Integral means of univalent functions on an annulus.

We study pointwise and integral analogues of the Schwarz lemma for holomorphic functions on an annulus. Counterexamples reveal that a pointwise version of the Schwarz lemma for holomorphic functions on an annulus would not be possible. If holomorphic functions on an annulus are either not univalent or not normalized by having zero constant coefficients in the Laurent series expansion then the pointwise analogue of the Schwarz lemma fails in a more dramatic fashion. We also examine an integral means version of the Schwarz lemma for univalent holomorphic functions of an annulus and its relation with normalization and univalence of the functions under consideration. It turns out that an integral means version of the Schwarz lemma also fails for an annulus even when the functions are univalent and normalized. If either the univalence condition or the normalization condition is dropped then the integral means version of the Schwarz lemma fails in a more dramatic way. Finally, we determine a par- tial analogue of the integral means version of the Schwarz lemma for univalent holomorphic functions on an annulus. We also show that an estimate obtained by Kirwan & Schober in is not valid. We also obtain estimates for the integral means of certain classes of univalent functions that need not be bounded on an annulus and estimates for the integral means of derivatives of such functions