Abstract
First, we consider integrals of the form
over the unit interval (0, 1) or the interval (1, ∞) or the half-line (0, ∞), wherea(x)≥0 and is integrable on the interval in question. These integrals are related to the Dirichlet series
, where the numbersa m ≥0. We survey certain known results in a new formulation in order to reveal the difference in behavior between the functions which are integrable on either (0, 1) or (1, ∞). Their proofs can be read out from the existing literature.
Second, we extend these results from single to double related integrals, while making distinction among the functionsa(x, y) which are integrable on either (0, 1)2 or (0, 1)×(1, ∞) or (1, ∞)×(0, 1) or (1, ∞)2. The case wherea(x, y) is integrable on (0, ∞)2 is also included.
Abstract
Сначала рассматриваутся интегралы вида
на единичном интервале (0,1), или на интервале (1,∞), или на полуоси (0,∞), где функция а(х)>0 и интегрируема на соответствуушем интервале. Эти интегралы родственны рядам Дирихле
где числаa m>-0. Мы даем обэор некоторых иэвестных реэулятатов в новои формулировке, чтобы покаэатя раэницу поведения функции, интегрируемых на (0,1) или на (1,∞). Ранее Это не отмечалося. Эатем мы распространяем Эти реэулятаты с одномерных на соответствуушие двоиные интегралы, где нузно раэличатя функцииa(x,y), интегрируемые на (0,1)2, или на (0,1)×(1,∞), или на (1,∞)×(0,1), или на (1,∞)2. Рассматривается такзе случаи, когдаa(x, y) интегрируема на (0,∞)2.
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References
L. Leindler, Generalizations of some theorems of Mulholland concerning Dirichlet series,Acta Sci. Math. (Szeged),57(1993), 401–408.
L. Leindler, Improvements of some theorems of Mulholland concerning Dirichlet series,Acta Sci. Math. (Szeged),58(1993), 281–297.
L. Leindler, Inequalities on Dirichlet series with positive coefficients and related integrals,Analysis Math.,24(1998), 201–211.
L. Leindler andA. Meir, Inequalities concerning Dirichlet series and integrals,Acta Sci. Math. (Szeged),59(1994), 209–220.
M. Mateljevic andM. Pavlovic,L p-behavior of power series with positive coefficients and Hardy spaces,Proc. Amer. Math. Soc.,87(1983), 309–316.
H. P. Mulholland, Some theorems on Dirichlet series with positive coefficients and related integrals,Proc. London Math. Soc.,29(1929), 281–292.
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This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant T 029094.
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Móricz, F., Мориц, Ф. Exact estimates for integrals related to dirichlet series. Anal Math 25, 87–102 (1999). https://doi.org/10.1007/BF02908428
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DOI: https://doi.org/10.1007/BF02908428