Let $alpha$ be a complex number with $Re alpha > 0$. Let the functions $f$ and $g$ be analytic in the unit disc $E={z: |z|<1}$ and normalized by the conditions $f(0)=g(0)=0$, $f'(0)=g'(0)=1$. In the present article, we study the differential subordinations of the forms $$alpha frac{z^2 f''(z)}{f(z)} +frac {zf'(z)}{f(z)} prec alpha frac{z^2 g''(z)}{g(z)} +frac {zg'(z)}{g(z)},~z in E, $$ and $$frac{z^2 f''(z)}{f(z)} prec frac{z^2 g''(z)}{g(z)},~z in E.$$As consequences, we obtain a number of sufficient conditions for starlikeness of analytic maps in the unit disc. Here, the symbol ` $ prec ~$' stands for subordination.

Keywords: univalent function, starlike function, convex function, differential subordination