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

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>