scholarly journals Growth estimates for the maximal term and central exponent of the derivative of a Dirichlet series

2020 ◽  
Vol 12 (2) ◽  
pp. 269-279
Author(s):  
S.I. Fedynyak ◽  
P.V. Filevych
Keyword(s):  
Continuous Function ◽  
Dirichlet Series ◽  
Conjugate Function ◽  
Maximal Term ◽  
Central Index ◽  
Nonnegative Sequence

Let $A\in(-\infty,+\infty]$, $\Phi:[a,A)\to\mathbb{R}$ be a continuous function such that $x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$ for every $x\in\mathbb{R}$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma):\sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ and $\Gamma(x)=(\widetilde{\Phi}(x)-\ln x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, and $F(s)=\sum\limits\limits_{n=0}^\infty a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximal term $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge0\}$ and central index $\nu(\sigma,F)=\max\{n\ge0:|a_n|e^{\sigma\lambda_n}=\mu(\sigma,F)\}$ are defined for all $\sigma<A$. It is proved that if $\ln\mu(\sigma,F)\le(1+o(1))\Phi(\sigma)$ as $\sigma\uparrow A$, then the inequalities $$ \varlimsup_{\sigma\uparrow A}\frac{\mu(\sigma,F')}{\mu(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1,\qquad \varlimsup_{\sigma\uparrow A}\frac{\lambda_{\nu(\sigma,F')}}{\Gamma^{-1}(\sigma)}\le1, $$ hold, and these inequalities are sharp.

scholarly journals Growth estimates for a Dirichlet series and its derivative

2020 ◽  
Vol 53 (1) ◽  
pp. 3-12
Author(s):  
S.I. Fedynyak ◽  
P.V. Filevych
Keyword(s):  
Continuous Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Conjugate Function ◽  
Nonnegative Sequence

Let $A\in(-\infty,+\infty]$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have$x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma)\colon \sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series absolutely convergent in the half-plane $\operatorname{Re}s<A$, $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re}s=\sigma\}$ and $G(\sigma,F)=\sum |a_n|e^{\sigma\lambda_n}$ for each $\sigma<A$. It is proved that if $\ln G(\sigma,F)\le(1+o(1))\Phi(\sigma)$, $\sigma\uparrow A$, then the inequality$$\varlimsup_{\sigma\uparrow A}\frac{M(\sigma,F')}{M(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1$$holds, and this inequality is sharp. % Abstract (in English)

scholarly journals Generalized types of the growth of Dirichlet series

2015 ◽  
Vol 7 (2) ◽  
pp. 172-187 ◽  
Author(s):  
T.Ya. Hlova ◽  
P.V. Filevych
Keyword(s):  
Dirichlet Series ◽  
Half Plane ◽  
Maximum Modulus ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Maximal Term ◽  
Necessary And Sufficient ◽  
Nonnegative Sequence

Let $A\in(-\infty,+\infty]$ and $\Phi$ be a continuously on $[\sigma_0,A)$ function such that $\Phi(\sigma)\to+\infty$ as $\sigma\to A-0$. We establish a necessary and sufficient condition on a nonnegative sequence $\lambda=(\lambda_n)$, increasing to $+\infty$, under which the equality$$\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln M(\sigma,F)}{\Phi(\sigma)}=\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln\mu(\sigma,F)}{\Phi(\sigma)},$$holds for every Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, $s=\sigma+it$, absolutely convergent in the half-plane ${Re}\, s<A$, where $M(\sigma,F)=\sup\{|F(s)|:{Re}\, s=\sigma\}$ and $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge 0\}$ are the maximum modulus and maximal term of this series respectively.

scholarly journals The growth of the maximal term of Dirichlet series

2018 ◽  
Vol 10 (1) ◽  
pp. 79-81
Author(s):  
P.V. Filevych ◽  
O.B. Hrybel
Keyword(s):  
Dirichlet Series ◽  
Continuous Functions ◽  
Maximum Term ◽  
Maximal Term ◽  
Alpha Beta

Let $\Lambda$ be the class of nonnegative sequences $(\lambda_n)$ increasing to $+\infty$, $A\in(-\infty,+\infty]$, $L_A$ be the class of continuous functions increasing to $+\infty$ on $[A_0,A)$, $(\lambda_n)\in\Lambda$, and $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series such that its maximum term $\mu(\sigma,F)=\max_n|a_n|e^{\sigma\lambda_n}$ is defined for every $\sigma\in(-\infty,A)$. It is proved that for all functions $\alpha\in L_{+\infty}$ and $\beta\in L_A$ the equality$$\rho^*_{\alpha,\beta}(F)=\max_{(\eta_n)\in\Lambda}\overline{\lim_{n\to\infty}}\frac{\alpha(\eta_n)}{\beta\left(\frac{\eta_n}{\lambda_n}+\frac{1}{\lambda_n}\ln\frac{1}{|a_n|}\right)}$$ holds, where $\rho^*_{\alpha,\beta}(F)$ is the generalized $\alpha,\beta$-order of the function $\ln\mu(\sigma,F)$, i.e. $\rho^*_{\alpha,\beta}(F)=0$ if the function $\mu(\sigma,F)$ is bounded on $(-\infty,A)$, and $\rho^*_{\alpha,\beta}(F)=\overline{\lim_{\sigma\uparrow A}}\alpha(\ln\mu(\sigma,F))/\beta(\sigma)$ if the function $\mu(\sigma,F)$ is unbounded on $(-\infty,A)$.

scholarly journals Note to the behavior of the maximal term of Dirichlet series absolutely convergent in half-plane

2021 ◽  
Vol 56 (2) ◽  
pp. 144-148
Author(s):  
M.M. Sheremeta
Keyword(s):  
Dirichlet Series ◽  
Half Plane ◽  
Absolute Convergence ◽  
Maximal Term ◽  
Necessary And Sufficient

By $S_0(\Lambda)$ denote a class of Dirichlet series $F(s)=\sum_{n=0}^{\infty}a_n\exp\{s\lambda_n\} (s=\sigma+it)$ withan increasing to $+\infty$ sequence $\Lambda=(\lambda_n)$ of exponents ($\lambda_0=0$) and the abscissa of absolute convergence $\sigma_a=0$.We say that $F\in S_0^*(\Lambda)$ if $F\in S_0(\Lambda)$ and $\ln \lambda_n=o(\ln |a_n|)$ $(n\to\infty)$. Let$\mu(\sigma,F)=\max\{|a_n|\exp{(\sigma\lambda_n)}\colon n\ge 0\}$ be the maximal term of Dirichlet series. It is proved that in order that $\ln (1/|\sigma|)=o(\ln \mu(\sigma))$ $(\sigma\uparrow 0)$ for every function $F\in S_0^*(\Lambda)$ it is necessary and sufficient that $\displaystyle \varlimsup\limits_{n\to\infty}\frac{\ln \lambda_{n+1}}{\ln \lambda_n}<+\infty. $For an analytic in the disk $\{z\colon |z|<1\}$ function $f(z)=\sum_{n=0}^{\infty}a_n z^n$ and $r\in (0, 1)$ we put $M_f(r)=\max\{|f(z)|\colon |z|=r<1\}$ and $\mu_f(r)=\max\{|a_n|r^n\colon n\ge 0\}$. Then from hence we get the following statement: {\sl if there exists a sequence $(n_j)$ such that $\ln n_{j+1}=O(\ln n_{j})$ and $\ln n_{j}=o(\ln |a_{n_{j}}|)$ as $j\to\infty$,  then the functions $\ln \mu_f(r)$ and $\ln M_f(r)$ are or not are slowly increasing simultaneously.

Generalized ʎ-Closed Sets and (ʎ-ʏ)*-Continuous Function

2017 ◽  
Vol 4 (ICBS Conference) ◽  
pp. 1-17 ◽  
Author(s):  
Alias Khalaf ◽  
Sarhad Nami
Keyword(s):  
Continuous Function ◽  
Closed Sets

Almost Somewhat Near Continuity and Near Regularity

2021 ◽  
Vol 7 (1) ◽  
pp. 88-99
Author(s):  
Zanyar A. Ameen
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
One To One ◽  
Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

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