Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 93–103

FUNCTIONAL IDENTITIES FOR SPECIAL FUNCTIONS OF QUASICONFORMAL THEORY

J. Zaja̧c

Polish Academy of Sciences, Institute of Mathematics ul. Narutowicza 56, PL-90-136 Łódź, Poland

Abstract. Let Φ_{K,n} be the distortion function for K -quasiconformal maps of B^{n} into B^{n}, n ≥ 2. We describe the family ℋ_{n} of all differentiable involutions h such that the functional identity h∘Φ_{K,n} = Φ_{1/K,n}∘h holds for every K > 0. An infinite sequence of elementary involutions of ℋ_{2} is explicitly constructed. A basic convergence theorem for Φ_{K,n}, K > 0, n = 2, 3, …, together with some other results for related functions are proved.

0. Introduction

The distortion function Φ_{K,n}: [0; 1] → [0; 1], in the n-dimensional (n ≥ 2) quasiconformal version of the Schwarz lemma (see [MRV] and [V]), is defined for K > 0, n = 2, 3, 4, …, by Φ_{K,n}(0) = 0, Φ_{K,n}(1) = 1, and

(0.1)  Φ_{K,n}(t) = M_{n}^{−1}(\frac{1}{K_{n}} M_{n}(t))

for 0 < t < 1, where K_{n} = K^{1/(n−1)} and M_{n} is given by

(0.2)  γ_{n}(1/t) = ω_{n−1}^{∘}M_{n}^{1−n}(t).

Here, γ_{n} denotes the conformal capacity of the Grötzsch condenser in R^{n}, ω_{n−1}^{∘} = (2/π)^{n−1}ω_{n−1} , where ω_{n−1} is the (n − 1)-dimensional surface area of the unit sphere S^{n−1} in R^{n}. Introducing any positive constant multiplier to the formula (0.2), defining M_{n}, we do not alter Φ_{K,n}. Thus, for our convenience, we normalize ω_{n−1} as above. For n = 2 the function M_{2} = μ, and

(0.3)  μ(t) = \frac{𝒦 (\sqrt{1 − t^{2}} )}{𝒦 (t)} ,

where

(0.4)  𝒦 (t) = ∫_{0}^{π/2}(1 − t^{2} sin^{2} u)^{−1/2} du

1991 Mathematics Subject Classification: Primary 30C60; Secondary 33A25, 33A70.

is the complete elliptic integral of the first kind. From this several functional identities follow (see [AVV1] and [V]).

For the higher-dimensional case n ≥ 3 neither such explicit expressions nor functional identities were known.

The main purpose of this paper is to show how one can obtain: one parameter family of identities satisfied by Φ_{K,n}, K > 0, n = 2, 3, …; equivalent identities for M_{n}, γ_{n}, τ_{n} and for 𝒦 , when n = 2; and a basic convergence theorem for Φ_{K,n}, as an application. Our idea can be realized since we know that M_{n} is differentiable in (0; 1) for n = 2, 3, …, (see [An2]).

The author wishes to thank M. Vuorinen for his assistance in preparing this material, Ken-ichi Sakan for useful remarks, the referee for helpful comments and suggestions, and to the Department of Mathematics of the University of Joensuu for hospitality while working on this paper.

1. Conjugate distortion functions

In the theory of quasiconformal mappings in R^{n}, n = 2, 3, …, we are particularly interested in two rings having extremal properties. The first is the Grötzsch ring R_{G,n}(s), s > 1, whose complementary components are the closed unit ball\overline{B}^{n} and the ray [se_{1}, ∞], where e_{1} is the first unit vector of the rectangular coordinates axes in R^{n}. The other is the Teichmüller ring R_{T,n}(t), t > 0, whose complementary components are the segment [−e_{1}, 0] and the ray [te_{1}, ∞]. The conformal capacities γ_{n}(s) and τ_{n}(t) of R_{G,n}(s) and R_{T,n}(t), respectively, are decreasing functions related by the following functional identity

(1.1)  γ_{n}(s) = 2^{n−1}τ_{n}(s^{2} − 1)  for s > 1.

(see [G]).

Let ℋ denote the family of all differentiable automorphisms of (0; 1), and ℋ_{n}, n = 2, 3, …, be the set of all involutions h ∈ ℋ such that

(1.2)  h ∘ Φ_{K,n} = Φ_{1/K,n} ∘ h

holds for each K > 0.

Moreover, let Q = (0; 1) × (0; 1). Then we prove

Theorem 1. Let n = 2, 3, …, be fixed. A function h ∈ ℋ_{n} if and only if there is L > 0 such that

(1.3)  h(t) = M_{n}^{−1}(L_{n}/M_{n}(t))

holds for 0 < t < 1 and L_{n} = L^{1/(n−1)} . Moreover, if (ξ, η) ∈ Q is an arbitrary point then there is L_{n}^{ξη} such that h(ξ) = η.

Proof. (⇒) Introducing M̃_{n} = M_{n} ∘ h ∘ M_{n}^{−1} , we see that (1.2) with (0.1) may be written down as

(1.4)  M̃_{n}(\frac{t}{K_{n}} ) = K_{n}M̃_{n}(t)

for 0 < t < ∞, K_{n} = K^{1/(n−1)} , K > 0 and n = 2, 3, . . .. By the definition of ℋ_{n}, and the regularity of M_{n} (see [An2]), it follows that M̃_{n} is differentiable in 0 < t < ∞. Hence, the well-known Euler identity implies that all the solutions of (1.4) can be written as

M̃_{n}(t) = \frac{L_{n}}{t}

for 0 < t < ∞, L_{n} = L^{1/(n−1)} , L > 0 and n = 2, 3, . . .. Thus

(1.5)  h(t) = M_{n}^{−1}(L_{n}/M_{n}(t)).

Let (ξ, η) ∈ Q be an arbitrary point. Setting L_{n}^{ξη} = M_{n}(ξ)M_{n}(η) we see that M_{n}^{−1}(L_{n}^{ξη}/M_{n}(ξ)) = η, so the second part of our theorem follows.

In the case (⇐) it is evident that each function of the form (1.5) belongs to ℋ_{n} for n = 2, 3, …, which ends the proof.

Let

(1.6)  ℋ ^{∞} = ⋃_{n=2}^{∞} ℋ_{n}.

We may additionally assume that each function of ℋ ^{∞} maps 0 and 1 onto 1 and 0, respectively. A function of ℋ ^{∞} class is called a conjugate distortion function. To justify the name let us consider the family of functions defined as

(1.7)  Ψ_{K,α,β}[ν, t] = ν^{−1}(K^{α}ν^{β}(t)),

for α, β ∈ R, β ≠ 0 and K > 0, where ν is a differentiable homeomorphism mapping (0; 1) onto (0; ∞). The family of functions defined by (1.7) forms a group of automorphisms of (0;1), under composition. With each automorphism Ψ_{K,α,β}[ν, ·] we associate the automorphism

(1.8)  Ψ_{K,α,β}^{∗}[ν, ·] = Ψ_{K,−α,−β}[ν, ·].

The correspondence Ψ_{K,α,β}[ν, ·] → Ψ_{K,α,β}^{∗}[ν, ·] we call a conjugation.

Setting ν = M_{n}, α = 1/(1 − n) and β = 1, into (1.7), we obtain Φ_{K,n}, whereas

(1.9)  Φ_{K,n}^{∗} := Ψ_{K,−1/(1−n),−1}^{∗}[M_{n}, ·]

is a conjugate distortion function.

For each n = 2, 3, …, let
(1.10)  ℱ_{n} = ⋃_{K>0} Φ_{K,n}.

Then we have

Theorem 2. Let n = 2, 3, … be fixed. Then ℱ_{n}^{∗} = ℋ_{n} and ℱ_{n} ∪ ℋ_{n} is a

group with composition.

The basic properties of the conjugate distortion functions we state as

Theorem 3. For each K, L > 0, n = 2, 3, …, we have:

(i) for every fixed L ∈ (0; ∞) Φ_{L,n}^{∗} is a decreasing automorphism of (0; 1) and for each t ∈ (0; 1) Φ_{L,n}^{∗} is a decreasing diffeomorphism of (0; ∞) onto (0; 1);

(ii) Φ_{L,n}^{∗} ∘ Φ_{K,n}^{∗} = Φ_{K/L,n}; (iii) Φ_{L,n}^{∗} ∘ Φ_{K,n} = Φ_{LK,n}^{∗}; (iv) Φ_{L,n}^{∗} = Φ_{1/L,n} ∘ Φ_{1,n}^{∗};

(v) Φ_{L,n}^{∗} ∘ Φ_{K,n} ∘ Φ_{L,n}^{∗} = Φ_{1/K,n}; (vi) Φ_{1,2}^{∗}(t) = \sqrt{1 − t^{2}} , Φ_{2,2}^{∗}(t) = (1 − t)/(1 + t), Φ_{4,2}^{∗}(t) = (1 − \sqrt{t})/(1 + \sqrt{t})^{2} ,

etc., 0 ≤ t ≤ 1.

Proof. The properties (i), …, (v) follow from the definition of Φ_{L,n}^{∗} and the properties of Φ_{K,n}. Since M_{2}(\sqrt{1 − t^{2}} ) = μ(\sqrt{1 − t^{2}} ) = 𝒦 (t)/𝒦 (\sqrt{1 − t^{2}} ) = 1/μ(t) then μ^{−1}(1/μ(t)) = \sqrt{1 − t^{2}} . Making use of (iii) we obtain: Φ_{2,2}^{∗}; Φ_{4,2}^{∗}; Φ_{8,2}^{∗}; etc.

The functional identities (1.2) have a natural geometric interpretation. To this let us consider the mapping

(1.11)  Φ̃_{K,n}(t, x) = (Φ_{K,n}(t), Φ_{1/K,n}(x))

that maps \overline{Q} onto itself for each K > 0 and n = 2, 3, . . .. We have

(1.12) Φ̃_{K,n}(t, Φ_{L,n}^{∗}(t)) = (Φ_{K,n}(t), Φ_{1/K,n} ∘ Φ_{L,n}^{∗}(t)) = (Φ_{K,n}(t), Φ_{L,n}^{∗}(Φ_{K,n}(t)))

for each 0 ≤ t ≤ 1, K, L > 0, and n = 2, 3, . . .. Thus every curve

(1.13)  Γ_{L,n} = { (t, x) : x = Φ_{L,n}^{∗}(t), 0 < t < 1 },  L > 0

is the invariant curve under each mapping from ℱ_{n}, for n = 2, 3, . . .. Moreover, by Theorem 1, we have

(1.14)  Q = ⋃_{L>0} Γ_{L,n}

for n = 2, 3, . . ..

As an immediate consequence of Theorem 1 we have

Theorem 4. Let n = 2, 3, … be fixed. The following identities:

(i) M_{n}(t)M_{n}(Φ_{L,n}^{∗}(t)) = L_{n}; (ii) γ_{n}(1/t)γ_{n}(1/Φ_{L,n}^{∗}(t)) = (ω_{n−1}^{∘})^{2}/L;

(iii) τ((1/t)^{2} − 1)τ_{n}((1/Φ_{L,n}^{∗}(t))^{2} − 1) = π^{2(1−n)}ω_{n−1}^{2}/L holds for 0 < t < 1 and each L > 0. Moreover, these three identities and the identity (1.2) are equivalent.

Proof. By (1.9), (0.1) and (1.2) we obtain (i). The identities (ii) and (iii) follow from (0.2) and (1.1).

In the case of n = 2 we additionally have

Corollary 1. The identity

(1.15)  μ = L\frac{𝒦 ∘ Φ_{L,2}^{∗}}{𝒦 ∘ Φ_{L,2}}

holds for each L > 0, where 𝒦 is the elliptic integral.

This observation is an immediate consequence of (0.3) and the identity (i) of Theorem 4. It is worth-while to note that (1.15) generalize the identities of Landen (cf. [AVV2, p. 6]).

Corollary 2. By Theorem 3 we have: Φ_{4,2}^{∗}(t) = ((1 − \sqrt{t})/(1 + \sqrt{t}))^{2} , Φ_{8,2}^{∗}(t) = ((\sqrt{1 + t}− ^{4} \sqrt{t})/(\sqrt{1 + t}+ ^{4} \sqrt{t}))^{2} , … and, generally, Φ_{2^{n},2}^{∗} = Φ_{1,2}^{∗} ∘Φ_{2,2}^{n}, n = 1, 2, …, where Φ_{2,2}(t) = 2\sqrt{t}/(1 + t), (see [V, p. 68]).

Corollary 3. By Theorem 3 and Corollary 2 we have:

(i) setting n = 2, then L = 1 or L = 2 into (1.2) we obtain the well-known identities (3.4), (3.5), …, (3.9), presented in [AVV1];

(ii) setting n = 2, then L = 4 with t = r^{2} into (1.2) we obtain (3.10) of [AVV1]; (iii) the identity (1.7) of [AVV1] follows from (v) of Theorem 3; (iv) setting n = 2, then L = 1 or L = 2 into (i) of Theorem 4 then, in view

of (0.2), we obtain the identities (5.57) of [V].

Corollary 4. For every K > 0 and 0 ≤ t ≤ 1, we have

(i) Φ_{K,2}^{∗}(t) = \sqrt{1 − Φ_{K,2}^{2}(t)};

(ii) Φ_{2K,2}^{∗}(t) = (1 − Φ_{K,2}(t))/(1 + Φ_{K,2}(t));

(iii) Φ_{4K,2}^{∗}(t) = (1 − \sqrt{Φ_{K,2}(t)} )^{2}/(1 + \sqrt{Φ_{K,2}(t)} )^{2} ;

(iv) Φ_{8K,2}^{∗}(t) = (\sqrt{1 + Φ_{K,2}(t)} − ^{4} \sqrt{4Φ_{K,2}(t)} )^{2}/(\sqrt{1 + Φ_{K,2}(t)} + ^{4} \sqrt{4Φ_{K2}(t)} ); etc.

Corollary 5. There exists no L > 0 such that Φ_{L,2}^{∗}(t) = 1 − t.

We recall that an explicit estimate

(1.16)  t^{1/K_{n}} ≤ Φ_{K,n}(t) ≤ λ_{n}^{1−1/K_{n}}t^{1/K_{n}}

holds for K ≥ 1, n = 2, 3, …, and 0 ≤ t ≤ 1, where K_{n} = K^{1/(n−1)} . The constant λ_{n} is known only when n = 2, in this case λ_{2} = 4 [LV, p. 62]. Generally, 2e^{0.76(n−1)} ≤ λ_{n} ≤ 2e^{n−1} , for n ≥ 3 (see [An1], [AF] and [V, p. 89]).

Now we prove

Theorem 5. For each L ≥ 1, n = 2, 3, …, the following inequalities

(1.17)  \{ \begin{matrix}λ_{n}^{1−L_{n}}(Φ_{1,n}^{∗}(t))^{L_{n}} & ≤ Φ_{L,n}^{∗}(t) ≤ (Φ_{1,n}^{∗}(t))^{L_{n}}, \\ (Φ_{1,n}^{∗}(t))^{1/L_{n}} & ≤ Φ_{1/L,n}^{∗}(t) ≤ λ_{n}^{1−1/L_{n}}(Φ_{1,n}^{∗}(t))^{1/L_{n}} \end{matrix}

holds for 0 ≤ t ≤ 1, with L_{n} = L^{1/(n−1)} .

Proof. Using (iv) of Theorem 3, with L = 1, then by the inequality (1.16) we obtain the second row of (1.17). Since Φ_{1/K,n} ∘ Φ_{1,n}^{∗} = Φ_{K,n}^{∗}, then the first row of (1.17) follows, if we apply inequality (1.16) to Φ_{K,n}^{−1} = Φ_{1/K,n}.

In the particular case n = 2 we have

Corollary 6. By (vi) of Theorem 3 then by Theorem 5, we have

(1.17′)  \{ \begin{matrix}4^{1−L}(1 − t^{2})^{L/2} & ≤ Φ_{L,2}^{∗}(t) ≤ (1 − t^{2})^{L/2}, \\ (1 − t^{2})^{1/2L} & ≤ Φ_{1/L,2}^{∗}(t) ≤ 4^{1−1/L}(1 − t^{2})^{1/2L}, \end{matrix}

for each 0 ≤ t ≤ 1 and L ≥ 1.

2. Applications

From (v) of Theorem 3 and (1.16), applied to Φ_{1/K,n} = Φ_{K,n}^{−1} , we see that

(2.1)  Φ_{L,n}^{∗}((Φ_{L,n}^{∗}(t))^{K_{n}}) ≤ Φ_{K,n}(t) ≤ Φ_{L,n}^{∗}(λ_{n}^{1−K_{n}}(Φ_{L,n}^{∗}(t))^{K_{n}})

holds for every 0 ≤ t ≤ 1, n = 2, 3, …, K ≥ 1 and L > 0, where K_{n} = K^{1/(n−1)} . Let

(2.2)  b_{n}[K, L](t) = Φ_{L,n}^{∗} ((Φ_{L,n}^{∗}(t))^{K_{n}})

and

(2.3)  B_{n}[K, L](t) = Φ_{L,n}^{∗} (λ_{n}^{1−K_{n}}(Φ_{L,n}^{∗}(t))^{K_{n}})

for K ≥ 1, L > 0, 0 ≤ t ≤ 1, n = 2, 3, …, and K_{n} = K^{1/(n−1)} . From the properties of Φ_{L,n}^{∗} and the inequality (1.16), it follows that

(2.4)  \{ \begin{matrix}b_{n}[K, L](t) & ≤ Φ_{K,n}(t) ≤ B_{n}[K, L](t), \\ Φ_{1,n}^{∗}(B_{n}[K, L](t)) & ≤ Φ_{K,n}^{∗}(t) ≤ Φ_{1,n}^{∗}(b_{n}[K, L](t)) \end{matrix}

holds for all L > 0, 0 ≤ t ≤ 1, n = 2, 3, …, and K ≥ 1.

Setting n = 2 then L = 2 into the first row of (2.4), we immediately obtain (vii) of [Z]. Setting n = 2 then L = 4, we see that (2.5)

(\frac{(1 + \sqrt{t})^{K} − (1 − \sqrt{t})^{K}}{(1 + \sqrt{t})^{K} + (1 − \sqrt{t})^{K}} )^{2} ≤ Φ_{K,2}(t) ≤ (\frac{(1 + \sqrt{t})^{K} − 2^{1−K}(1 − \sqrt{t})^{K}}{(1 + \sqrt{t})^{K} + 2^{1−K}(1 − \sqrt{t})^{K}} )^{2}

holds for 0 ≤ t ≤ 1 and K ≥ 1. The right-hand inequality is [AVV3, Theorem 5.7].

By Corollary 2 then (2.4), we can improve (2.5) when taking L = 2^{i}, i ≥ 3. It can be easily checked by computer, which is also useful to illustrate (2.2), (2.3) and (2.4).

To explain the nature of (2.4) and the idea of the conjugate distortion functions we prove first

Lemma. For each K > 0 and n = 2, 3, …,
(2.6)  lim_{L→∞} Φ_{L,n}^{∗} ∘ φ ∘ Φ_{L,n}^{∗}(t) = Φ_{K,n}(t)

for 0 ≤ t ≤ 1, where φ: [0; 1] → [0; 1] is any function such that

(2.7)  lim_{t→0^{+}} \frac{log φ(t)}{log t} = K_{n}.

Proof. Let K > 0, n = 2, 3, … be arbitrary. It follows from (2.7) and [P, Theorem 3.1] that for 0 ≤ t ≤ 1,

lim_{L→∞} Φ_{L,n} ∘ φ ∘ Φ_{L,n}(t) = Φ_{1/K,n}(t).

Hence, and by Theorem 3, we get

Φ_{L,n}^{∗} ∘ φ ∘ Φ_{L,n}^{∗}(t) = Φ_{1,n}^{∗} ∘ (Φ_{L,n} ∘ φ ∘ Φ_{L,n}) ∘ Φ_{1,n}^{∗}(t)
→ Φ_{1,n}^{∗} ∘ Φ_{1/K,n} ∘ Φ_{1,n}^{∗}(t) = Φ_{K,n}(t) as L → ∞

for 0 ≤ t ≤ 1, which ends the proof.

Now we can prove

Theorem 6. For each K ≥ 1, n = 2, 3, … and 0 ≤ t ≤ 1,
(2.8) \{ \begin{matrix}lim_{L→∞} b_{n}[K, L](t) = lim_{L→∞} B_{n}[K, L](t) = Φ_{K,n}(t), \\ lim_{L→∞} Φ_{1,n}^{∗}(b_{n}[K, L](t)) = lim_{L→∞} Φ_{1,n}^{∗}(B_{n}[K, L](t)) = Φ_{K,n}^{∗}(t). \end{matrix}

Moreover, b_{n} is an increasing function of L whereas B_{n} is a decreasing function of L ∈ (0, ∞).

Proof. Setting φ_{1}(t) = t^{K_{n}} and φ_{2}(t) = λ_{n}^{1−K_{n}}t^{K_{n}}, we have

lim_{t→0^{+}} \frac{log φ_{1}(t)}{log t} = lim_{t→0^{+}} \frac{log φ_{2}(t)}{log t} = K_{n}, K_{n} = K^{1/(n−1)}.

This, in view of the Lemma and (iii) of Theorem 3 then by (2.2) and (2.3), gives (2.8). The second statement is derived from parallel properties of the distortion function Φ_{K,n}.

By Theorem 3 and Theorem 6 it follows that the sequences b_{2}[K, 2^{i}] and B_{2}[K, 2^{i}], i = 1, 2, …, of elementary functions converge to Φ_{K,2} uniformly. This gives a new, pure numerical, method to estimate Φ_{K,2} and any functional of it.

It seems worth-while to note that the basic approximation Theorem 6 was previously a conjecture, cf. [Z]. By a significant result obtained by D. Partyka [P, Theorem 3.1], relevant to this matter, it was possible to prove (2.8).

For every 0 ≤ t < 1, n = 2, 3, …, and K, L > 0, set
(2.9)  λ_{n}[K, L](t) = \frac{Φ_{K,n}(t)}{Φ_{L,n}^{∗}(t)}.

It satisfies the following functional identities

(2.10)  λ_{n}[K, L](Φ_{M,n}(t)) = λ_{n}[KM, LM](t)

and

(2.11)  λ_{n}[K, L](Φ_{M,n}^{∗}(t)) = 1/λ_{n}[M/L, M/K](t).

By Theorem 3, Corollary 6 and Corollary 4, the following inequalities

(2.12)  \frac{b_{n}[K, M](t)}{Φ_{1,n}^{∗}(b_{n}[L, M](t))} ≤ λ_{n}[K, L](t) ≤ \frac{B_{n}[K, M](t)}{Φ_{1,n}^{∗}(B_{n}[L, M](t))}

hold for every 0 ≤ t < 1, K, L ≥ 1 and each M > 0.

In connection with study of quasisymmetric functions of the real line [LV] and the unit circle [K] the distortion function λ(K) introduced by Lehto, Virtanen and Väisälä (see [LV, (6.4), p. 81]), has found applications. A generalization of this, introduced by Agard [Ag], namely λ(K, t), has been studied by Vamanamurthy and Vuorinen [VV].

We have

(λ_{2}[K, K](t))^{2} = λ(K, t)

and

(λ_{2}[K, K](1/\sqrt{2} ))^{2} = λ(K).

Setting in (2.12) M = 4 and L = K, we get

(2.13)  \begin{matrix}\frac{(1/(1 − t))^{K}[(1 + \sqrt{t})^{K} − (1 − \sqrt{t})^{K}]^{4}}{8[(1 + \sqrt{t})^{2K} + (1 − \sqrt{t})^{2K}]} ≤ λ(K, t) \\ ≤ \frac{(2/(1 − t))^{K}[(1 + \sqrt{t})^{K} − 2^{1−K}(1 − \sqrt{t})^{K}]^{4}}{16[(1 + \sqrt{t})^{2K} + 4^{1−K}(1 − \sqrt{t})^{2K}]} . \end{matrix}

By (2.12) and Theorem 6 we see that λ_{2}[K, L](t) can be approximated by elementary functions.

Other functionals of Φ_{K,n} and Φ_{L,n}^{∗}, with applications, will be considered in an additional paper.

A sharp estimation for max_{0≤t≤1} [Φ_{K,2}(t) − t], K ≥ 1, has been obtained by the author [Z, Theorem 2]. It says that for each K ≥ 1,

(2.14)  max_{0≤t≤1} [Φ_{K,2}(t) − t] ≤ \{ \begin{matrix}1 − \frac{1 + 4^{1−K}}{2K} , & 1 ≤ K ≤ K_{0}, \\ \frac{1 − 4^{1−K}}{1 + 4^{1−K}} , & K > K_{0}, \end{matrix}

where K_{0} satisfies the equation (1 + 4^{1−K})^{2} = K4^{2−K}, 2.481 < K_{0} < 2.482. Taking advantage of (2.4) we improve (2.14) obtaining

Theorem 7. For each K ≥ 1,

(2.15)  max_{0≤t≤1} [Φ_{K,2}(t) − t] ≤ B_{2}[K, 4](t_{0}) − t_{0}

where t_{0} is such that B′_{2}[K, 4](t_{0}) = 1.

Proof. At first we show that B_{2}[K, 4] is concave. To this end let us differentiate B_{2}[K, 4](t) with respect to t, 0 < t < 1, we obtain

\begin{aligned}B′_{2}[K, 4](t) & = K2^{3−K} \frac{(1 − t)^{K−1}}{[(1 + \sqrt{t})^{K} + 2^{1−K}(1 − \sqrt{t})^{K}]^{2}} × \\ & × \frac{1}{\sqrt{t}} \frac{1 − 2^{1−K}((1 − \sqrt{t})/(1 + \sqrt{t}))^{K}}{1 + 2^{1−K}((1 − \sqrt{t})/(1 + \sqrt{t}))^{K}} . \end{aligned}

Introducing x = (1 − \sqrt{t})/(1 + \sqrt{t}), and considering

f(x) = \frac{1 + x}{1 − x} \frac{1 − 2^{1−K}x^{K}}{1 + 2^{1−K}x^{K}} , 0 < x < 1

we can see that

[ ln f(x)]′ = \frac{2}{1 − x^{2}} − K2^{1−K}x^{K−1} \frac{2}{1 − 4^{1−K}x^{2K}} ≥ 0

for 0 < x < 1 and K ≥ 1.

We shall prove that

(2.16) 1 − 4^{1−K}x^{2K} ≥ K2^{1−K}x^{K−1}(1 − x^{2})  for 0 ≤ x ≤ 1 and K ≥ 1.

Note, that for K = 1 the inequality attains the equality sign. Because

∂_{1} = \frac{∂}{∂K} (1 − 4^{1−K}x^{2K}) = −8(\frac{x}{2})^{2K} ln \frac{x}{2} > 0 for 0 ≤ x ≤ 1, K ≥ 1

and

∂_{2} = \frac{∂}{∂K} (K2^{1−K}x^{K−1}(1 − x^{2})) = (\frac{x}{2})^{K−1}(1 − x^{2}) (1 + K ln \frac{x}{2}) ≤ 0

holds for 0 < x < 2/e and K ≥ 1, then (2.16) remains true for 0 ≤ x ≤ 2/e and K ≥ 1.

Let 2/e ≤ x ≤ 1 and 1 ≤ K ≤ 3/2. Hence

\begin{aligned}8(\frac{x}{2})^{2K} & ≥ \frac{8}{e^{K+1}} (\frac{x}{2})^{K−1} ≥ \frac{8}{e^{5/2}} (\frac{x}{2})^{K−1} \\ & ≥ (\frac{x}{2})^{K−1}(1 − \frac{4}{e^{2}} ) ≥ (\frac{x}{2})^{K−1}(1 − x^{2}) \end{aligned}

and

(2.17)  − ln \frac{x}{2} > 1 + K ln \frac{x}{2}  ⇔  (K + 1) ln \frac{x}{2} < −1.

Thus ∂_{1} − ∂_{2} ≥ 0 for 2/e ≤ x ≤ 1 and 1 ≤ K ≤ 3/2. By this we see that (2.16) holds for 2/e ≤ x ≤ 1 and 1 ≤ K ≤ 3/2.

Suppose now that K > 3/2 and 0 ≤ x ≤ 1. Then

(2.18)  1 + K ln \frac{x}{2} ≤ 1 + K ln \frac{1}{2} ≤ 1 + ln \frac{1}{2^{3/2}} < 0

and thus ∂_{2} < 0 for K ≥ 3/2. In [Z, p. 7] it is proved that the first ratio of B′_{2}[K, 4](t) is decreasing. This fact, with our considerations on f, shows that B_{2}[K, 4] is concave, and our proof is complete.

Functional identities for special functions of quasiconformal theory 103

The theory of the conjugate distortion functions presented in this paper afford us to state the following:

(i) b_{n}[K, L] and B_{n}[K, L] are concave as functions of variable t ∈ [0; 1] for every K > 1, L > 0 and n = 2, 3, …;

(ii) b_{n}[K, L] is increasing whereas B_{n}[K, L] is decreasing as functions L, L > 0 for every K > 1 and n = 2, 3, . . .. The convexity and concavity of the conjugate distortion functions seems to

be an interesting topic for investigation on special functions.

References

[Ag] Agard, S.: Distortion theorems for quasiconformal mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 413, 1968, 1–12.

[An1] Anderson, G.D.: Dependence on dimension of a constant related to the Grötzsch ring. - Proc. Amer. Math. Soc. 61, 1976, 77–80.

[An2] Anderson, G.D.: Derivatives of the conformal capacity of extremal rings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 1985, 29–46.

[AF] Anderson, G.D., and J.S. Frame: Numerical estimates for a Grötzsch ring constant. - Constr. Approx. 4, 1988, 223–242.

[AVV1] Anderson, G.D., M.K. Vamanamurthy, and M. Vuorinen: Distortion function for plane quasiconformal mappings. - Israel J. Math. 62, 1, 1988, 1–16.

[AVV2] Anderson, G.D., M.K. Vamanamurthy, and M. Vuorinen: Conformal invariants quasiconformal maps and special functions. - Preprint, 1991.

[AVV3] Anderson, G.D., M.K. Vamanamurthy, and M. Vuorinen: Inequalities for quasiconformal mappings in the plane and in space. - Preprint, 1991.

[G] Gehring, F.W.: Symmetrization of rings in space. - Trans. Amer. Math. Soc. 101, 1961, 499–519.

[K] Krzyż, J.G.: Harmonic analysis and boundary correspondence under quasiconformal mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 14, 1989, 225–242.

[LV] Lehto, O., and K.I. Virtanen: Quasiconformal mappings in the plane. - Grundlehren der Mathematischen Wissenschaften 126, 2nd ed., Springer-Verlag, New York–Heidelberg–Berlin, 1973.

[MRV] Martio, O., S. Rickman, and J. Väisälä: Distortion and singularities of quasiregular mappings. - Ann. Acad. Sci. Fenn. Ser. A I Math. 465, 1970, 1–13.

[P] Partyka, D.: Approximation of the Hersch–Pfluger distortion function. - Preprint, 1992.

[V] Vuorinen, M.: Conformal geometry and quasiregular mappings. - Lecture Notes in Mathematics 1319. Springer-Verlag, Berlin, 1988.

[VV] Vamanamurthy, M.K., and M. Vuorinen: Functional inequalities for special functions in quasiconformal theory. - Manuscript, 1991.

[Z] Zaja̧c, J.: The distortion function Φ_{K} and quasihomographies. - CTAFT, 1992 (to appear).

Received 11 February 1992