Regular polygon

Convex function

Geometry

Combinatorics

Mathematics

Prime (order theory)

Bulletin of the Australian Mathematical Society

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 $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}&gt;0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|&lt;1\},\end{eqnarray}$$ where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} &amp; a_{2} &amp; a_{3}\\ a_{2} &amp; a_{3} &amp; a_{4}\\ a_{3} &amp; a_{4} &amp; a_{5}\end{array}\right|.\end{eqnarray}$$

THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS