In this paper the authors show that when there exist points on the boundary of the unit disc for which the inequalities \[ \text{Re}\left(z\,\frac{f'(z)}{f(z)}\right)>0,\quad \text{Re}\left(1+z\,\frac{f''(z)}{f'(z)}\right) >0 \] reduce to equalities, then the function \(f(z)\) reduces to a polynomial or to an entire function. In the last section of this paper, the authors criticize the result by using the Schwarzian derivative test (or Nehari test) which one obtained in section 2 and in section 3.