The truncation technique for hyperbolic densities in combination with a version of the Ahlfors-Schwarz lemma [\textit{D. Minda}, Complex Variables, Theory Appl. 8, 129-144 (1987; Zbl 0576.30022)] is used in order to prove some classical covering and distortion results in the theory of analytic functions. Let \(H(D)\) be the family of all holomorphic functions in the unit disk \(D\) and let \(B_1=\{f\in H(D):f'(0)=1\) and \(|f'(z)|(1-|z |^2)\leq 1\), \(z\in D\}\). Denote by \(D(a,r)\) the open disk of radius \(r\) centered at \(\mathbb{C}\). Theorem. For any \(f\in B_1\), \(D(f(0),1/2)\subseteq f(D)\) and \(D(f(0),\pi/4)\subseteq co(f(D))\). The constants \(1/2\), \(\pi/4\) are the best possible. Let \(f^\#(w)={|f'(x)|\over 1+|f(w)|^2}\) and denote by \(D_s(a,r)\) the open spherical disk of radius \(r\) centered at \(a\in \mathbb{C}\). Theorem. Suppose \(f:D\to\widehat C\) is meromorphic, \(f^\#(0)=\alpha\), \(f^\#(z) (1-|z|^2)\leq\alpha\) \((z\in D)\), \(\alpha>0\). Then \(D_s(f(0)\), \(\arctan \alpha)\subseteq f(D)\). Theorem. Let \(f:D\to \mathbb{C}\) be holomorphic, \(f^\#(z)(1-|z|^2)\leq\alpha\) \((z\in D)\), \(\alpha>0\) and \(t\in(1,\infty)\) is the solution of \(\alpha= 2t/(t^2-1)\). If \(|f(0)|\geq t\) then \(|f'(0)|\leq 2|f(0)|(\log{|f(0)|\over t}+{t^2+1\over t^2-1})\). If \(|f(0)|\leq t\) then \(|f'(0) |\leq \alpha(1+|f(0)|^2)\).