Remarks on a Paper by Lipman Bers

to show that certain estimates by Nehari are sharp. The range of the complex parameter a for which these functions are schlicht in the unit disk can be treated as an example of a one-dimensional subspace in the Teichmfiller space of an annulus, and as such can be used to illustrate some of the ideas developed by Bers in [1]. In these remarks we carry out the details of this point of view. If, to fit into the framework, the setting is translated from the unit disk to the lower half-plane L, the corresponding family of mappings arise as solutions to the schwarzian differential equation { W, z} = ( for ( in the subspace spanned by the element l/z2 in B2(L, r), r a cyclic fuchsian group generated by a loxodromic transformation. The intersection of this subspace with the Teichmiiller space T(r) can be described explicity, and we write out formulas for the various mappings and group homomorphisms associated to each element 9 in or on the boundary of T(r). These elements are then interpreted, both qualitatively and quantitatively, as deformations of an annulus, and we point out a natural correspondence between this deformation space and the Teichmiiller space of a torus or a punctured torus. Unfortunately the correspondence with the space of a punctured torus, in the form that we would like, is not explicit enough, and we have left it at equation (11). A solution of this equation, explicit enough to answer the question posed there, would be of interest in that it would lead to the first non-trivial example of a Teichmfiller space in the setting of the preceding paper. The example does illustrate that a cusp on the boundary can be reached by "twisting" without "pinching," showing along the way that the expression in (8), shown by Bers in [1, ? 6] to have a universal upper bound, does not have a positive lower bound. Although the example itself is trivial, the deformations involved have analogues for general Riemann surfaces, where we might expect the qualitative behavior to be the same. The "twisting" and "pinching" of the annulus, for example, can be carried out near a boundary curve of any open surface; while the corresponding deformation of the torus