伪星形和伪凸狄利克雷级数

M. Sheremeta
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引用次数: 0

摘要

对于形式为$F(s)=e^{-sh}+\sum_{j=1}^{n}a_j\exp\{-sh_j\}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$的Dirichlet级数,引入了阶$\alpha\in [0,\,1)$和类型$\beta\in (0,\,1]$的伪星形和阶$\alpha$和类型$\beta$的伪凸性的概念,其中$h>h_n>\dots>h_1\ge 1$和$(\lambda_k)$是一个增加到$+\infty$的正数序列。用系数证明了伪星形和伪凸性的判据。将所得结果应用于洛朗级数的亚纯星形和凸性研究\break$f(s)=1/z^p+\sum_{j=1}^{p-1}a_j/z^j+\sum_{k=1}^{\infty}f_kz^k$,研究了微分方程$w''+\gamma w'+(\delta e^{2sh}+\tau)w=0$具有$\alpha$阶和$\beta=1$型伪星形或伪凸解的条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
ON PSEUDOSTARLIKE AND PSEUDOCONVEX DIRICHLET SERIES
The concepts of the pseudostarlikeness of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ and the pseudoconvexity of the order $\alpha$ and type $\beta$ are introduced for Dirichlet series of the form $F(s)=e^{-sh}+\sum_{j=1}^{n}a_j\exp\{-sh_j\}+\sum_{k=1}^{\infty}f_k\exp\{s\lambda_k\}$, where $h>h_n>\dots>h_1\ge 1$ and $(\lambda_k)$ is an increasing to $+\infty$ sequence of positive numbers. Criteria for pseudostarlikeness and pseudoconvexity in terms of coefficients are proved. The obtained results are applied to the study of meromorphic starlikeness and convexity of the Laurent series \break $f(s)=1/z^p+\sum_{j=1}^{p-1}a_j/z^j+\sum_{k=1}^{\infty}f_kz^k$. Conditions, under which the differential equation $w''+\gamma w'+(\delta e^{2sh}+\tau)w=0$ has a pseudostarlike or pseudoconvex solution of the order $\alpha$ and the type $\beta=1$ are investigated.
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