The author derives some stronger inequalities than those given by the classical Schwarz lemma. In particular, let \(f\) be an analytic function mapping the unit disk into itself such that \(f(0)= 0\). Under these assumptions, the classical Schwarz lemma yields the inequality \(|f(z)|\leq|z|\). Here, the author proves the stronger inequality \[ |f(z)|\leq|z|{|z|+|f'(0)|\over 1+|f'(0)||z|}. \] (Note that the Schwarz lemma gives \(|f'(0)|\leq 1\).) If we assume, in addition, that there exists a point \(b\), \(|b|= 1\) such that \(|f(b)|= 1\), then it is an easy consequence of the Schwarz lemma that \(|f'(b)|\geq 1\). (Here, \(f'(b)\) can be interpreted in some weak senses, for example, as a radial derivative, and the inequality will still be valid.) The author improves this inequality, showing that \[ |f'(b)|\geq {2\over 1+|f'(0)|}. \] If the hypothesis that \(f(0)= 0\) is removed, then the inequality \[ |f'(b)|\geq {2\over 1+|F'(0)|} {1-|f(0)|\over 1+|f(0)|} \] is obtained, where \[ F(z)= {f(z)- f(0)\over 1-\overline{f(0)}f(z)}. \] Both of these inequalities are sharp for \(f(z)= z^2\). The proofs are both short and elementary.