Borel type asymptotic relation for entire Dirichlet series and $h$-measure of an exceptional sets
Анотація
There are presented sufficient conditions for the entire Dirichlet series with monotonically increasing sequence of exponents providing validity of Borel-type relation outside some set $E$ of finite $h$-measure, i.e. $m_h E=\int_E dh(x)<+\infty$ with a positive continuously differentiable on $[0,+\infty)$ function $h,$ whose derivative $h'$ increases to infinity.
The corresponding Borel-type relation states that the logarithm of supremum of the Dirichlet series along an imaginary line behaves like as logarithm of maximal term of the Dirichlet series.
The conditions are given as the convergence of some auxiliary series constructed from the values of the function $h'$ and the sequence of exponents.
Посилання
O.B. Skaskiv, Behavior of the maximum term of a Dirichlet series that defines an entire function, Math. Notes, 37 (1985), no.1, 24–28. https://doi.org/10.1007/BF01652509
O.B. Skaskiv, On certain relations between the maximum modulus and the maximal term of an entire
Dirichlet series, Math. Notes, 66 (1999), no.2, 223–232. https://doi.org/10.1007/BF02674881
T.M. Salo, O.B. Skaskiv, O.M. Trakalo, On the best possible description of exceptional set in Wiman–Valiron theory for entire function, Mat. Stud., 16 (2001), no.2, 131–140.
P.V. Filevych, Asymptotic relations between the means of Dirichlet series and their application, Mat. Stud., 19 (2003), no.2, 127–140. https://doi.org/10.30970/ms.19.2.127-140
O.B. Skaskiv, O.V. Zrum, On an exceptional set in the Wiman inequalities for entire functions, Mat. Stud., 21 (2004), no.1, 13–24. https://doi.org/10.30970/ms.21.1.13-24 (in Ukrainian)
A.O. Kuryliak, O.B. Skaskiv, Wiman’s type inequality for analytic and entire functions and h-measure of an exceptional sets, Carpathian Math. Publ., 12 (2020), no.2, 492–498. https://doi.org/10.15330/cmp.12.2.492-498
O.B. Skaskiv, P.V. Filevych, On the size of an exceptional set in the Wiman theorem, Mat. Stud., 12 (1999), no.1, 31–36. (in Ukrainian) http://matstud.org.ua/texts/1999/12_1/12_1_031-036.pdf
T.M. Salo, O.B. Skaskiv, O.M. Trakalo, On the best possible description of exeptional set in Wiman-Valiron theory for entire function, Mat. Stud., 16 (2001), no.2, 131–140. http://matstud.org.ua/texts/2001/16 2/131-140.pdf
O.B. Skaskiv, O.M. Trakalo, On exeptional set in Borel relation for multiple entire Dirichlet series, Mat. Stud., 15 (2001), no.2, 163–172. (in Ukainian) http://matstud.org.ua/texts/2001/15 2/163-172.pdf
P.V. Filevych, An exact estimate for the measure of the exceptional set in the Borel relation for entire functions, Ukrainian Math. J., 53 (2001), no.2, 328–332. https://doi.org/10.1023/A:1010489609188
O.B. Skaskiv, Estimates of measures of exeptional sets in the Wiman-Valiron theory, Nonlinear. bound. probl. Collect. sc. proc., 11 (2001), Donetsk, 2001, 186–190.
O.B. Skaskiv, O.M. Trakalo, Sharp estimate of exceptional set in Borel’s relation for entire functions of several complex variables, Mat. Stud., 18 (2002), no.1, 53–56. (in Ukainian) https://doi.org/10.30970/ms.18.1.53-56
O.B. Skaskiv, D.Yu. Zikrach, On the best possible description of an exceptional set in asymptotic estimates for Laplace–Stieltjes integrals, Mat. Stud., 35 (2011), no.2, 131–141. https://doi.org/10.30970/ms.35.2.131-141
T.M. Salo, O.B. Skaskiv, Minimum modulus of lacunary power series and h-measure of exceptional sets, Ufa Math. J., 9 (2017), no.4, 135–144. https://doi.org/10.13108/2017-9-4-135
A.O. Kuryliak, I.E. Ovchar, O.B. Skaskiv, Wiman’s inequality for Laplace integrals, Int. Journal of Math. Analysis, 8 (2014), no.8, 381–385. https://doi.org/10.12988/ijma.2014.4232
A.O. Kuryliak, O.B. Skaskiv, O.V. Zrum, Levy’s phenomenon for entire functions of several variables, Ufa Math. J., 6 (2014), no.2, 111–120. https://doi.org/10.13108/2014-6-2-111
A.O. Kuryliak, O.B. Skaskiv, L.O. Shapovalovska, Wiman-type inequality for functions analytic in a polydisk, Ukr. Math. J., 68 (2016), no.1, 83–93. https://doi.org/10.1007/s11253-016-1210-9
A. Kuryliak, O. Skaskiv, S. Skaskiv, Levy’s phenomenon for analytic functions on a polydisk, European J. Math., 6 (2020), no.1, 138–152. https://doi:10.1007/s40879-019-00363-2
A.O. Kuryliak, O.B. Skaskiv, S.I. Panchuk, Bitlyan-Gol’dberg type inequality for entire functions and diagonal maximal term, Mat. Stud., 54 (2020), no.2, 135–145. https://doi.org/10.30970/ms.54.2.135-145
A.O. Kuryliak, O.B. Skaskiv, Wiman-type inequality in multiple-circular domains: Lévy’s phenomenon and exceptional sets, Ukr. Math. J., 74 (2022), no.5, 743–756. https://doi.org/10.1007/s11253-022-02098-y
A. Bandura, O. Skaskiv, I. Tymkiv, Composition of entire and analytic functions in the unit ball, Carpathian Math. Publ., 14 (2022), no.1, 95–104. https://doi.org/10.15330/cmp.14.1.95-104
A. Bandura, O. Skaskiv, L. Smolovyk, Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction, Dem. Math., 52 (2019), no.1, 482–489. https://doi.org/10.1515/dema-2019-0043
M.N. Sheremeta, The Wiman-Valiron method for Dirichlet series, Ukr. Math. J., 30 (1978), no.4, 376–383. https://doi.org/10.1007/BF01085861
M.N. Sheremeta, Asymptotic properties of entire functions defined by Dirichlet series and of their derivatives, Ukr. Math. J., 31 (1979), no.6, 558–564. https://doi.org/10.1007/BF01092538
M.N. Sheremeta, Analogues of Wiman’s theorem for Dirichlet series, Math. USSR-Sb., 38 (1981), no.1, 95–107. https://doi.org/10.1070/SM1981v038n01ABEH001322
O.B. Skaskiv, T.M. Salo, Entire Dirichlet series of rapid growth and new estimates for the measure of exceptional sets in theorems of the Wiman–Valiron type, Ukr. Math. J., 53 (2001), no.6, 978–991. https://doi.org/10.1023/A:1013308103502
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