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

• Date: 2025
• Main Authors: Sheremeta, M., Mulyava, O., Medvedev, M., Шеремета, Мирослав, Мулява, Оксана, Медвєдєв, Микола
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2025
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/8555
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.