Conformal invariants, quasiconformal maps, and special functions

Distortion in the spherical metric under quasiconformal mappings

“…Theorem 2.7. Let λ n denote the Grötzsch constant (see [1], (8.38)); let η K,n be as in Lemma 3.1 andM 2 (K, n, R) be as in Lemma 4.4. Then…”

Hypergeometric Functions and Hyperbolic Metric

“…In this section, we recall some definitions and properties of special functions [R], and state two results that are particularly useful in proving monotonicity of a quotient of two functions [AVV1], [HVV].…”

“…In particular, Φ(1) = 0 and hence Φ(x) = d 0,1 (−x, −1) for x ≥ 1. Note also that Φ can be expressed by 2Φ(x) = − log(2µ(r)/π), where µ(r) denotes the well-known modulus of the Grötzsch ring D \ [0, r] given by µ(r) = (π/2)K(r )/K(r), for r ∈ (0, 1), and where D is the unit disk {z : |z| < 1} in the complex plane C (see [LV] or [AVV1] for details).…”

“…[AVV1,(3.13)]), inequalities for hypergeometric functions will lead to estimates for the hyperbolic metric.…”

Conical limit sets and continued fractions

“…The analogous result for m = 1 is that that the class CL(1) is invariant under quasisymmetric homeomorphisms of S 1 ; the proof is similar and we omit it. For the necessary background on quasisymmetric and quasiconformal mappings, see [2] or [14]. Because conical limit sets are locally determined, and locally quasiconformal homeomorphisms are locally extensible to globally quasiconformal homeormorphisms, we can deduce from Theorem 1.3 (iii) that CL(m) is invariant under locally quasiconformal homeomorphisms.…”