Hankel and Toeplitz determinants for a subclass of analytic functions

Abstract

Let the function $f\left( z \right) =z+\sum_{k=2}^{\infty}a{_{k}}z {^{k}}\in A$ be locally univalent for $z \in \mathbb{D}%
:=\{z \in \mathbb{C}:{|}z {|}<1\}$ and $0\leq\alpha<1$.
Then, $f$\textit{\ }$\in $ $M(\alpha )$ if and only if
\begin{equation*}
\Re\Big( \left( 1-z ^{2}\right) \frac{f(z )}{z }\Big) >\alpha,\quad
z \in \mathbb{D}.
\end{equation*}%
Due to their geometrical characteristics, this class has a significant
impact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant
\begin{equation*}
\left\vert H_{2}\left( 2\right) \left( f\right) \right\vert =\left\vert
a_{2}a_{4}-{a_{3}^{2}}\right\vert
\end{equation*}
and some Toeplitz determinants
\begin{equation*}
\left\vert {T}_{3}\left( 1\right) \left( f\right) \right\vert =\left\vert 1-2%
{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}\right\vert,\ \
\left\vert {T}_{3}\left( 2\right) \left( f\right) \right\vert =\left\vert {%
a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}\right\vert
\end{equation*}
of a subclass of analytic functions $M(\alpha )$ in the open unit disk $%
\mathbb{D}$.