On Baernstein’s theorem for minimal surfaces

Abstract This paper is devoted to the development of Beckenbach’s theory of the meromorphic minimal surfaces. We consider the relationship between the number of separated maximum modulus points of an entire minimal surface S and the Lebesgue measure of the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>θ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi>π</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>π</m:mi> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>log</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mi>𝐱</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>θ</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∥</m:mo> </m:mrow> </m:mrow> <m:mo>&gt;</m:mo> <m:mrow> <m:mi>α</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mo>⁡</m:mo> <m:mi>M</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{\theta\in[-\pi,\pi]:\log{\|\mathbf{x}(re^{i\theta})\|&gt;\alpha\log{M(r,S)}}\}} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>α</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0\leq\alpha&lt;1} ). The results of Arima and Baernstein are generalized. We also give examples showing that the obtained estimate is sharp.