On the growth of a class of Dirichlet series absolutely convergent in half-plane

“…F ) for any Dirichlet series F ∈ D 0 (λ), were found in many works (see, for example, [5,6,7,8,9,10,11,12]). In this article, Problem 1 is completely solved, in particular, in the case when α ∈ L is an arbitrary function slowly varying at the point +∞ and β ∈ Ω 0 is an arbitrary function regularly varying at the point 0 with index ρ > 0.…”

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confidence: 96%

“…Let's choose the number t > 0 so that the inequality h(t) > c holds. Then, according to (8), there exists σ 0 < 0 such that tΦ(σ/t) ≥ α −1 (cβ(σ))/γ(σ) for all σ ∈ [σ 0 , 0). Therefore, Φ(σ) → +∞ as σ ↑ 0, and hence Φ ∈ Ω 0 .…”

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confidence: 99%

Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

Filevych,

Hrybel

2024

Mat. Stud.

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Let $\lambda=(\lambda_n)_{n\in\mathbb{N}_0}$ be a non-negative sequence increasing to $+\infty$, $\tau(\lambda)=\varlimsup_{n\to\infty}(\ln n/\lambda_n)$, and $\mathcal{D}_0(\lambda) $ be the class of all Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_n(F)e^{s\lambda_n}$ absolutely convergent in the half-plane $\operatorname{Re}s<0$ with $a_n(F)\not=0$ for at least one integer $n\ge0$. Also, let $\alpha$ be a continuous function on $[x_0,+\infty)$ increasing to $+\infty$, $\beta$ be a continuous function on $[a,0)$ such that $\beta(\sigma)\to+\infty$ as $\sigma\uparrow0$, and $\gamma$ be a continuous positive function on $[b,0)$. In the article, we investigate the growth of a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ depending on the behavior of the sequence $(|a_n(F)|)$ in terms of its $\alpha,\beta,\gamma$-orders determined by the equalities$$R^*_{\alpha,\beta,\gamma}(F)=\varlimsup_{\sigma\uparrow0}\frac{\alpha(\max\{x_0,\gamma(\sigma)\ln\mu(\sigma)\})}{\beta(\sigma)},$$$$R_{\alpha,\beta,\gamma}(F)=\varlimsup_{\sigma\uparrow0}\frac{\alpha(\max\{x_0,\gamma(\sigma)\ln M(\sigma)\})}{\beta(\sigma)},$$where $\mu(\sigma)=\max\{|a_n(F)|e^{\sigma\lambda_n}\colon n\ge0\}$ and $M(\sigma)=\sup\{|F(s )|\colon \operatorname{Re}s=\sigma\}$ are the maximal term and the supremum modulus of the series $F$, respectively. In particular, if for every fixed $t>0$ we have $\alpha(tx)\sim \alpha(x)$ as $x\to+\infty$, $\beta(t\sigma)\sim t^{-\rho}\beta(\sigma)$ as $\sigma\uparrow0$ for some fixed $\rho>0$, $0<\varliminf_{\sigma\uparrow0}\gamma(t\sigma)/\gamma(\sigma) \le\varlimsup_{\sigma\uparrow0}\gamma(t\sigma)/\gamma(\sigma)<+\infty$,$\Phi(\sigma)=\alpha^{-1}(\beta(\sigma))/\gamma(\sigma)$ for all $\sigma\in[\sigma_0,0)$,$\widetilde{\Phi}(x)=\max\{x\sigma-\Phi(\sigma)\colon \sigma\in[\sigma_0,0)\}$ for all $x\in\mathbb{R}$, and $\Delta_\Phi(\lambda)=\varlimsup_{n\to\infty}( -\ln n/\widetilde{\Phi}(\lambda_n))$, then: (a) for each Dirichlet series $F\in\mathcal{D}_0(\lambda)$ we have$$R^*_{\alpha,\beta,\gamma}(F)=\varlimsup_{n\to +\infty}\left(\frac{\ln^+|a_n(F)|}{-\widetilde{\Phi }(\lambda_n)}\right)^\rho;$$ (b) if $\tau(\lambda)>0$, then for each $p_0\in[0,+\infty]$ and any positive function $\Psi$ on $[c,0)$ there exists a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ such that $R^*_{\alpha,\beta,\gamma}(F)=p_0$ and $M(\sigma,F)\ge \Psi(\sigma)$ for all $\sigma\in[\sigma_0,0)$; (c) if $\tau(\lambda)=0$, then $(R_{\alpha,\beta,\gamma} (F))^{1/\rho}\le (R^*_{\alpha,\beta,\gamma}(F))^{1/\rho}+\Delta_\Phi(\lambda)$ for every Dirichlet series\linebreak $F\in\mathcal{D}_0(\lambda)$; (d) if $\tau(\lambda)=0$, then for each $p_0\in[0,+\infty]$ there exists a Dirichlet series $F\in\mathcal{D}_0(\lambda)$ such that $R^*_{\alpha,\beta,\gamma}(F)=p_0$ and $(R_{\alpha,\beta,\gamma}(F))^{1/\rho}=(R ^*_{\alpha,\beta,\gamma}(F))^{1/\rho}+\Delta_\Phi(\lambda)$.

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“…In the paper [6] the results of E. G. Calys are generalized on the case of entire Dirichlet series of finite generalized orders, moreover instead of two entire functions m ≥ 2 entire Dirichlet series were considered.…”

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confidence: 99%

On entire Dirichlet series similar to Hadamard compositions

Mulyava

Sheremeta

2023

Mat. Stud.

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A function $F(s)=\sum_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with $0\le\lambda_n\uparrow+\infty$ is called the Hadamard composition of the genus $m\ge 1$ of functions $F_j(s)=\sum_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where$P(x_1,...,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}x_1^{k_1}\cdot...\cdot x_p^{k_p}$ is a homogeneous polynomial of degree $m\ge 1$. Let $M(\sigma,F)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$ and functions $\alpha,\,\beta$ be positive continuous and increasing to $+\infty$ on $[x_0, +\infty)$. To characterize the growth of the function $M(\sigma,F)$, we use generalized order $\varrho_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\alpha(\ln\,M(\sigma,F))}{\beta(\sigma)}$, generalized type$T_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\ln\,M(\sigma,F)}{\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}$and membership in the convergence class defined by the condition$\displaystyle \int_{\sigma_0}^{\infty}\frac{\ln\,M(\sigma,F)}{\sigma\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}d\sigma<+\infty.$Assuming the functions $\alpha, \beta$ and $\alpha^{-1}(c\beta(\ln\,x))$ are slowly increasing for each $c\in (0,+\infty)$ and $\ln\,n=O(\lambda_n)$ as $n\to \infty$, it is proved, for example, that if the functions $F_j$ have the same generalized order $\varrho_{\alpha,\beta}[F_j]=\varrho\in (0,+\infty)$ and the types $T_{\alpha,\beta}[F_j]=T_j\in [0,+\infty)$, $c_{m0...0}=c\not=0$, $|a_{n,1}|>0$ and $|a_{n,j}|= o(|a_{n,1}|)$ as $n\to\infty$ for $2\le j\le p$, and $F$ is the Hadamard composition of genus$m\ge 1$ of the functions $F_j$ then $\varrho_{\alpha,\beta}[F]=\varrho$ and $\displaystyle T_{\alpha,\beta}[F]\le \sum_{k_1+\dots+k_p=m}(k_1T_1+...+k_pT_p).$It is proved also that $F$ belongs to the generalized convergence class if and only ifall functions $F_j$ belong to the same convergence class.

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On the growth of a class of Dirichlet series absolutely convergent in half-plane

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On Dirichlet series like to compositions of Hadamard

On Dirichlet series like to compositions of Hadamard

Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

On entire Dirichlet series similar to Hadamard compositions

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