Abstract: Let \mathcalA be a family of analytic functions with are the form f(z)=z+\sum\limits_n=2^\infty a_n z^n on the open unit disk D. A class of analytic functions S_e^* which are defined on the open unit cicle D and associated with exponential function is introduced, that is S_e^*=\left\f \mid \fracz f^\prime(z)f(z)\prec\mathrme^z \quad(f \in \mathcalA, z \in D)\right\. And the upper bound of the fourth-order Hankel determinant H₄(1) for this function class S_e^* associated with exponential function is given.