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ON CERTAIN CLOSE-TO-CONVEX FUNCTIONS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 17 July 2023, pp. 365-375
- Print publication:
- April 2024
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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 
$\mathcal {K}_u$.
INTEGRAL MEANS OF UNIVALENT FUNCTIONS ON AN ANNULUS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 103 / Issue 3 / June 2021
- Published online by Cambridge University Press:
- 21 September 2020, p. 512
- Print publication:
- June 2021
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