(\alpha)$ is investigated.