A value-distribution criterion for the class <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mspace width="3.33333pt"/> <mml:mi>log</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> and some related questions

Abstract

We give a necessary and sufficient condition for an analytic function in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> to have real part in class <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">log</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> . This condition contains the classical one of Zygmund; other variants are also given.