On Strongly Starlike Functions Related to the Bernoulli Lemniscate
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Abstract
Let $\mathcal{S}^{\ast}_{L}(\lambda)$ be the class of functions $f$, analytic in the unit disc $\Delta=\{z:|z|<1\}$, with the normalization $f(0)=f'(0)-1=0$, which satisfy the condition
\begin{equation*}
\frac{zf'(z)}{f(z)}\prec \left(1+z\right)^{\lambda},
\end{equation*}
where $\prec$ is the subordination relation. The class $\mathcal{S}^{\ast}_{L}(\lambda)$ is a subfamily of the known class of strongly starlike functions of order $\lambda$. In this paper,
the relations between $\mathcal{S}^{\ast}_{L}(\lambda)$ and other classes geometrically defined are considered. Also, we obtain some characteristics such as, bounds for coefficients, radius of convexity, the Fekete-Szeg\"{o} inequality, logarithmic coefficients and the second Hankel determinant inequality for functions belonging to this class. The univalent functions $f$ which satisfy the condition
\begin{equation*}
\Re\left\{1+\frac{zf''(z)}{f'(z)}\right\}<1+\frac{\lambda}{2},\qquad
(z \in \Delta)\end{equation*}
are also considered here.
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