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A refinement of the Sobolev imbedding theorem, due to Trudinger, is shown to be optimal in a natural sense.

For the twice-punctured unit disc U p = {z : |z| < 1, 2 ^ ±p} , where 0 < p < 1, we obtain precise descriptions for p near 0 of various parameters associated with the uniformisation of n p by the upper half-plane U = {T : Im T > 0}. These parameters include the hyperbolic length of the geodesic surrounding ±p, the so-called "accessory parameters", and the "proximity parameter" which determines the behaviour of the hyperbolic density near the punctures of fi p .

ABSTRACT. For the plane regions Oi = {\z\ < R,z / 0,1} with R > 1, and 0.2 = C \ {0, l,p} with \p\ = R > 1, we describe, as R -» oo, the hyperbolic lengths of the geodesies surrounding 0 and 1. Upper and lower bounds for the lengths are also stated, and these results are used to obtain inequalities, which are precise in a certain sense, for the length of the geodesic surrounding 0 and 1 in an arbitrary plane region 0 satisfying 121 c 0 C 0,2.