By Pick's invariant form of Schwarz's lemma, an analytic function B (z) which is bounded by one in the unit disk D = {z: |z| < 1} satisfies the inequality
![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022081612632-0210:S0025579300000085_inline1.gif?pub-status=live)
at each point α of D. Recently, several authors [2, 10, 11] have obtained more general estimates for higher order derivatives. Best possible estimates are due to Ruscheweyh [12]. Below in §2 we use a Hilbert space method to derive Ruscheweyh's results. The operator method applies equally well to operator-valued functions, and this generalization is outlined in §3.