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Exact estimates for integrals involving Dirichlet series with nonnegative coefficients

Author(s): Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 127 (1999), 2417-2422.
MSC (1991): Primary 30B50; Secondary 30B10
Posted: April 9, 1999
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Abstract: We consider the Dirichlet series

\begin{displaymath}\sum^\infty _{k=2} a_k k^{-1-x} =: f(x), \quad x>0, \end{displaymath}

with coefficients $a_k \ge 0$ for all $k$. Among others, we prove exact estimates of certain weighted $L^p$-norms of $f$ on the unit interval $(0,1)$ for any $0<p<\infty$, in terms of the coefficients $a_k$. Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for integrals involving the former one by relying on exact estimates for integrals involving the latter one. As a by-product, we obtain an analogue of the Cauchy-Hadamard criterion of (absolute) convergence of the more general Dirichlet series

\begin{displaymath}\sum^\infty _{k=1} c_k k^{-z}, \quad z:= x+iy, \end{displaymath}

with complex coefficients $c_k$.

References:

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L. Leindler, Improvements of some theorems of Mulholland concerning Dirichlet series, Acta Sci. Math. (Szeged), 58(1993), 281-297. MR 95g:40002
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L. Leindler, On power series with positive coefficients, Analysis Math. 20 (1994), 205-211. MR 95k:40010
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L. Leindler and A. Meir, Inequalities concerning Dirichlet series and integrals, Acta Sci. Math. (Szeged) 59(1994), 209-220. MR 95h:26025
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M. Mateljevic and M. Pavlovic, $L^p$-behavior of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc. 87 (1983), 309-316. MR 84g:30034
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Additional Information:

Ferenc Móricz
Affiliation: Bolyai Institute, University of Szeged, Aradi Vertanuk Tere 1, 6720 Szeged, Hungary
Email: moricz@math.u-szeged.hu

DOI: 10.1090/S0002-9939-99-04851-0
PII: S 0002-9939(99)04851-0
Keywords: Dirichlet series, power series, Cauchy-Hadamard criterion for Dirichlet series, line of convergence, $L^p$-behavior, weight function, slowly decreasing function, Cauchy condensation principle
Received by editor(s): November 19, 1997
Posted: April 9, 1999
Additional Notes: This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant T 016 393.
Communicated by: Frederick W. Gehring
Copyright of article: Copyright 1999, American Mathematical Society

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