THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS
By B. Kowalczyk, Adam Lecko, Young Jae Sim. Bulletin of the Australian Mathematical Society, 2018
145 citations recorded by the source corpus.
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 $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$ where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/482469aaffd6f155e74cf6de00f6486fef82c91d
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛* (2019, background): “[10] obtained the sharp bound of |H3(1)| for the class T (α) := {f ∈ A : Re(f(z)/z) > α, α ∈ [0, 1)} and in [11] establish the sharp bound of the same for the class of convex functions.”
- Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions of Order Alpha (2021, background): “Recently the second and the third Hankel determinants H2,2( f ) and H3,1( f ) have been examined for selected subclasses of A and particularly of S (see [5, 12] for further references).”