Let $\mathcal{ A}$ denote the class of analytic functions $f$ on the unit disc $\mathbb{ D}=\{ z\in\mathbb{ C}:\;|z|<1\}$ normalized by $f(0)=0$ and $f^{ \prime}(0)=1$. In the present article, we consider and $\mathcal{ F}(c)$ the subclasses of $\mathcal{ A}$ are defined by \begin{ align*} \mathcal{ F}(c)=\bigg\{ f\in\mathcal{ A}:\;{ \rm Re}\;\bigg(1+\frac{ zf^{ \prime\prime}(z)}{ f^{ \prime}(z)}\bigg)>1-\frac{ c}{ 2},\;\;\mbox{ for some}\;c\in(0,3]\bigg\}, \end{ align*} and derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in and $\mathcal{ F}(c)$ expressed in terms of their value $f^{ \prime\prime}(0)$, in particular, when the quantity is equal to zero. Moreover, we obtain sharp bounds for distortion and growth theorems for functions in the class $\mathcal{ F}(c)$.
Categories: math.CV