Growth, Distortion, and Schwarzian Norm Estimates for Exponentially Convex Functions

In this paper, we investigate the growth, distortion, pre-Schwarzian and Schwarzian norms of functions in the exponentially convex class ${\mathcal{C}}_{e^{\lambda }}$, $0<\lambda \le \pi \mathrm{/}2$, defined by $1+z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)\prec e^{\lambda z}$. By representing the associated Schwarz function explicitly, we derive parameter-dependent estimates for $f$, $f^{\mathrm{\prime }}$, and the pre-Schwarzian derivative. We further obtain Schwarzian norm estimates under both the general normalization and the additional condition $f^{\mathrm{\prime }\mathrm{\prime }}(0)=0$. The corresponding extremal problems are analyzed through suitable Schwarz functions, and the dependence of the resulting bounds on the exponential parameter is made explicit.