Majorization problems for certain starlike functions associated with the exponential function

"Let $\mathcal{S}^*_e$ and $\mathcal{S}^*_B$ denote the class of analytic functions $f$ in the open unit disc normalized by $f(0)=0=f'(0)-1$ and satisfying, respectively, the following subordination relations: $$ \frac{zf'(z)}{f(z)}\prec e^z\quad{\rm and}\quad\frac{zf'(z)}{f(z)}\prec e^{e^z-1}.$$ In this article, we investigate majorization problems for the classes $\mathcal{S}^*_e$ and $\mathcal{S}^*_B$ without acting upon any linear or nonlinear operators."