ON CERTAIN CLOSE-TO-CONVEX FUNCTIONS

Abstract

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 $\mathcal {K}_u$ .