Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 425–444

TWO-POINT DISTORTION THEOREMS FOR BOUNDED UNIVALENT FUNCTIONS

William Ma and David Minda

Pennsylvania College of Technology, Integrated Studies Division Williamsport, PA 17701-5799, U.S.A.; wma@pct.edu University of Cincinnati, Department of Mathematical Sciences Cincinnati, OH 45221-0025, U.S.A.; David.Minda@math.uc.edu

Abstract. One-parameter families of sharp two-point distortion theorems are established for nonnormalized bounded univalent functions f, that is, univalent functions f defined on the unit disk D = {z : |z| < 1} with f(D) ⊂ D. These theorems provide sharp upper and lower bounds on d_{D}(f(a), f(b)), the hyperbolic distance between f(a) and f(b), in terms of d_{D}(a, b) and the “hyperbolic derivatives” (1 − |a|^{2})|f′(a)|/(1 − |f(a)|^{2}), (1 − |b|^{2})|f′(b)|/(1 − |f(b)|^{2}) for arbitrary a, b ∈ D. The weakest upper and lower bounds obtained are invariant versions of the classical growth theorems for bounded univalent functions that are due to Pick. The lower bounds are also sufficient conditions to imply that f is univalent in D. As part of establishing these results, a new, sharp coefficient inequality for bounded univalent functions is derived.

1. Introduction

Many classical growth and distortion theorems for univalent functions f defined on the unit disk D = {z : |z| < 1} are established under the assumption that f is normalized (f(0) = 0 and f′(0) = 1). Blatter [B1] obtained a sharp two-point distortion theorem for nonnormalized univalent functions defined on D. It is especially interesting that Blatter’s distortion theorem is also sufficient for univalence; that is, if a holomorphic function f defined on D satisfies Blatter’s distortion inequality, then f is univalent, or constant. In contrast, many classical growth and distortion theorems for normalized univalent functions are also satisfied by nonunivalent functions. Kim and Minda [KM] extended the work of Blatter; they established a one-parameter family of sharp two-point distortion theorems for univalent functions on D. This enabled them to establish a connection between Blatter’s distortion theorem and a classical growth theorem of Koebe for univalent functions. They also obtained similar results for convex univalent functions. Recently, Ma and Minda [MM_{2} ] obtained various one parameter families of sharp two-point distortion theorems for strongly close-to-convex functions. All of these two-point distortion theorems yield comparison theorems between hyperbolic and euclidean gometry on simply connected regions.

1991 Mathematics Subject Classification: Primary 30C45; Secondary 30F45.

The research of the second author partially supported by a National Science Foundation Grant.

It is natural to consider the class of bounded univalent functions. By the phrase “bounded univalent function” we always mean a univalent function f defined on D with f(D) ⊂ D. In addition, we always assume that f(D) is a proper subset of D. If f(D) = D, then f is a Möbius transformation. Pick [Pi] established a number of sharp results for bounded univalent functions f, often using the assumption that f is normalized (f(0) = 0 and f′(0) = α ∈ (0, 1]). We shall obtain one-parameter families of sharp two-point distortion theorems for nonnormalized bounded univalent functions. The classical growth theorems of Pick for bounded univalent functions are the weakest cases of our results. Since our results are rather complicated to state, we shall not give them here. But we shall give the flavor of our work. Let d_{D}(a, b) denote the hyperbolic distance between a, b ∈ D. For a bounded univalent function f we prove sharp upper and lower bounds on d_{D}(f(a), f(b)) in terms of d_{D}(a, b) and the value of the “hyperbolic derivative” of f at a and b. Our technique of proof uses second-order linear differential inequalities for real-valued functions together with coefficient inequalities for bounded univalent functions. As part of our work we establish a new coefficient inequality for bounded univalent functions. Our approach is similar to that employed in [MM_{2} ].

2. Preliminaries

Hyperbolic geometry plays an important role in this paper. The hyperbolic metric on D is

λ_{D}(z)|dz| = \frac{|dz|}{1 − |z|^{2}} .

A region Ω in C is called hyperbolic if C\Ω contains at least two points. In this paper all regions will be subsets of D and so automatically hyperbolic. The density of the hyperbolic metric on a hyperbolic region Ω is derived from

λ_{Ω}(f(z))|f′(z)| = \frac{1}{1 − |z|^{2}} ,

where f: D → Ω is any holomorphic universal covering projection. The hyperbolic metric is independent of the choice of the covering projection of D onto Ω. If Ω is simply connected, then a covering f: D → Ω is a conformal mapping. The distance function induced on Ω by the hyperbolic metric is

d_{Ω}(A, B) = inf ∫_{γ} λ_{Ω}(w) |dw|,

where the infimum is taken over all paths γ in Ω joining A and B. There always exists a path δ in Ω connecting A and B such that

d_{Ω}(A, B) = ∫_{δ} λ_{Ω}(w) |dw|;

such a path δ is called a hyperbolic geodesic arc between A and B. When Ω is not simply connected, there can be more than one hyperbolic geodesic joining A and B. For the unit disk we have the explicit formula

d_{D}(a, b) = artanh |\frac{b − a}{1 − \overline{a}b}|.

In the unit disk hyperbolic geodesics are arcs of circles orthogonal to the unit circle. When Ω is simply connected the hyperbolic geodesics are the images of these circular arcs under a conformal mapping f: D → Ω.

Several invariant differential operators will be used. Suppose f: D → D is holomorphic. Define

D_{h1}f(z) = \frac{(1 − |z|^{2})f′(z)}{1 − |f(z)|^{2}},
D_{h2}f(z) = \frac{(1 − |z|^{2})^{2}f′′(z)}{1 − |f(z)|^{2}} + \frac{2(1 − |z|^{2})^{2}\overline{f(z)}f′(z)^{2}}{(1 − |f(z)|^{2})^{2}} − \frac{2\overline{z}(1 − |z|^{2})f′(z)}{1 − |f(z)|^{2}},

D_{h3}f(z) = \frac{(1 − |z|^{2})^{3}f′′′(z)}{1 − |f(z)|^{2}} + \frac{6(1 − |z|^{2})^{3}\overline{f(z)}f′(z)f′′(z)}{1 − |f(z)|^{2}}

− \frac{6\overline{z}(1 − |z|^{2})^{2}f′′(z)}{1 − |f(z)|^{2}} + \frac{6\overline{z}^{2}(1 − |z|^{2})f′(z)}{1 − |f(z)|^{2}}

− \frac{12\overline{z}(1 − |z|^{2})^{2}\overline{f(z)}f′(z)^{2}}{(1 − |f(z)|^{2})^{2}} + \frac{6(1 − |z|^{2})^{3}\overline{f(z)^{2}}f′(z)^{3}}{(1 − |f(z)|^{2})^{3}} .

For simplicity we write D_{j}f in place of D_{hj}f (j = 1, 2, 3) throughout this paper. The reader should note that D_{j}f has a different meaning in [KM], [MM_{1} ] and [MM_{2} ]. For a locally univalent holomorphic function f: D → D set

Q_{f}(z) = \frac{D_{2}f(z)}{D_{1}f(z)} = (1 − |z|^{2})\frac{f′′(z)}{f′(z)} + \frac{2(1 − |z|^{2})\overline{f(z)}f′(z)}{1 − |f(z)|^{2}} − 2\overline{z}

and note that

\frac{D_{3}f(z)}{D_{1}f(z)} − \frac{3}{2}(\frac{D_{2}f(z)}{D_{1}f(z)})^{2} = (1 − |z|^{2})^{2}S_{f}(z),

where

S_{f}(z) = \frac{f′′′(z)}{f′(z)} − \frac{3}{2}(\frac{f′′(z)}{f′(z)})^{2}

is the Schwarzian derivative of f. These operators are invariant in the sense that |D_{j}(S ∘ f ∘ T)| = |D_{j}f| ∘ T, whenever S and T are conformal automorphisms of D.

3. Coefficient inequalities

We begin by recalling two known coefficient inequalities for bounded univalent functions and then expressing them in the invariant form we require. In addition, we derive a new sharp coefficient inequality for bounded univalent functions. Throughout this section we let g(z) = a_{1}z + a_{2}z^{2} + a_{3}z^{3} + ⋯ denote a normalized (g(0) = 0) univalent function with g(D) ⊂ D, while f will designate an arbitrary bounded univalent function.

Pick [Pi] established the sharp inequality

|a_{2}| ≤ 2|a_{1}|(1 − |a_{1}|)

with equality if and only if |a_{1}| = α ∈ (0, 1] and g is a rotation of

\begin{aligned}k_{α}(z) & = \frac{(1 − z)^{2} + 2αz − (1 − z)\sqrt{(1 − z)^{2} + 4αz}}{2αz} \\ & = αz + 2α(1 − α)z^{2} + α(1 − α)(5 − 3α)z^{3} + · · · . \end{aligned}

The function k_{α} satisfies

(\frac{1 + k_{α}(z)}{1 − k_{α}(z)})^{2} − 1 = α[(\frac{1 + z}{1 − z} )^{2} − 1],

or k_{α}(z) = k^{−1}(αk(z)), where k(z) = z/(1 − z)^{2} is the Koebe function. The image of D under k_{α} is the unit disk with the slit (−1, −r_{α}] removed, where r_{α} = −k_{α}(−1) = (2 − α − 2\sqrt{1 − α})/α. The function k_{α} is extremal for a number of problems involving bounded univalent functions. A simple proof of Pick’s inequality is contained in [FO]. The invariant formulation of Pick’s inequality is

|D_{2}f(z)| ≤ 4|D_{1}f(z)|(1 − |D_{1}f(z)|)

or

|Q_{f}(z)| ≤ 4(1 − |D_{1}f(z)|)

for any bounded univalent function f. Equality holds at z_{0} ∈ D if and only if |D_{1}f(z_{0})| = α and f = S ∘ k_{α} ∘ T, where S, T are conformal automorphisms of D and T (z_{0}) = 0. In fact, for α ∈ (0, 1) and z ∈ D,

\frac{|D_{2}k_{α}(z)|}{|D_{1}k_{α}(z)|(1 − |D_{1}k_{α}(z)|)} ≤ 4

with equality if and only if z ∈ (−1, 1). Therefore, if f = S ∘ k_{α} ∘ T, then equality holds along the entire hyperbolic geodesic T^{−1}(−1, 1) while strict inequality is valid off this geodesic.

For a normalized bounded univalent function g Nehari [Ne] proved

|\frac{a_{3}}{a_{1}} − (\frac{a_{2}}{a_{1}} )^{2}| ≤ 1 − |a_{1}|^{2}

with equality if and only if g is a rotation of k_{α}. This result is a consequence of certain general inequalities of Nehari for bounded univalent functions; a simple proof is given in [MM_{1} ]. The invariant formulation is

(1 − |z|^{2})^{2}|S_{f}(z)| ≤ 6(1 − |D_{1}f(z)|^{2})

with equality at z_{0} if and only if |D_{1}f(z_{0})| = α and f = S ∘ f ∘ T, where S, T are conformal automorphisms of D and T (z_{0}) = 0. More precisely, for α ∈ (0, 1) and z ∈ D,

(1 − |z|^{2})^{2}|S_{k_{α}}(z)| ≤ 6(1 − |D_{1}k_{α}(z)|^{2})

with equality if and only if z ∈ (−1, 1). Hence, if f = S ∘ k_{α} ∘ T, then equality holds on the hyperbolic geodesic T^{−1}(−1, 1) and strict inequality is valid off the geodesic.

Theorem 1. Suppose g(z) = a_{1}z + a_{2}z^{2} + a_{3}z^{3} + ⋯ is univalent in D and g(D) ⊂ D. Then for p ≥ \frac{3}{2}

|3[\frac{a_{3}}{a_{1}} − (\frac{a_{2}}{a_{1}} )^{2}] + \frac{p + |a_{1}|}{1 − |a_{1}|}(\frac{a_{2}}{a_{1}} )^{2}| + \frac{p + |a_{1}|}{1 − |a_{1}|}|\frac{a_{2}}{a_{1}} |^{2} ≤ (8p − 3 + 5|a_{1}|)(1 − |a_{1}|).

Proof. There is no harm in assuming a_{1} > 0. Define

G(z) = \frac{g(z)}{a_{1}} = z + (\frac{a_{2}}{a_{1}} )z^{2} + (\frac{a_{3}}{a_{1}} )z^{3} + ⋯ = z + A_{2}z^{2} + A_{3}z^{3} + ⋯

and a_{1} = e^{−τ}. Then G ∈ S, the class of normalized univalent functions and G is bounded, |G(z)| < e^{τ} for z ∈ D. We want to prove

|3(A_{3} − A_{2}^{2}) + \frac{p + e^{−τ}}{1 − e^{−τ}} A_{2}^{2}| + \frac{p + e^{−τ}}{1 − e^{−τ}} |A_{2}|^{2} ≤ (8p − 3 + 5e^{−τ})(1 − e^{−τ}).

It is sufficient to prove

L(G) ≤ (8p − 3 + 5e^{−τ})(1 − e^{−τ}),

where

\begin{aligned}L(G) & = Re\{3(A_{3} − A_{2}^{2}) + \frac{p + e^{−τ}}{1 − e^{−τ}} A_{2}^{2}\} + \frac{p + e^{−τ}}{1 − e^{−τ}} |A_{2}|^{2} \\ & = 3 Re\{A_{3} − A_{2}^{2}\} + \frac{2(p + e^{−τ})}{1 − e^{−τ}} Re^{2}\{A_{2}\}. \end{aligned}

We shall establish this result by embedding G in a Löwner chain. There is a Löwner chain

G(z, t) = ∑_{n=1}^{∞} A_{n}(t)z^{n}

defined for t ≥ 0 such that G(z, 0) = G(z), G(z, t) = e^{t}z for t ≥ τ and

\frac{∂G(z, t)}{∂t} = z \frac{∂G(z, t)}{∂z}p(z, t),

where

p(z, t) = 1 + ∑_{n=1}^{∞} C_{n}(t)z^{n}

has positive real part in D [Po, Section 6.1]. From G(z, t) = e^{t}z for t ≥ τ, it follows that C_{n}(t) = 0 for t ≥ τ and n = 1, 2, …. Then

A_{2} = A_{2}(0) = − ∫_{0}^{τ} e^{−t}C_{1}(t) dt,
A_{3} = A_{3}(0) = − ∫_{0}^{τ} e^{−2t}C_{2}(t) dt + (∫_{0}^{τ} e^{−t}C_{1}(t) dt)^{2},

and

A_{3} − A_{2}^{2} = − ∫_{0}^{τ} e^{−2t}C_{2}(t) dt,

so that

L(G) = −3 ∫_{0}^{τ} e^{−2t} Re\{C_{2}(t)\} dt + \frac{2(p + e^{−τ})}{1 − e^{−τ}}(∫_{0}^{τ} e^{−t} Re\{C_{1}(t)\} dt)^{2}.

Since [Po, p. 166]

− Re{C_{2}(t)} ≤ 2 − Re^{2}{C_{1}(t)},

we obtain

L(G) ≤ 3(1−e^{−2τ})−3 ∫_{0}^{τ} e^{−2t} Re^{2}\{C_{1}(t)\} dt+ \frac{2(p + e^{−τ})}{1 − e^{−τ}}(∫_{0}^{τ}e^{−t} Re\{C_{1}(t)\} dt)^{2}.

The Cauchy–Schwarz inequality gives

\begin{aligned}(∫_{0}^{τ} e^{−t} Re\{C_{1}(t)\} dt)^{2} & ≤ ∫_{0}^{τ} e^{−t} dt ∫_{0}^{τ} e^{−t} Re^{2}\{C_{1}(t)\} dt \\ & = (1 − e^{−τ}) ∫_{0}^{τ} e^{−t} Re^{2}\{C_{1}(t)\} dt, \end{aligned}

so

L(G) ≤ 3(1 − e^{−2τ}) + ∫_{0}^{τ} [2(p + e^{−τ}) − 3e^{−t}]e^{−t} Re^{2}{C_{1}(t)} dt.

Since p ≥ \frac{3}{2} , 2(p + e^{−τ}) − 3e^{−t} ≥ 0. Also | Re\{C_{1}(t)\}| ≤ 2 since p(z, t) has positive real part. Therefore,

L(G) ≤ 3(1 − e^{−2τ}) + 8(p + e^{−τ})(1 − e^{−τ}) − 6(1 − e^{−2τ}) = (8p − 3 + 5e^{−τ})(1 − e^{−τ}),

which is the desired result.

Corollary 1. Suppose g(z) = a_{1}z + a_{2}z^{2} + a_{3}z^{3} + ⋯ is univalent in D and g(D) ⊂ D. Then for p ≥ \frac{3}{2}

|3[\frac{a_{3}}{a_{1}} − (\frac{a_{2}}{a_{1}} )^{2}] + \frac{p + |a_{1}|}{1 − |a_{1}|}(\frac{a_{2}}{a_{1}} )^{2}| + \frac{1 + p}{1 − |a_{1}|}|\frac{a_{2}}{a_{1}} |^{2} ≤ (8p + 1 + |a_{1}|)(1 − |a_{1}|).

Equality holds if and only if g is a rotation of k_{α}.

Proof. Use the identity

\frac{1 + p}{1 − |a_{1}|}|\frac{a_{2}}{a_{1}} |^{2} = \frac{p + |a_{1}|}{1 − |a_{1}|}|\frac{a_{2}}{a_{1}} |^{2} + |\frac{a_{2}}{a_{1}} |^{2}

in conjunction with Pick’s inequality and the theorem in order to obtain the corollary. The equality statement follows from the case of equality in Pick’s result.

Corollary 2. Suppose f is univalent in D and f(D) ⊂ D. Then for p ≥ \frac{3}{2}

\begin{aligned}|(1 − |z|^{2})^{2}S_{f}(z)+ & \frac{p + |D_{1}f(z)|}{2(1 − |D_{1}f(z)|)}Q_{f}(z)^{2}| + \frac{1 + p}{2(1 − |D_{1}f(z)|)}|Q_{f}(z)|^{2} \\ & ≤ 2(8p + 1 + |D_{1}f(z)|)(1 − |D_{1}f(z)|). \end{aligned}

Equality holds at z_{0} ∈ D if and only if |D_{1}f(z_{0})| = α and f = S ∘ k_{α} ∘ T, where S, T are conformal automorphisms of D with T (z_{0}) = 0.

Corollary 2 is just the invariant formulation of Corollary 1. Also, for the function k_{α} equality holds in Corollary 2 for all z ∈ (−1, 1) with strict inequality off this interval. Therefore, if f = S ∘k_{α} ∘T, then equality holds on the hyperbolic geodesic T^{−1}(−1, 1) with strict inequality off this geodesic.

4. Differential inequalities

In this section we establish integral inequalities which are elementary consequences of second-order linear differential inequalities.

Proposition 1. Suppose u, v ∈ C^{2}[a, b], k, p > 0 and v′′ ≤ k^{2}p^{2}v, u′′ = k^{2}p^{2}u. If v(a) ≥ u(a) and v(b) ≥ u(b), then either v = u or else v > u on (a, b).

Proof. See [MM_{2} ].

Proposition 2. Suppose v ∈ C^{2}[−L, L], v > 0, k > 0, p ≥ 1, |v′| ≤ kpv and v′′ ≤ k^{2}p^{2}v. Then

(i) \frac{1}{k} log \frac{[1 + exp(−2kpL)]^{1/p} + [v(L) + v(−L)]^{1/p}}{[1 + exp(−2kpL)]^{1/p} + [v(L) + v(−L)]^{1/p} exp(−2kL)}

≤ ∫_{−L}^{L} \frac{v(s)^{1/p}}{1 + v(s)^{1/p}} ds

and equality holds if and only if v(s) = Ae^{±kps}, A > 0.

(ii) ∫_{−L}^{L} \frac{ds}{1 + v(s)^{1/p}} ≤ \frac{1}{k} log \frac{[1 + exp(−2pkL)]^{1/p} exp(2kL) + [v(L) + v(−L)]^{1/p}}{[1 + exp(−2pkL)]^{1/p} + [v(L) + v(−L)]^{1/p}}

and equality holds if and only if v(s) = Ae^{±kps}, A > 0.

Proof. We begin by determining u ∈ C^{2}[−L, L] so that u′′ = k^{2}p^{2}u and u satisfies the boundary conditions u(−L) = v(−L), u(L) = v(L). The general solution of u′′ = k^{2}p^{2}u is u(s) = A cosh(kps)+B sinh(kps) where A, B ∈ R. The boundary conditions yield

A = \frac{v(L) + v(−L)}{2 cosh(kpL)}, B = \frac{v(L) − v(−L)}{2 sinh(kpL)}.

If

τ = \frac{B}{A} = \frac{v(L) − v(−L)}{v(L) + v(−L)} · \frac{cosh(kpL)}{sinh(kpL)},

then u(s) = A[cosh(kps) + τ sinh(kps)]. Integration of the double inequality −kp ≤ v′/v ≤ kp over the interval [−L, L] results in

e^{−2kL} ≤ \frac{v(L)}{v(−L)} ≤ e^{2kL}.

Because h(t) = (t − 1)/(t + 1) is increasing for t > −1, we deduce that

−\frac{sinh(kpL)}{cosh(kpL)} ≤ \frac{v(L) − v(−L)}{v(L) + v(−L)} ≤ \frac{sinh(kpL)}{cosh(kpL)},

or τ ∈ [−1, 1]. From A > 0 and τ ∈ [−1, 1] it follows that u > 0 and |u′| ≤ kpu.

(i) Since t ↦ t/(1 + t) is strictly increasing,

∫_{−L}^{L} \frac{v(s)^{1/p}}{1 + v(s)^{1/p}} ds ≥ ∫_{−L}^{L} \frac{u(s)^{1/p}}{1 + u(s)^{1/p}} ds

= ∫_{−L}^{L} \frac{A^{1/p}[cosh(kps) + τ sinh(kps)]^{1/p}}{1 + A^{1/p}[cosh(kps) + τ sinh(kps)]^{1/p}} ds = I(τ).

Direct calculation shows that

I(1) = I(−1) = \frac{1}{k} log(\frac{1 + A^{1/p}e^{kL}}{1 + A^{1/p}e^{−kL}} )

and

I′′(τ) = \frac{1}{p^{2}} ∫_{−L}^{L} \frac{A^{2} sinh^{2}(kps)u(s)^{1/p−2}[(1 − p) − (1 + p)u(s)^{1/p}]}{[1 + u(s)^{1/p}]^{3}} ds.

Since p ≥ 1, I′′(τ) < 0, or I(τ) is strictly concave down on [−1, 1]. Therefore, I(τ) ≥ I(±1) with strict inequality unless τ = ±1. This proves (i) and shows that strict inequality holds unless u(s) = Ae^{±kps}.

(ii) In order to establish (ii), just note that

∫_{−L}^{L} \frac{ds}{1 + v(s)^{1/p}} = 2L − ∫_{−L}^{L} \frac{v(s)^{1/p}}{1 + v(s)^{1/p}} ds

and make use of (i).

5. Main results

We now establish some differential identities that will be used later. Assume γ : z = z(s), −L ≤ s ≤ L, is a smooth path in D parametrized by hyperbolic arclength; this means that z′(s) = (1−|z(s)|^{2})e^{iθ(s)} , where θ = arg z′(s), or e^{iθ(s)} is a unit tangent to γ at z(s), and 2L is the hyperbolic length of γ. Let f be any locally univalent function defined on D with f(D) ⊂ D. We always assume f is not a conformal automorphism of D, so |D_{1}f(z)| < 1 for all z ∈ D.

It is straightforward to verify that

\frac{d}{ds}|D_{1}f(z(s))| = |D_{1}f(z(s))| Re\{e^{iθ(s)}Q_{f}(z(s))\}.

Then

\frac{d}{ds} \frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|} = \frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|} \frac{Re\{e^{iθ(s)}Q_{f}(z(s))\}}{1 − |D_{1}f(z(s))|}

434 William Ma and David Minda and for any real number p

\frac{d}{ds}(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} = p(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} \frac{Re\{e^{iθ(s)}Q_{f}(z(s))\}}{1 − |D_{1}f(z(s))|}.

Since

\frac{d}{ds}(\frac{1}{1 − |D_{1}f(z(s))|}) = \frac{|D_{1}f(z(s))| Re\{e^{iθ(s)}Q_{f}(z(s))\}}{(1 − |D_{1}f(z(s))|)^{2}},

we obtain

\frac{d^{2}}{ds^{2}} [(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p}] = p^{2}(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} \frac{Re^{2}\{e^{iθ(s)}Q_{f}(z(s))\}}{(1 − |D_{1}f(z(s))|^{2})^{2}}

+ p(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} \frac{Re\{(d/ds)[e^{iθ(s)}Q_{f}(z(s))]\}}{1 − |D_{1}f(z(s))|}

+ p(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} Re\{e^{iθ(s)}Q_{f}(z(s))\} \frac{d}{ds}(\frac{1}{1 − |D_{1}f(z(s))|})

= p(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} \frac{1}{(1 − |D_{1}f(z(s))|)^{2}}

× [(p + |D_{1}f(z(s))|) Re^{2}{e^{iθ(s)}Q_{f}(z(s))}

+ (1 − |D_{1}f(z(s))|) Re\{\frac{d}{ds}[e^{iθ(s)}Q_{f}(z(s))]\}].

A lengthy, but straightforward, calculation produces

\begin{aligned}e^{iθ(s)} \frac{d}{ds}Q_{f}(z(s)) & = e^{2iθ(s)}(1 − (z(s))^{2})^{2}S_{f}(z(s)) + \frac{1}{2}(e^{iθ(s)}Q_{f}(z(s)))^{2} \\ & + (e^{2iθ(s)}\overline{z}(s) − z(s))Q_{f}(z(s)) − 2 + 2|D_{1}f(z(s))|^{2}. \end{aligned}

The hyperbolic curvature of γ at z(s) is

\begin{aligned}κ_{h}(z(s), γ) & = (1 − |z(s)|^{2})κ_{e}(z(s), γ) + Im\{2e^{iθ(s)}\overline{z}(s)\} \\ & = (1 − |z(s)|^{2})κ_{e}(z(s), γ) − i(e^{iθ(s)}\overline{z}(s) − e^{−iθ(s)}z(s)), \end{aligned}

where

κ_{e}(z(s), γ) = \frac{1}{|z′(s)|} Im\{\frac{z′′(s)}{z′(s)}\}

is the euclidean curvature of γ at z(s). Since γ is parametrized by hyperbolic arclength,

κ_{e}(z(s), γ) = \frac{dθ(s)/ds}{1 − |z(s)|^{2}}

Two-point distortion theorems for bounded univalent functions 435 so that

κ_{h}(z(s), γ) = \frac{dθ(s)}{ds} − i(e^{iθ(s)}\overline{z}(s) − e^{−iθ(s)}z(s))

and

ie^{iθ(s)}κ_{h}(z(s), γ) = \frac{d}{ds}e^{iθ(s)} + (e^{2iθ(s)}\overline{z}(s) − z(s)).

Therefore,

Q_{f}(z(s))\frac{d}{ds}e^{iθ(s)} = iκ_{h}(z(s), γ)e^{iθ(s)}Q_{f}(z(s)) − (e^{2iθ(s)}\overline{z}(s) − z(s))Q_{f}(z(s)).

Next,

\begin{aligned}\frac{d}{ds}(e^{iθ(s)}Q_{f}(z(s))) & = e^{2iθ(s)}(1 − |z(s)|^{2})^{2}S_{f}(z(s)) + \frac{1}{2}(e^{iθ(s)}Q_{f}(z(s)))^{2} − 2 \\ & + iκ_{h}(z(s), γ)e^{iθ(s)}Q_{f}(z(s)) + 2|D_{1}f(z(s))|^{2}. \end{aligned}

From this expression we obtain

\frac{d^{2}}{ds^{2}} [(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p}]

= p(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} \frac{1}{(1 − |D_{1}f(z(s))|)^{2}}

× [(p + |D_{1}f(z(s))|) Re^{2}{e^{iθ(s)}Q_{f}(z(s))} + (1 − |D_{1}f(z(s))|)(Re\{e^{2iθ(s)}(1 − |z(s)|^{2})^{2}S_{f}(z(s)) + \frac{1}{2}(e^{iθ(s)}Q_{f}(z(s)))^{2}\} − 2 − κ_{h}(z(s), γ) Im{e^{iθ(s)}Q_{f}(z(s))} + 2|D_{1}f(z(s))|^{2})].

It is useful to have this second derivative expressed in terms of the hyperbolic curvature of f ∘γ rather than the hyperbolic curvature of γ. A formula connecting the hyperbolic curvature of f ∘ γ to the hyperbolic curvature of γ is

κ_{h}(f(z(s)), f ∘ γ)|D_{1}f(z(s))| = κ_{h}(z(s), γ) + Im{e^{iθ(s)}Q_{f}(z(s))}, so that

\frac{d^{2}}{ds^{2}} [(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p}]

= p(\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p} \frac{1}{(1 − |D_{1}f(z(s))|)^{2}}

× {(p + |D_{1}f(z(s))|) Re^{2}{e^{iθ(s)}Q_{f}(z(s))} + (1 − |D_{1}f(z(s))|)[Im^{2}{e^{iθ(s)}Q_{f}(z(s))} + Re\{e^{2iθ(s)}(1 − |z(s)|^{2})^{2}S_{f}(z(s)) + \frac{1}{2}(e^{iθ(s)}Q_{f}(z(s)))^{2}\} − 2 − κ_{h}(f(z(s)), f ∘ γ)|D_{1}f(z(s))| Im{e^{iθ(s)}Q_{f}(z(s))} + 2|D_{1}f(z(s))|^{2}]}.

Theorem 2. Suppose f is univalent in D and f(D) ⊂ D.

(i) For a, b ∈ D and p ≥ \frac{3}{2} ,

\frac{1}{4} log([[1 + exp(−4pd_{D}(a, b))]^{1/p} + [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{p}]^{1/p}]

/ [[1 + exp(−4pd_{D}(a, b))]^{1/p}

+ exp(−4d_{D}(a, b))[(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{p}]^{1/p}])

≤ d_{D}(f(a), f(b)).
(ii) For a, b ∈ D and p ≥ 1,

d_{D}(f(a), f(b))

≤ \frac{1}{4} log([exp(4d_{D}(a, b))[1 + exp(−4pd_{D}(a, b))]^{1/p}

+ [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{−p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{−p}]^{1/p}]

/ [[1 + exp(−4pd_{D}(a, b))]^{1/p}

+ [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{−p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{−p}]^{1/p}]).

In either inequality equality holds for distinct points a, b ∈ D if and only if f = S ∘ k_{α} ∘ T, where S, T are conformal automorphisms of D and a, b ∈ T^{−1}(−1, 1).

Proof. (i) Fix a, b ∈ D. Intially we assume that the hyperbolic geodesic arc [f(a), f(b)]_{D} = Γ lies in f(D) = Ω and set γ = f^{−1} ∘ Γ. Then γ is a smooth arc in D joining a and b and we suppose γ : z = z(s), −L ≤ s ≤ L, is a hyperbolic arclength parametrization of γ. Then 2L ≥ d_{D}(a, b) with equality if and only if γ is the hyperbolic geodesic arc joining a and b. For p > 0 define

v(s) = (\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{p}.

Then

v′(s) = pv(s)\frac{Re\{e^{iθ(s)}Q_{f}(z(s))\}}{1 − |D_{1}f(z(s))|},

Two-point distortion theorems for bounded univalent functions 437 so the invariant form of Pick’s inequality yields

|v′(s)| ≤ 4pv(s).

Since κ_{h}(f(z(s)), f ∘ γ) = 0, we obtain

v′′(s) = \frac{pv(s)}{1 − |D_{1}f(z(s))|}[\frac{p + |D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|} Re^{2}\{e^{iθ(s)}Q_{f}(z(s))\}

+ Im^{2}{e^{iθ(s)}Q_{f}(z(s))}

+ Re\{e^{2iθ(s)}(1 − |z(s)|^{2})^{2}S_{f}(z(s)) + \frac{1}{2}(e^{iθ(s)}Q_{f}(z(s)))^{2}\}

− 2 + 2|D_{1}f(z(s))|^{2}]

= \frac{pv(s)}{1 − |D_{1}f(z(s))|}[\frac{1 + p}{2(1 − |D_{1}f(z(s))|)}|Q_{f}(z(s))|^{2}

+ Re\{e^{2iθ(s)}((1 − |z(s)|^{2})^{2}S_{f}(z(s)) + \frac{p + |D_{1}f(z(s))|}{2(1 − |D_{1}f(z(s))|)}Q_{f}(z(s))^{2})\}

− 2 + 2|D_{1}f(z(s))|^{2}]

≤ \frac{pv(s)}{1 − |D_{1}f(z(s))|}[\frac{1 + p}{2(1 − |D_{1}f(z(s))|)}|Q_{f}(z(s))|^{2}

+ |(1 − |z(s)|^{2})^{2}S_{f}(z(s)) + \frac{p + |D_{1}f(z(s))|}{2(1 − |D_{1}f(z(s))|)}Q_{f}(z(s))^{2}|

− 2 + 2|D_{1}f(z(s))|^{2}]. By making use of Corollary 2 of Theorem 1 we obtain

v′′(s) ≤ pv(s)[2(8p + 1 + |D_{1}f(z(s))| − 2(1 + |D_{1}f(z(s))|] = 16p^{2}v(s) since p ≥ \frac{3}{2} .

We have shown that v satisfies the hypotheses of Proposition 2 for k = 4 and p ≥ \frac{3}{2} . Now

d_{D}(f(a), f(b)) = ∫_{f∘γ} λ_{D}(w) |dw| = ∫_{γ} \frac{|f′(z)|}{1 − |f(z)|^{2}} |dz|

= ∫_{−L}^{L} \frac{|f′(z(s))|}{1 − |f(z(s))|^{2}} (1 − |z(s)|^{2}) ds = ∫_{−L}^{L}|D_{1}f(z(s))| ds

= ∫_{−L}^{L} \frac{v(s)^{1/p}}{1 + v(s)^{1/p}} ds.

From part (i) of Proposition 2 we conclude that

∫_{−L}^{L} \frac{v(s)^{1/p}}{1 + v(s)^{1/p}} ds ≥ \frac{1}{4} log \frac{[1 + exp(−8pL)]^{1/p} + C}{[1 + exp(−8pL)]^{1/p} + C exp(−8L)},

where

C = C(a, b) = [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{p}]^{1/p},

and equality implies v′′ = 16p^{2}v on [−L, L]. The right-hand side of this inequality is a strictly decreasing function of L when p ≥ 1. Therefore,

\frac{1}{4} log \frac{[1 + exp(−8pL)]^{1/p} + C}{[1 + exp(−8pL)]^{1/p} + C exp(−8L)}

≥ \frac{1}{4} log \frac{[1 + exp(−4pd_{D}(a, b))]^{1/p} + C}{[1 + exp(−4pd_{D}(a, b))]^{1/p} + C exp(−4d_{D}(a, b))}

with strict inequality unless 2L = d_{D}(a, b), that is, unless γ is the hyperbolic geodesic arc between a and b. This proves (i) in the special case [f(a), f(b)]_{D} ⊂ f(D).

Let us check when equality holds in (i) in the special situation [f(a), f(b)]_{D} ⊂ f(D). Equality implies v′′ = 16p^{2}v on [−L, L]. But this means that equality must hold in Corollary 2 of Theorem 1 along the hyperbolic geodesic arc γ. This forces f = S ∘ k_{α} ∘ T, where S, T are conformal automorphisms of D and a, b ∈ T^{−1}(−1, 1).

Now, we turn to the situation in which the geodesic arc [f(a), f(b)]_{D} is not entirely in f(D). Then there exist α, β ∈ ∂Ω such that [f(a), α)_{D} and (β, f(b)]_{D} are disjoint, contained in f(D) and their union is in [f(a), f(b)]_{D} . If z ∈ D, and f(z) ∈ [f(a), α]_{D} , then the first part of the proof gives

d_{D}(f(a), f(z)) ≥ \frac{1}{4} log \frac{[1 + exp(−4pd_{D}(a, z))]^{1/p} + C}{[1 + exp(−4pd_{D}(a, z))]^{1/p} + C exp(−4d_{D}(a, z))}

where C = C(a, z). Since

C(a, z) ≥ \frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|}

we get

d_{D}(f(a), f(z))≥\frac{1}{4} log \frac{[1 + exp(−4pd_{D}(a, z))]^{1/p} + \frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|}}{[1 + exp(−4p_{D}(a, z))]^{1/p} + \frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|} exp(−4d_{D}(a, z))}.

If f(z) → α along [f(a), α)_{D} , then the point z → ∂D and so d_{D}(a, z) → ∞. Therefore, we obtain

d_{D}(f(a), α) ≥ \frac{1}{4} log(1 + \frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|}).

Similarly,

d_{D}(β, f(b)) ≥ \frac{1}{4} log(1 + \frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|}),

so that

d_{D}(f(a), f(b)) ≥ d_{D}(f(a), α) + d_{D}(β, f(b))

≥ \frac{1}{4} log[(1 + \frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})(1 + \frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})]

> \frac{1}{4} log[1 + \frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|} + \frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|}]

≥ \frac{1}{4} log[1 + [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{p}]^{1/p}]

= \frac{1}{4} log[1 + C(a, b)]

> \frac{1}{4} log \frac{[1 + exp(−4pd_{D}(a, b))]^{1/p} + C(a, b)}{[1 + exp(−4pd_{D}(a, b))]^{1/p} + C(a, b) exp(−4d_{D}(a, b))}.

Thus, strict inequality holds in this situation.

(ii) Fix a, b ∈ D and let γ be the hyperbolic geodesic arc between a and b, say γ : z = z(s), −L ≤ s ≤ L, is a hyperbolic arclength parametrization of γ. In this situation d_{D}(a, b) = 2L and κ_{h}(z(s), γ) = 0. If p > 0 and

v(s) = (\frac{|D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|})^{−p},

then

v′(s) = −pv(s)\frac{Re\{e^{iθ(s)}Q_{f}(z(s))\}}{1 − |D_{1}f(z(s))|}

and so

|v′(s)| ≤ 4pv(s)

since f is univalent. Since κ_{h}(z(s), γ) = 0, we have

v′′(s) = \frac{pv(s)}{1 − |D_{1}f(z(s))|}[\frac{p − |D_{1}f(z(s))|}{1 − |D_{1}f(z(s))|} Re^{2}\{e^{iθ(s)}Q_{f}(z(s))\}

− Re\{e^{2iθ(s)}(1 − |z(s)|^{2})^{2}S_{f}(z(s)) + \frac{1}{2}(e^{iθ(s)}Q_{f}(z(s)))^{2}\}

+ 2 − 2|D_{1}f(z(s))|^{2}]

= \frac{pv(s)}{1 − |D_{1}f(z(s))|}[\frac{1}{2} |Q_{f}(z(s))|^{2} + \frac{p − 1}{1 − |D_{1}f(z(s))|} Re^{2}\{e^{iθ(s)}Q_{f}(z(s))\}

− Re{e^{2iθ(s)}(1 − |z(s)|^{2})^{2}S_{f}(z(s))} + 2 − 2|D_{1}f(z(s))|^{2}]

≤ \frac{pv(s)}{1 − |D_{1}f(z(s))|}[\frac{2p − 1 − |D_{1}f(z(s))|}{2(1 − |D_{1}f(z(s))|)}|Q_{f}(z(s))|^{2}

+ (1 − |z(s)|^{2})^{2}|S_{f}(z(s))| + 2(1 − |D_{1}f(z(s)))^{2})].

By using the invariant forms of the inequalities of Pick and Nehari we get

\begin{aligned}v′′(s) & ≤ pv(s)[8(2p − 1 − |D_{1}f(z(s))|) + 6(1 + |D_{1}f(z(s))|) \\ & + 2(1 + |D_{1}f(z(s))|)] = 16p^{2}v(s). \end{aligned}

This shows that v satisfies the hypotheses of Proposition 2 with k = 4 and p ≥ 1. Because f ∘ γ is a path joining f(a) and f(b),

d_{D}(f(a), f(b)) ≤ ∫_{f∘γ} λ_{D}(w) |dw| = ∫_{γ} \frac{|f′(z)|}{1 − |f(z)|^{2}} |dz|

= ∫_{−L}^{L} \frac{|f′(z(s))|}{1 − |f(z(s))|^{2}} (1 − |z(s)|^{2}) ds = ∫_{−L}^{L}|D_{1}f(z(s))| ds

= ∫_{−L}^{L} \frac{ds}{1 + v(s)^{1/p}}

with equality if and only if f ∘ γ is the hyperbolic geodesic joining f(a) and f(b). From part (ii) of Proposition 2 we have

∫_{−L}^{L} \frac{ds}{1 + v(s)^{1/p}} ≤ \frac{1}{4} log \frac{[1 + exp(−8pL)]^{1/p} exp(8L) + D}{[1 + exp(−8pL)]^{1/p} + D} ,

where

D = D(a, b) = [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{−p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{−p}]^{1/p}

and equality implies v′′ = 16p^{2}v on [−L, L]. Since 2L = d_{D}(a, b), this establishes part (ii) of the theorem. Equality forces v′′ = 16p^{2}v on [−L, L], which means that equality must hold in the invariant forms of both the Pick and the Nehari inequalities along the hyperbolic geodesic γ. This implies that f = S ∘ k_{α} ∘ T, where S, T are conformal automorphisms of D and a, b ∈ T^{−1}(−1, 1).

The lower bound in part (i) is a decreasing function of p while the upper bound in (ii) is an increasing function of p. Therefore, the cases p = ∞ of both inequalities are the weakest inequalities contained in Theorem 2. We state these results as a corollary.

Corollary 3. Suppose f is univalent in D and f(D) ⊂ D. Then for a, b ∈ D

max\{\frac{1}{4} log(\frac{1}{1 − |D_{1}f(a)| + |D_{1}f(a)| exp(−4d_{D}(a, b))}),
\frac{1}{4} log(\frac{1}{1 − |D_{1}f(b)| + |D_{1}f(b)| exp(−4d_{D}(a, b))})\}

≤ d_{D}(f(a), f(b))

≤ min\{\frac{1}{4} log(1 − |D_{1}f(a)| + |D_{1}f(a)| exp(4d_{D}(a, b))),

\frac{1}{4} log(1 − |D_{1}f(b)| + |D_{1}f(b)| exp(4d_{D}(a, b)))\}.

This corollary is nothing more than an invariant version of the classical growth theorem for bounded univalent functions. Pick [Pi] proved that if g(z) = αz + a_{2}z^{2} + ⋯ is univalent in D with g(D) ⊂ D, then

−k_{α}(−|z|) ≤ |g(z)| ≤ k_{α}(|z|)

with equality only for rotations of k_{α}. If we choose a = 0, b = z in the corollary, then we obtain

\frac{1}{(1 − α) + α((1 − |z|)/(1 + |z|))^{2}} ≤ (\frac{1 + |g(z)|}{1 − |g(z)|})^{2} ≤ (1 − α) + α(\frac{1 + |z|}{1 − |z|})^{2}.

These inequalities are equivalent to Pick’s.

Remarks. Part (i) of Theorem 2 is also sufficient for univalence. More precisely, if a holomorphic function f: D → D satisfies the inequality (i) for some p ≥ \frac{3}{2} and all a, b ∈ D, then either f is univalent in D, or f is constant. The proof is similar to that given in [KM]. Also, the inequalities in Theorem 2 are linearly invariant in the sense that they are unchanged when f is replaced by S ∘ f ∘ T, where S, T are conformal automorphisms of D.

Theorem 2 bounds d_{D}(f(a), f(b)) above and below in terms of d_{D}(a, b). By slightly rewriting the conclusion of Theorem 2 we can make the bounds more explicit and relate Theorem 2 to the Schwarz–Pick lemma. If f is univalent in D and f(D) ⊂ D, then for p ≥ \frac{3}{2}

d_{D}(a, b) − \frac{1}{4} log([[1 + exp(−4pd_{D}(a, b))]^{1/p} exp(4d_{D}(a, b))

+ [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{p}]^{1/p}]

/ [[1 + exp(−4pd_{D}(a, b))]^{1/p}

+ [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{p}]^{1/p}])

≤ d_{D}(f(a), f(b)),

while for p ≥ 1

d_{D}(f(a), f(b)) ≤ d_{D}(a, b)

− \frac{1}{4} log([[1 + exp(−4pd_{D}(a, b))]^{1/p}

+ [(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{−p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{−p}]^{1/p}]

/ [[1 + exp(−4pd_{D}(a, b))]^{1/p}

+ exp(−4pd_{D}(a, b))[(\frac{|D_{1}f(a)|}{1 − |D_{1}f(a)|})^{−p} + (\frac{|D_{1}f(b)|}{1 − |D_{1}f(b)|})^{−p}]^{1/p}]).

For p = ∞ we obtain

d_{D}(a, b) − min\{\frac{1}{4} log(|D_{1}f(a)| + (1 − |D_{1}f(a)|) exp(4d_{D}(a, b))),

\frac{1}{4} log(|D_{1}f(b)| + (1 − |D_{1}f(b)|) exp(4d_{D}(a, b)))\}

≤ d_{D}(f(a), f(b))

≤ d_{D}(a, b) − max\{\frac{1}{4} log(\frac{1}{|D_{1}f(a)| + (1 − |D_{1}f(a)|) exp(−4d_{D}(a, b))}),

\frac{1}{4} log(\frac{1}{|D_{1}f(b)| + (1 − |D_{1}f(b)|) exp(−4d_{D}(a, b))})\}.

For a general holomorphic function f: D → D Beardon and Carne [BC] proved a refinement of the Schwarz–Pick lemma:

2d_{D}(f(a), f(b)) ≤ log[cosh(2d_{D}(a, b)) + |D_{1}f(a)| sinh(2d_{D}(a, b))].

(They used 2λ_{D}(z)|dz| as the hyperbolic metric.) This inequality is not symmetric in a and b. It can be written more symmetrically as

d_{D}(f(a), f(b))

≤ d_{D}(a, b) − max\{\frac{1}{2} log(\frac{2}{1 + |D_{1}f(a)| + (1 − |D_{1}f(a)|) exp(−4d_{D}(a, b))}),

\frac{1}{2} log(\frac{2}{1 + |D_{1}f(b)| + (1 − |D_{1}f(b)|) exp(−4d_{D}(a, b))})\}.

Our upper bounds for univalent functions are stronger and we also obtain lower bounds.

Two-point distortion theorems for univalent functions on D can be recast as two-point comparison theorems between hyperbolic and euclidean geometry on a simply connected region Ω ≠ C [KM]. Similarly, the two-point distortion theorems for bounded univalent functions in Theorem 2 can be reformulated as comparison theorems between hyperbolic geometry on a simply connected region Ω ⊂ D and hyperbolic geometry on the ambient space D. For the sake of brevity we do not explicitly state all of these comparison results, but only give one comparison theorem that follows from Corollary 3.

Corollary 4. Suppose Ω ⊂ D is a simply connected region, Ω ≠ D. Then for w ∈ Ω

\frac{λ_{D}(w)}{1 − exp(−4ε_{Ω}(w))} ≤ λ_{Ω}(w),

where ε_{Ω}(w) = inf{d_{D}(w, ω) : ω ∈ ∂Ω}.

Proof. Let f: D → Ω be a conformal mapping. Then for a ∈ D, |D_{1}f(a)| = λ_{D}(f(a))/λ_{Ω}(f(a)), so it suffices to show

|D_{1}f(a)| ≤ 1 − exp(−4ε_{Ω}(f(a)))

for a ∈ D. Fix a ∈ D. Then take a sequence {b_{n}}_{n=1}^{∞} in D with |b_{n}| → 1 and d_{D}(f(a), f(b_{n})) → ε_{Ω}(f(a)). Now, Corollary 3 gives

\frac{1}{4} log \frac{1}{1 − |D_{1}f(a)| + |D_{1}f(a)| exp(−4d_{D}(a, b_{n}))} ≤ d_{D}(f(a), f(b_{n})).

By letting n → ∞ we obtain

\frac{1}{4} log \frac{1}{1 − |D_{1}f(a)|} ≤ ε_{Ω}(f(a))

since d_{D}(a, b_{n}) → ∞. This is equivalent to the desired result.

444 William Ma and David Minda

Remark. Corollary 4 is an invariant version of a classical covering theorem for bounded univalent functions. If g(z) = αz + a_{2}z^{2} + ⋯ is univalent in D with g(D) ⊂ D, then Pick [Pi] proved that

g(D) ⊃ \{w : |w| < −k_{α}(−1) = \frac{2 − α − 2\sqrt{1 − α}}{α} \}.

Suppose Ω_{α} = k_{α}(D). Then λ_{Ω_{α}}(0) = 1/α. Also,

ε_{Ω_{α}}(0) = d_{D}(0, −k_{α}(−1)) = \frac{1}{4} log \frac{1}{1 − α},

so that

\frac{λ_{D}(0)}{1 − exp(−4ε_{Ω_{α}}(0))} = \frac{1}{α} = λ_{Ω_{α}}(0).

This shows that the inequality in Corollary 4 is best possible. In fact, Corollary 4 is equivalent to Pick’s covering theorem. This is the analog of the fact that the Koebe \frac{1}{4} -theorem for univalent functions is equivalent to the inequality λ_{Ω}(w) ≥ 1/(4δ_{Ω}(w)) for a simply connected region Ω ≠ C, where δ_{Ω}(w) = inf{|w − ω| : ω ∈ ∂Ω}.

References

[BC] Beardon, A.F., and K. Carne: A strengthening of the Schwarz–Pick inequality. - Amer. Math. Monthly 99, 1992, 216–217.

[B1] Blatter, C.: Ein Verzerrungssatz für schlichte Funktionen. - Comment. Math. Helv. 53, 1978, 651–659.

[FO] Flinn, B., and B. Osgood: Hyperbolic curvature and conformal mapping. - Bull. London Math. Soc. 18, 1986, 272–276.

[KM] Kim, S., and D. Minda: Two-point distortion theorems for univalent functions. - Pacific J. Math. 163, 1994, 137–157.

[MM_{1} ] Ma, W., and D. Minda: Euclidean linear invariance and uniform local convexity. - J.

Austral. Math. Soc. Ser. A 52, 1992, 401–418.

[MM_{2} ] Ma, W., and D. Minda: Two-point distortion theorems for strongly close-to-convex

functions. - Complex Variables Theory Appl. (to appear).

[Ne] Nehari, Z.: Some inequalities in the theory of functions. - Trans. Amer. Math. Soc. 75, 1953, 256–286.

[Pi] Pick, G.: Über die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschränktes Gebiet. - S.-B. Kaiserl. Akad. Wiss. Wien, Math.-Naturwiss. Kl. 126, 1917, 247–263.

[Po] Pommerenke, Ch.: Univalent Functions. - Vandenhoeck and Ruprecht, Göttingen, 1975.

Received 16 February 1996