Schwarzian Norm Estimates for Analytic Functions Associated with Convex Functions

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disc $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ normalized by $f(0)=0$ and $f^{\mathrm{\prime }}(0)=1$. In the present article, we consider and $\mathcal{F}(c)$ the subclasses of $\mathcal{A}$ are defined by \begin{align*} $\mathcal{F}$©=\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 $f^{\mathrm{\prime }\mathrm{\prime }}(0)$, in particular, when the quantity is equal to zero. Moreover, we obtain sharp bounds for distortion and growth theorems for functions in the class $\mathcal{F}(c)$.