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On partial solutions to conjectures for radius problems involving lemniscate of Bernoulli

Gurpreet Kaur
강원경기수학회 (The Kangweon-Kyungki Mathematical Society)|2023.12|한국수학논문집 (Korean Journal of Mathematics)|제31권 제4호|pp. 433-444

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참고문헌

키워드

convex functionstarlike functionlemniscate of Bernoulliradius problempositive real part
주제분류수학

초록

Given a function $f$ analytic in open disk centred at origin of radius unity and satisfying the condition $|f(z)/g(z)-1|<1$ for a analytic function $g$ with certain prescribed conditions in the unit disk, radii constants $R$ are determined for the values of $Rzf'(R z)/f(R z)$ to lie inside the domain enclosed by the curve $|w^2-1|=1$ (lemniscate of Bernoulli). This, in turn, provides a partial solution to the conjectures and problems for determination of sharp bounds $R$ for such functions $f$.

Abstract

Given a function $f$ analytic in open disk centred at origin of radius unity and satisfying the condition $|f(z)/g(z)-1|<1$ for a analytic function $g$ with certain prescribed conditions in the unit disk, radii constants $R$ are determined for the values of $Rzf'(R z)/f(R z)$ to lie inside the domain enclosed by the curve $|w^2-1|=1$ (lemniscate of Bernoulli). This, in turn, provides a partial solution to the conjectures and problems for determination of sharp bounds $R$ for such functions $f$.

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