Advertisement

Ukrainian Mathematical Journal

, Volume 56, Issue 6, pp 1015–1022 | Cite as

On zeros, singular boundary functions, and modules of angular boundary values for one class of functions analytic in a half-plane

  • B. V. Vinnitskii
  • V. L. Sharan
BRIEF COMMUNICATIONS
  • 16 Downloads

Abstract

We obtain a description of zeros, singular boundary functions, and modules of angular boundary values of the functions f ≢ 0 that are analytic in the half-plane ℂ{in+} {= {{itz}:Re{itz} > 0}and satisfy the condition <Equation ID=”IE1”> <EquationSource Format=”MATHTYPE”> <![CDATA[ % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaada % qadaqaaiabgcGiIGGaaiab-v7aLjab-5da+iab-bdaWaGaayjkaiaa % wMcaamaabmaabaGaey4aIqIaam4yamaaBaaaleaacaaIXaaabeaaki % abg6da+iaaicdaaiaawIcacaGLPaaadaqadaqaaiabgcGiImXvP5wq % onvsaeHbfv3ySLgzaGqbciab+Pha6jabgIGioprr1ngBPrwtHrhAYa % qehuuDJXwAKbstHrhAGq1DVbacgaGae0NaHm0aaSbaaSqaaiabgUca % RaqabaaakiaawIcacaGLPaaacaGG6aWaaqWaaeaacqGFMbGzdaqada % qaaiab+Pha6bGaayjkaiaawMcaaaGaay5bSlaawIa7aiabgsMiJkaa % dogadaWgaaWcbaGaaGymaaqabaGcciGGLbGaaiiEaiaacchadaqada % qaamaabmaabaGae83WdmNae83kaSIae8xTdugacaGLOaGaayzkaaWa % aqWaaeaacqGF6bGEaiaawEa7caGLiWoacqWF3oaAdaqadaqaamaaem % aabaGae4NEaOhacaGLhWUaayjcSdaacaGLOaGaayzkaaaacaGLOaGa % ayzkaaaaaa!7C75! ]]></EquationSource> <EquationSource Format=”TEX”> <![CDATA[ \left( {\forall \varepsilon > 0} \right)\left( {\exists c_1 > 0} \right)\left( {\forall z \in \mathbb{C}_+} \right):\left| {f\left( z \right)} \right| \leqslant c_1 \exp \left( {\left( {\sigma + \varepsilon} \right)\left| z \right|\eta \left( {\left| z \right|} \right)} \right) ]]></EquationSource></Equation> where 0 ≤ σ \s< + ∞ is a given number and ηis a positive function continuously differentiable on [0; + ∞]and such that <Equation ID=”IE2”> <EquationSource Format=”MATHTYPE”> <![CDATA[ % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaada % WcgaqaaiaadshacuaH3oaAgaqbamaabmaabaGaamiDaaGaayjkaiaa % wMcaaaqaaiabeE7aOnaabmaabaGaamiDaaGaayjkaiaawMcaaiabgk % ziUkaaicdaaaaaaa!4481! ]]></EquationSource> <EquationSource Format=”TEX”> <![CDATA[ {{t\eta ‘\left( t \right)} \mathord{\left/ {\vphantom {{t\eta ‘\left( t \right)} {\eta \left( t \right) \to 0}}} \right. \kern-\nulldelimiterspace} {\eta \left( t \right) \to 0}} ]]></EquationSource></Equation>

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Vynnytskyi, B. V., Sharan, V. L. 1999On the zeros of functions of improved formal order that are analytic in the half-planeUkr. Mat. Zh.51904909Google Scholar
  2. 2.
    Sharan, V. L. 1998On singular boundary functions for certain classes of functions analytic in the half-planeContemporaryProblems in MathematicsInstitute of Mathematics, Ukrainian Academy of SciencesKiev208210[in Ukrainian], Abstracts of the International Conference, Part 3Google Scholar
  3. 3.
    Grishin, A. F. 1990First-order functions that are subharmonic in a half-plane and a Tauberian theoremTeor. Funkts. Funktsional. Anal. Prilozhen.53879458, No. 6, 554–559 (1992)Google Scholar
  4. 4.
    Grishin, A. F. 1992First-Order Subharmonic FunctionsAuthor’s Abstract of Doctoral-Degree Thesis (Physics and Mathematics)Kharkovin RussianGoogle Scholar
  5. 5.
    Koosis, P. 1984Introduction to HpSpacesMirMoscowRussian translationGoogle Scholar
  6. 6.
    Govorov, N. V. 1986Riemann Boundary-Value Problem with Infinite IndexNaukaMoscowin RussianGoogle Scholar
  7. 7.
    Vynnytskyi, B. V., Sharan, V. L. 2000On factorization of one class of functions analytic in the half-planeMat. Stud.144148Google Scholar
  8. 8.
    Seneta, E. 1985Regularly Varying FunctionsNaukaMoscowRussian translationGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • B. V. Vinnitskii
    • 1
  • V. L. Sharan
    • 1
  1. 1.Drohobych Pedagogic UniversityDrohobych

Personalised recommendations