The Poincaré Metric on the Twice Punctured Plane and the Theorems of Landau and Schottky

Estimates of the hyperbolic metric on the twice punctured plane

“…On the unit circle, this is stronger than the fact that λ 01 (e iθ ) is decreasing on (0, π] given by Hempel [5] and Weitsman [14]. In Section 3, we use these estimates to get better lower bound on λ 01 (z) than (1.3).…”

“…The properties (a) and (d) are due to Hempel [5] and (b) was first proved by Bermant [2] (also see Solynin and Vuorinen [12]), and (c) was given by both Hempel [5] and Weitsman [14].…”

Topics in special functions. II

“…The hyperbolic metric of the twice-punctured plane and other types of plane domains [BEFS], [Bea1], [Bea2], [Ha], [Hem1], [Hem2], [ASVV], [SolV2], [SuV], [KY], [Ya], and [BHS] involves, for example, the constant a from (3.7). The function η K occurs, for instance, in [Vas1], [Vas2], [KY], and [Ya].…”

“…Several authors, including J. A. Hempel [Hem1], [Hem2], obtained bounds for Ψ(t, r). In 1997, G. Martin [Mart] proved, using the technique of holomorphic motions, that the sharp upper bound in Schottky's theorem is the same as Agard's distortion function η K (t), where K = (1+r)/(1−r), r = |z|.…”

The Distortion Theorem for quasiconformal mappings, Schottky’s Theorem and holomorphic motions

“…The referee points out that instead of stretching Teichmüller's argument one can also refer to Hempel's paper [11], where the maximal growth of the hyperbolic metric is studied. This together with Teichmüller's Theorem can also be used to prove Theorem 1.1.…”