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

Abstract

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.

Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients

“…then differential equation (1) has solution (14) which is pseudostarlike in Π 0 of order α and type β.…”

“…Therefore, as in [14] the conformal function (2) in Π 0 is called pseudostarlike of order α ∈ [0, 1) and type β ∈ (0, 1] if…”

Pseudostarlike and pseudoconvex in a direction multiple Dirichlet series

“…In view of (6), as in [25], we call conformal function (3) in Π 0 pseudostarlike of the order α ∈ [0, 1) and the type β ∈ (0, 1] if…”

“…1){((n + 2)β + n)(h, b) − 2βα}|f (n) | ≤ ≤ 2{(3β + 1)(h, b) − 2βα}|f (1) | + 3{(4β + 2)(h, b) − 2βα}|f (2) |+ + +∞ ∥(n)∥=∥(2)∥ (n + 2){((n + 3)β + n + 1)(h, b) − 2βα} |γ 1 ||f ((n) | (n + 1)(h, b) 2 = = +∞ ∥(n)∥=∥(1)∥ (n + 3){((n + 4)β + n + 2)(h, b) − 2βα} |γ 0 ||f ((n) | (n + 2)(h, b) 2 ≤ ≤ 4{(3β + 1)(h, b) − 2βα}|γ 1 | + 3{(4β + 2)(h, b) − 2βα}|γ 0 | 2(h, b) 2 + + +∞ ∥(n)∥=∥(1)∥ 4 |γ 1 |(n + 1){((n + 2)β + n)(h, b) − 2βα} (n + 1)(h, b) 2 + + 6 |γ 0 |(n + 1){((n + 2)β + n)(h, b) − 2βα} (n + 2)(h, b) 2 |f ((n) | ≤ ≤ 4{(3β + 1)(h, b) − 2βα}|γ 1 | + 3{(4β + 2)(h, b) − 2βα}|γ 0 | 2(h, b) 2 + +2 |γ 1 | + |γ 0 | (h, b) 2 +∞ ∥(n)∥=∥(1)∥ (n + 1){((n + 2)β + n)(h, b) − 2βα}|f ((n) |and in view of(25) we get(27), i.e. function (21) is pseudoconvex of the order α and the type β in the direction b ≥ 0.…”

Neighborhoods of Dirichlet series absolutely convergent in a half-plane

“…It is known [10] that each function F ∈ SD(0) is non-univalent in Π 0 , but there exist conformal in Π 0 functions (2), and if (2) in Π 0 is said to be pseudostarlike if Re{F (s)/F (s)} > 0 and is said to be pseudoconvex if Re{F (s)/F (s)} > 0 for s ∈ Π 0 . In [10] (see also [11, p. 139 [12] to be pseudostarlike of the order α ∈ [0, 1) if Re{F (s)/F (s)} > α for s ∈ Π 0 , i.e., F (s)…”