TY - JOUR
AU - M. Sheremeta
AU - O. Mulyava
AU - M. Medvedev
PY - 2024/09/30
Y2 - 2024/11/03
TI - On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 76
IS - 9
SE - Research articles
DO - 10.3842/umzh.v76i9.8555
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/8555
AB - UDC 517.537In his study of the geometric properties of functions analytic in a disk ${\Bbb D} = \{z\colon |z|<1\},$ 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{\Bbb 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 ${\Bbb D}$ 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.
ER -