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INVARIANCE OF THE COEFFICIENTS OF STRONGLY CONVEX FUNCTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 95 / Issue 3 / June 2017
- Published online by Cambridge University Press:
- 23 November 2016, pp. 436-445
- Print publication:
- June 2017
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- Article
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- You have access
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Let the function
$f$ be analytic in 
$\mathbb{D}=\{z:|z|<1\}$ and given by 
$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. For 
$0<\unicode[STIX]{x1D6FD}\leq 1$, denote by 
${\mathcal{C}}(\unicode[STIX]{x1D6FD})$ the class of strongly convex functions. We give sharp bounds for the initial coefficients of the inverse function of 
$f\in {\mathcal{C}}(\unicode[STIX]{x1D6FD})$, showing that these estimates are the same as those for functions in 
${\mathcal{C}}(\unicode[STIX]{x1D6FD})$, thus extending a classical result for convex functions. We also give invariance results for the second Hankel determinant 
$H_{2}=|a_{2}a_{4}-a_{3}^{2}|$, the first three coefficients of 
$\log (f(z)/z)$ and Fekete–Szegö theorems.
