CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 11, Pages 250–270 (November 8, 2007) S 1088-4173(07)00168-3 TOPICS IN SPECIAL FUNCTIONS. II G. D. ANDERSON, M. K. VAMANAMURTHY, AND M. VUORINEN Dedicated to Seppo Rickman and Jussi V¨ais¨al¨a. Abstract. In geometric function theory, conformally invariant extremal prob- lems often have expressions in terms of special functions. Such problems occur, for instance, in the study of change of euclidean and noneuclidean distances under quasiconformal mappings. This fact has led to many new results on special functions. Our goal is to provide a survey of such results. 1. Introduction In our earlier survey [AVV5] we studied classical special functions, such as the gamma function, the hypergeometric function 2F1, and the complete elliptic inte- grals. In this survey our focus is different: We will concentrate on special functions that occur in the distortion theory of quasiconformal maps. The definitions of these functions involve the aforementioned functions, and therefore these two surveys are closely connected. However, we hope to minimize overlap with the earlier survey. Extremal problems of geometric function theory lead naturally to the study of special functions. L. V. Ahlfors’s book [Ah] on quasiconformal mappings, where the extremal length of a curve family is a crucial tool, discusses among other things the extremal problems of Gr¨otzsch and Teichm¨uller and provides methods for studying the boundary correspondence under quasiconformal automorphisms of the upper half plane or the unit disk. These problems have led to many ramifications and have greatly stimulated later research. Reading [Ah], it becomes apparent that special functions, such as elliptic functions and integrals, are crucial for such applications. Recall that the modulus of a plane ring domain is equal to log R if the ring can be mapped conformally onto the annulus {z ∈ C :1< |z| <R}. The capacity of the ring is defined as 2π/ log R. The problem of Gr¨otzsch asks to determine the maximal conformal modulus µ(r) of all ring domains separating the points {0,r}, 0 <r<1, on the real axis from the unit circle. The extremal ring, the so-called Gr¨otzsch ring, is the unit disk with the segment [0,r] removed, with modulus given by π K(r) π/2 dt (1.1) µ(r) ≡ , K(r) ≡ . K 2 2 (r) 0 1 − r2 sin t Received by the editors March 30, 2007. 2000 Mathematics Subject Classification. Primary 30C62; Secondary 33E05, 33E99. Key words and phrases. Schwarz Lemma, elliptic integral, distortion function, quasiconformal, Schottky, monotonicity, convexity, Teichm¨uller capacity. The authors thank the Finnish Mathematical Society, the Finnish Academy of Sciences, and the Academy of Finland (grant no. 107317) for their support of this research. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 250 TOPICS IN SPECIAL FUNCTIONS. II 251 √ Here r = 1 − r2,0<r<1, and K = K(r) is called the complete elliptic integral of the first kind. For later use, for K>0 we define two homeomorphisms ϕK :[0, 1] → [0, 1],ϕK (0) = 0, and ηK :[0, ∞) → [0, ∞),ηK (0) = 0, by −1 (1.2) ϕK (r) ≡ µ (µ(r)/K), 0 <r≤ 1, 2 ≡ ϕK (s) (1.3) ηK (t) ,t>0,s= t/(1 + t) . ϕ1/K (s ) Teichm¨uller’s problem is to find the maximal modulus, or equivalently, the min- imal capacity p(z), of all ring domains separating the points {0, 1} from {z,∞}, where z ∈ C \{0, 1} . In the general case, p(z) is expressible in terms of elliptic integrals with complex argument [Kuz1, p. 192], [Kuz2]. If z>1isreal,theex- tremal ring is called the Teichm¨uller ring; it has complementary components [0, 1] and [z,∞), and p(z) can be expressed in terms of the above function µ (see [LV]). For example, the Gr¨otzsch and Teichm¨uller problems in the plane lead to the special function µ in (1.1), the related function τ (see Section 5), and their n- dimensional analogs. Such problems and special functions have been studied by Ahlfors and Beurling [AhB], Hersch and Pfluger [HP], Fuglede [Fu], Gehring and V¨ais¨al¨a[GV],V¨ais¨al¨a[V¨a], Lehto and Virtanen [LV], and the present authors [AVV2], among others. Hersch and Pfluger [HP] generalized the classical Schwarz Lemma for analytic functions to the class QCK (B),K ≥ 1, of K-quasiconformal mappings of the unit disk B = {z ∈ C : |z| < 1} into itself fixing the origin. They proved that, for r ∈ (0, 1), (1.4) UK (r) ≡ sup{|f(z)| : f ∈ QCK (B), |z|≤r} has the expression (1.5) UK (r)=ϕK (r) . The Hersch-Pfluger result in (1.5) is known as the Schwarz Lemma for quasi- conformal mappings (cf. [LV, p. 64]). Next, for K ≥ 1andt>0, let (1.6) VK (t) ≡ sup{|f(z)| : |z|≤t} , where the supremum is taken over all plane K-quasiconformal mappings fixing 0, 1, and ∞. In 1968, S. Agard [Ag] proved that (1.7) VK (t)=ηK (t) , where ηK (t) is as in (1.3). For t =1,wesetηK (1) = λ(K). The special case t =1 of (1.7) was proved by Lehto, Virtanen, and V¨ais¨al¨a [LVV]. The function λ(K)is of particular interest for quasiconformal mappings, since the boundary values of a K-quasiconformal homeomorphism w of the upper half plane onto itself satisfy the inequality 1 w(x + t) − w(x) ≤ ≤ λ(K) λ(K) w(x) − w(x − t) for all real x and t, t = 0 [LV, p. 81], [BAh]. Accordingly, several results in distortion theory of quasiconformal mappings depend on λ(K). For t>0, let A(t)={f : B → C\{0, 1}|f is analytic with |f(0)| = t}. 252 G. D. ANDERSON, M. K. VAMANAMURTHY, AND M. VUORINEN In 1904, F. Schottky (cf. [Hi, pp. 261–262], [Ha, p. 702]) proved that there is a func- tion Ψ(t, r) such that |f(z)|≤Ψ(t, |z|) for all f ∈A(t)andz ∈ B (wetakeΨ(t, r) to be the smallest such function). 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 theo- rem is the same as Agard’s distortion function ηK (t), where K =(1+r)/(1−r),r= |z|. As pointed out by S.-L. Qiu [An], this result is also implicit in the earlier works of Agard [Ag], Teichm¨uller [T], and Lehto, Virtanen, and V¨ais¨al¨a[LVV].Akey idea is to reduce the problem to the study of the hyperbolic density of the twice- punctured plane. Recent results for these extremal functions ϕ, λ,andη will be reviewed in Sec- tions 2, 3, and 4, respectively. The methods used in these studies are based on classical analysis. Some of the technical tools are the l’Hˆopital Monotone Rule and the Power Series Quotient Monotone Rule (see Section 2). As a general rule, if results are well known, they will be stated without proof. Graphing of functions and computer experiments in general played an important role in our work. The software included with the book [AVV4] provides useful computer programs for such experiments. Notation 1.8. Throughout this paper the hyperbolic tangent function and its inverse will be denoted by th and arth, respectively. Likewise, the hyperbolic cosine function, and its inverse, will be denoted by ch and arch. 2. The distortion function ϕK (r) We first discuss useful tools from classical analysis. The l’Hˆopital Monotone Rule also has numerous applications in various fields; see [Ch, p. 124, Lemma 3.1] and [Pi] for further references. Lemma 2.1 (l’Hˆopital Monotone Rule [AVV3], [AVV4, Theorem 1.25]). For −∞ < a<b<∞,letg and h be real-valued functions that are continuous on [a, b] and differentiable on (a, b),withh(x) =0 . If g/h is (strictly) increasing (resp. decreasing) on (a, b), then the functions g(x) − g(a) g(x) − g(b) and h(x) − h(a) h(x) − h(b) are also (strictly) increasing (resp. decreasing) on (a, b). Lemma 2.2 (Power Series Quotient Monotone Rule [BK] and [HVV]). For R>0, ≡ ∞ n ≡ ∞ n let the two real power series g(x) n=0 anx and h(x) n=0 bnx be conver- gent on the interval (−R, R). If the sequence {an/bn} is increasing (decreasing), and bn > 0 for all n, then the function ∞ n g(x) n=0 anx ∞ f(x)= = n h(x) n=0 bnx is also increasing (decreasing) on (0,R). In fact, the function h(x)g(x) − g(x)h(x) has positive (negative) Maclaurin coefficients. TOPICS IN SPECIAL FUNCTIONS. II 253 We begin with the Hersch-Pfluger distortion function ϕK given in (1.2), because the other distortion functions studied here are expressible in terms of this function. It satisfies the composition rule (2.3) ϕAB(r)=ϕA(ϕB(r)),A,B>0,r∈ (0, 1), −1 so that ϕ1/K (r)=ϕK (r) (cf. [AVV4, (10.2)]). It was stated by H¨ubner [H¨u] and proved by He [He] (see also [LV, pp. 64–65]) that, for K>1, −1/K 1−1/K lim r ϕK (r)=4 . r→0 The next result is an improvement of this limit relation. −1/K Theorem 2.4. (1) For K>1, the function r ϕK (r) is strictly decreasing in 1−1/K −1/K r from (0, 1) onto (1, 4 ).For0 <K<1, r ϕK (r) is strictly increasing − in r from (0, 1) onto (41 1/K , 1).