Sharp bounds for the second Hankel determinant of logarithmic coefficients for parabolic starlike and uniformly convex functions of order alpha

Document Type : Original Article

Author

Serap Bulut

Kocaeli University, Faculty of Aviation and Space Sciences, Arslanbey Campus, 41285 Kartepe-Kocaeli, Turkey

10.22034/kjm.2023.395878.2856

Abstract

Let $\mathcal{A}$ denote the class of analytic functions $f$ in the open unit disk $\mathbb{U}$ normalized by $f(0)=f^{\prime }(0)-1=0,$ and let$\mathcal{S}$ be the class of all functions $f\in \mathcal{A}$ that are univalent in $\mathbb{U}$. For a function $f\in \mathcal{S}$, the logarithmic coefficients $\delta _{n}\,\left( n=1,2,3,\ldots \right) $ are defined by
\begin{equation*}
\log \frac{f(z)}{z}=2\sum_{n=1}^{\infty }\delta _{n}z^{n}\qquad \left( z\in
\mathbb{U}\right) .
\end{equation*}
For $0\leq \alpha <1,$ let $\mathcal{S}_{p}\left( \alpha \right) $ and $
\mathcal{UCV}\left( \alpha \right) $ denote the classes of functions $f\in
\mathcal{A}$ such that
\begin{equation*}
\left\vert \frac{zf^{\prime }(z)}{f(z)}-1\right\vert <\left( 1-2\alpha
\right) +\Re \left( \frac{zf^{\prime }(z)}{f(z)}\right) \qquad \left( z\in
\mathbb{U}\right)
\end{equation*}
and
\begin{equation*}
\left\vert \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right\vert <2\left(
1-\alpha \right) +\Re \left( \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}
\right) \qquad \left( z\in \mathbb{U}\right),
\end{equation*}
respectively. In the present paper, we determine the sharp upper bound for $\left\vert \delta _{n}\right\vert \;\left( n=1,2,3,\ldots \right) $ of functions $f$ belonging to the classes $\mathcal{S}_{p}\left( \alpha \right) $. Also, we obtain upper bounds for $\left\vert \delta _{n}\right\vert \;\left(n=1,2,3\right) $ of functions belonging to the class $\mathcal{UCV}\left(\alpha \right)$.

Keywords

Analytic function

univalent function

logarithmic coefficients

subordination

parabolic-starlike functions

Subjects

• 30 Functions of a complex variable

---