On  meromorphically starlike functions of order $\alpha$ and type $\beta$, which satisfy Shah's differential equation

Trukhan, Yu.S.; Mulyava, O. M.

doi:10.15330/cmp.9.2.154-162

Cited by 3 publications

(1 citation statement)

References 5 publications

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“…Using the Alexander criterion, S. Shah [4] indicated the conditions for real parameters γ 0 , γ 1 , γ 2 , under which a differential equation z 2 w ′′ +zw ′ +(γ 0 z 2 +γ 1 z +γ 2 )w = 0 has an entire transcendental solution f such that the function f and all its derivatives are close-to-convex in D. Many authors (see, for example, [5][6][7][8]) continued Shah's research.…”

Section: Introduction An Analytic Functionmentioning

confidence: 99%

On close-to-pseudoconvex Dirichlet series

Mulyava,

Sheremeta,

Medvediev

2024

Mat. Stud.

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For a Dirichlet series of form $F(s)=\exp\{s\lambda_1\}+\sum\nolimits_{k=2}^{+\infty}f_k\exp\{s\lambda_k\}$ absolutely convergent in the half-plane $\Pi_0=\{s\colon \mathop{\rm Re}s<0\}$ new sufficient conditionsfor the close-to-pseudoconvexity are found and the obtained result is applied to studying of solutions linear differential equations of second order with exponential coefficients. In particular, are proved the following statements: 1) Let $\lambda_k=\lambda_{k-1}+\lambda_1$ and $f_k>0$ for all $k\ge 2$. If $1\le\lambda_2f_2/\lambda_1\le 2$ and $\lambda_kf_k-\lambda_{k+1}f_{k+1}\searrow q\ge 0$ as $k\to+\infty$ then function of form {\bf(1)} is close-to-pseudoconvex in $\Pi_0$ (Theorem 3). This theorem complements Alexander's criterion obtained for power series.2) If either $-h^2\le\gamma\le0$ or $\gamma=h^2$ then differential equation $(1-e^{hs})^2w''-h(1-e^{2hs})w'+\gamma e^{2hs}=0$ $(h>0, \gamma\in{\mathbb R})$ has a solution $w=F$ of form {\bf(1)} with the exponents $\lambda_k=kh$ and the the abscissa of absolute convergence $\sigma_a=0$ that is close-to-pseudoconvex in $\Pi_0$ (Theorem 4).

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Section: Introduction An Analytic Functionmentioning

confidence: 99%

On close-to-pseudoconvex Dirichlet series

Mulyava,

Sheremeta,

Medvediev

2024

Mat. Stud.

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Pseudostarlike and pseudoconvex Dirichlet series of the order $\alpha$ and the type $\beta$

Sheremeta

2020

Mat. Stud.

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The concepts of the pseudostarlikeness of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ and the pseudoconvexity of order $\alpha$ and type $\beta$ are introduced for Dirichlet series with null abscissa of absolute convergence. In terms of coefficients, the pseudostarlikeness and the pseudoconvexity criteria of order $\alpha$ and type $\beta$ are proved.Let $h\ge 1$, $\Lambda=(\lambda_k)$ be an increasing to $+\infty$ sequence of positive numbers ($\lambda_1>h$. We call a conformal function of the form $F(s)=e^{sh}+\sum\nolimits_{k=1}^{\infty}f_k\exp\{s\lambda_k\}, \ s=\sigma+it,$in $\Pi_0=\{s\colon \, \text{Re}\,s<0\}$ pseudostarlike of order $\alpha\in [0,\,1)$ and type$\beta \in (0,\,1]$ if\begin{equation*}\left|\frac{F'(s)}{F(s)}-h\right|<\beta\left|\frac{F'(s)}{F(s)}-(2\alpha-h)\right|,\quad s\in \Pi_0.\end{equation*}The main results of the article are contained in Theorems 1 and 2. Theorem 1 states: \textit{If $\alpha \in [0, \, 1)$ and $\beta \in (0, \, 1]$ such that\begin{equation*}\sum\limits_{k=1}^{\infty}\{(1+\beta)\lambda_k -2\beta\alpha -h(1-\beta)\}|f_k|\le 2\beta (h-\alpha)\label{t7}\end{equation*}then the function $F$ is pseudostarlike of order $\alpha$ and type $\beta$.}The corresponding results for Hadamard compositions of such series are also established.

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On Pseudostarlike and Pseudoconvex Dirichlet Series

Sheremeta

2021

BMJ

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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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On meromorphically starlike functions of order $\alpha$ and type $\beta$, which satisfy Shah's differential equation

Cited by 3 publications

References 5 publications

On close-to-pseudoconvex Dirichlet series

On close-to-pseudoconvex Dirichlet series

Pseudostarlike and pseudoconvex Dirichlet series of the order $\alpha$ and the type $\beta$

On Pseudostarlike and Pseudoconvex Dirichlet Series

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