In this paper, we find sharp upper bounds for the higher-order Schwarzian functionals \(\sigma _n(f)(0)\) , for \(n=3,4\) , in two subclasses of univalent functions defined using the exponential function. These subclasses are the exponential starlike class \(\begin{aligned} \mathcal {S}_\textrm{e}^*=\left\{ f\in \mathcal {S}:\frac{zf'(z)}{f(z)}\prec \textrm{e}^z\right\} \end{aligned}\) and the exponential convex class \(\begin{aligned} \mathcal {C}_\textrm{e}=\left\{ f\in \mathcal {S}:1+\frac{zf''(z)}{f'(z)}\prec \textrm{e}^z\right\} . \end{aligned}\) For functions in \(\mathcal {C}_\textrm{e}\) , we prove that \(\begin{aligned} |\sigma _3(f)(0)|\le 1,\quad |\sigma _4(f)(0)|\le 2. \end{aligned}\) For functions in \(\mathcal {S}_\textrm{e}^*\) , we show that \(\begin{aligned} |\sigma _3(f)(0)|\le 3,\quad |\sigma _4(f)(0)|\le 16\sqrt{\tfrac{3}{11}}. \end{aligned}\) As application, the sharp Schwarzian bounds induce corresponding sharp estimates for the initial Grunsky coefficients \(b_{n,m}\) , for exponentially subordinate functions.
