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THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 04 April 2018, pp. 435-445
- Print publication:
- June 2018
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- Article
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We prove the sharp inequality
$|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions 
$f$ with 
$a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$, such that where
$$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$
$H_{3,1}(f)$ is the third Hankel determinant 
$$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$
