Let \(A\) be the class of functions analytic in the unit disc \(\mathbb{U}= \{z\in\mathbb{C}:|z|< 1\}\). We say that a function \(f\in A\) is subordinate to \(g\in A\) if there exists a function \(w\in A\) such that \(|w(z)|< 1\) and \(f(z)= g(w(z))\) for \(z\in \mathbb{U}\). In this case we write \(f\prec g\) or \(f(z)\prec g(z)\). Miller and Mocanu investigated many problems concerning so called ``differential subordination''. Using the methods of differential subordination the authors prove some new results. For example, the following implications are derived: \[ \Biggl|{f(z) f''(z)\over[f'(z)]^2}\Biggr|= \Biggl|\Biggl(1- {f(z) f''(z)\over [f'(z)]^2}\Biggr)- 1\Biggr|< M\Rightarrow \Biggl|{f(z)\over zf'(z)}- 1\Biggr|< M; \] \[ \Biggl|\Biggl\{\alpha\Biggl(1- {f(z) f''(z)\over [f'(z)]^2}\Biggr)+ (1-\alpha) {f(z)\over zf'(z)}\Biggr\}- 1\Biggr|< M\Rightarrow\Biggl|{f(z)\over zf'(z)}- 1\Biggr|< M. \]