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ON CERTAIN CLOSE-TO-CONVEX FUNCTIONS

Published online by Cambridge University Press:  17 July 2023

MD FIROZ ALI*
Affiliation:
National Institute of Technology Durgapur, Mahatma Gandhi Road, Durgapur, Durgapur-713203, West Bengal, India e-mail: ali.firoz89@gmail.com
MD NUREZZAMAN
Affiliation:
National Institute of Technology Durgapur, Mahatma Gandhi Road, Durgapur, Durgapur-713203, West Bengal, India e-mail: nurezzaman94@gmail.com

Abstract

Let $\mathcal {K}_u$ denote the class of all analytic functions f in the unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$, normalised by $f(0)=f'(0)-1=0$ and satisfying $|zf'(z)/g(z)-1|<1$ in $\mathbb {D}$ for some starlike function g. Allu, Sokól and Thomas [‘On a close-to-convex analogue of certain starlike functions’, Bull. Aust. Math. Soc. 108 (2020), 268–281] obtained a partial solution for the Fekete–Szegö problem and initial coefficient estimates for functions in $\mathcal {K}_u$, and posed a conjecture in this regard. We prove this conjecture regarding the sharp estimates of coefficients and solve the Fekete–Szegö problem completely for functions in the class $\mathcal {K}_u$.

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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