The sharp bound of the third Hankel determinant for starlike functions
Bogumiła Kowalczyk, Adam Lecko, Derek K. Thomas
Abstract
Abstract In this paper, we prove the sharp inequality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mn>4</m:mn> <m:mo>/</m:mo> <m:mn>9</m:mn> </m:mrow> </m:mrow> </m:math> \lvert H_{3,1}(f)\rvert\leq 4/9 for starlike functions 𝑓, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>f</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> H_{3,1}(f) is the third Hankel determinant, thus solving a long-standing problem.