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On certain relations between the maximum modulus and the maximal term of an entire dirichlet series

Authors:

O. B. Skaskiv

Affiliation:

(1) I. Franko Lvov State University, USSR

Abstract:

For an entire Dirichlet series $F(z){text{ = }}sumnolimits_{{text{n = 0}}}^{{text{ + }}infty } {ane^{{text{z}}lambda {text{n}}} ,{text{ }}0 leqslant lambda n uparrow + infty {text{ }}(n to + infty ),}$ , sufficient conditions on the exponents $$lambda _n $$

are established such that the following relations hold outside a set of finite measure asx→+∞:

$psi (In supleft{ {|F(x + iy)|:y in mathbb{R}} right}) = (1 + o(1))psi (In max{ |a_n |e^{xlambda n} :n geqslant } ),$

, where ψ(x) is a function increasing to +∞ and such thatx≤ψ(x)≤e ^x (x≥0). Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 282–292, August, 1999

Keywords:

entire Dirichlet series maximum modulus of a Dirichlet series maximal term of a Dirichlet series set of finite measure Chebyshev’ s inequality

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