THE GROWTH OF DIRICHLET SERIES ABSOLUTELY CONVERGENT IN A HALF-PLANE IN TERMS OF GENERALIZED TYPES
Keywords:
Dirichlet series, maximum modulus, maximal term, generalized type.Abstract
Let$A\in(−\infty,+\infty)$. We establish a necessary and sufficient condition on a nonnegative sequence $(\lambda_n),$ increasing to $+\infty$, under which there exists a function $\Phi$, convex and increasing to $+\infty$ on $(-\infty,A)$, such that for every Dirichlet series of the form $F(s) =\sum a_n e^{s\lambda_n},$ $s =\sigma+it,$ absolutely convergent in the half-plane $\mathrm{Re} s<A$ we have $\varlimsup_{\sigma\up A}\frac{\ln\sup\{|F(s)|: \mathrm{Re}\ s = \sigma\}}{\Phi(\sigma)}=\varlimsup_{\sigma\up A} \frac{\ln\max\{|a_n|e^{\sigma\lambda_n: \ n\ge 0}\}}{\Phi(\sigma)}.$
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Published
2019-02-18
How to Cite
Глова, Т. Я. (2019). THE GROWTH OF DIRICHLET SERIES ABSOLUTELY CONVERGENT IN A HALF-PLANE IN TERMS OF GENERALIZED TYPES. PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, (1(33), 103–109. Retrieved from https://pvntsh.nung.edu.ua/index.php/number/article/view/106
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Section
Mathematics and Mechanics
