CHARACTERIZATIONS OF HYPERBOLICALLY CONVEX REGIONS

SEONG-A KIM AND TOSHIYUKI SUGAWA

Abstract. Let X be a simply connected and hyperbolic subregion of the complex plane ℂ. A proper subregion Ω of X is called hyperbolically convex in X if for any two points A and B in Ω, the hyperbolic geodesic arc joining A and B in X is always contained in Ω. We establish a number of characterizations of hyperbolically convex regions Ω in X in terms of the relative hyperbolic density ρ_{Ω}(w) of the hyperbolic metric of Ω to X, that is the ratio of the hyperbolic metric λ_{Ω}(w)|dw| of Ω to the hyperbolic metric λ_{X}(w)|dw| of X. Introduction of hyperbolic differential operators on X makes calculations much simpler and gives analogous results to some known characterizations for euclidean or spherical convex regions. The notion of hyperbolic concavity relative to X for realvalued functions on Ω is also given to describe some sufficient conditions for hyperbolic convexity.

1. Introduction

The Riemann sphere ℙ = ℂ ∪ {∞}, the complex plane ℂ and the unit disk 𝔻 = {z ∈ ℂ : |z| < 1} have the canonical metrics λ_{ℙ}(z)|dz|, λ_{ℂ}(z)|dz| and λ_{𝔻}(z)|dz| of constant curvature +4, 0 and −4, respectively. Here λ_{ℙ}(z) = 1/(1 + |z|^{2}), λ_{ℂ}(z) = 1 and λ_{𝔻}(z) = 1/(1 − |z|^{2}).

It is well known that a region Ω in the Riemann sphere ℙ with #(ℙ \ Ω) ≥ 3 admits a (holomorphic) universal covering projection p of the unit disk 𝔻 onto Ω and the region is called hyperbolic. Since the metric λ_{𝔻}(z)|dz| is invariant under the covering transformation group of p, the region Ω carries the unique metric λ_{Ω}(w)|dw| determined from λ_{Ω}(p(z))|p′(z)| = λ_{𝔻}(z) for all z ∈ 𝔻 where w = p(z). The metric is independent of the particular choice of p. We call λ_{Ω}(w)|dw| the hyperbolic metric of Ω and λ_{Ω} the hyperbolic density of the hyperbolic metric of Ω. Note that the density λ_{Ω} is real analytic and hence it is smooth. Also, the hyperbolic metric has constant Gaussian curvature −4, that is, −Δ log λ_{Ω} = −4λ_{Ω}^{2}.

The quantity λ_{Ω}(w) can be regarded as the ratio of the hyperbolic metric λ_{Ω}(w)|dw| to the euclidean metric λ_{ℂ}(w)|dw|. Thus, it is sometimes called the euclidean density of the hyperbolic metric of Ω.

Let X be a hyperbolic region in ℙ or ℙ itself or ℂ and A, B ∈ X. We denote by d_{X}(A, B) the distance between A and B measured by the metric λ_{X}(w)|dw|, namely,

d_{X}(A, B) = inf ∫_{γ} λ_{X}(w)|dw|,

Date: February 7, 2004.

1991 Mathematics Subject Classification. Primary 30F45; Secondary 52A55, 53A35.

Key words and phrases. hyperbolically convex, hyperbolically concave function, hyperbolic metric.

Research supported by Woosuk University, Korea.

where the infimum is taken over all paths γ in X joining A and B. It is known that there exists an arc δ joining two points A and B in X such that d_{X}(A, B) = ∫_{δ} λ_{X}(w)|dw|. We call such an arc δ a geodesic arc joining A and B in X. Note that the geodesic arc joining A and B in X is uniquely determined by A and B when X is a simply connected hyperbolic region.

For example,

d_{ℙ}(A, B) = arctan |\frac{A − B}{1 + \overline{A}B} | and d_{𝔻}(A, B) = arctanh |\frac{A − B}{1 − \overline{A}B} | .

A geodesic arc is the shorter arc of the great circle joining two distinct points in ℙ in the case of ℙ and the part of the circular arc joining two distinct points in 𝔻 which is perpendicular to the boundary of 𝔻 in the case of 𝔻.

Characterizations of (euclidean) convex regions Ω in the complex plane ℂ have been given as analytic or geometric properties of the density λ_{Ω}(w) of the hyperbolic metric of Ω ([2], [3], [11], [13], [14], and [15]). We state some of those characterizations in the next theorem (see [3] for a unified, geometric approach).

Theorem A.

Suppose that Ω is a hyperbolic region in ℂ. Then the following are equivalent:

(i) Ω is convex.

(ii) |∂ \frac{1}{λ_{Ω}} | ≤ 1.

(iii) Δ\frac{1}{λ_{Ω}} ≤ 0.

(iv) 1/λ_{Ω} is concave on Ω.

(v) \frac{1}{λ_{Ω}} |∂^{2} \frac{1}{λ_{Ω}} | + |∂ \frac{1}{λ_{Ω}} |^{2} ≤ 1.

Analogous results for spherically convex regions Ω in the Riemann sphere ℙ were also obtained in terms of the spherical density

μ_{Ω}(w) = \frac{λ_{Ω}(w)|dw|}{λ_{ℙ}(w)|dw|} = (1 + |w|^{2})λ_{Ω}(w)

of the hyperbolic metric of Ω by Kim and Minda [4]. Here, a region in ℙ is called spherically convex if it is convex relative to spherical geometry determined by λ_{ℙ}(w)|dw|. In the spherical case, the differential operators ∂, ∂^{2} and Δ should be replaced by the spherical ones ∂_{ℙ}, ∂_{ℙ}^{2} and Δ_{ℙ}, respectively, and the notion of concavity should be modified to that of spherical concavity. Precise definitions of those notions will be given in Sections 2 and 3.

The following characterizations of spherically convex regions are due to Kim and Minda [4] except for (ii) which was essentially found by Ma and Minda ([6], Theorem 4) earlier.

Theorem B.

Suppose that Ω is a hyperbolic region in ℙ. Then the following are equivalent:

(i) Ω is spherically convex.

(ii) |∂_{ℙ} \frac{1}{μ_{Ω}} | ≤ 1.

(iii) |∂_{ℙ} \frac{1}{μ_{Ω}} |^{2} ≤ 1 − \frac{1}{μ_{Ω}^{2}} .

(iv) Δ_{ℙ} \frac{1}{μ_{Ω}} ≤ −\frac{8}{μ_{Ω}} .

(v) 1/μ_{Ω} is spherically concave on Ω.

(vi) \frac{1}{μ_{Ω}} |∂_{ℙ}^{2} \frac{1}{μ_{Ω}} | + |∂_{ℙ} \frac{1}{μ_{Ω}} |^{2} ≤ 1 − \frac{1}{μ_{Ω}^{2}} .

On the other hand, hyperbolically convex regions in the unit disk 𝔻 which are convex relative to the hyperbolic geometry of 𝔻 appear quite naturally, for instance, as Dirichlet fundamental regions of Fuchsian groups acting on 𝔻. Recently, conformal homeomorphisms of the unit disk 𝔻 onto hyperbolically convex regions in 𝔻, which are called hyperbolically convex functions, were intensively studied by Ma and Minda ([7], [9]) and by Mejía and Pommerenke [10]. Meanwhile, it seems that intrinsic characterizations of hyperbolic regions are less known.

In this article, we define the notion of hyperbolic concavity for a real-valued function on a subregion Ω of a simply connected hyperbolic region X in ℂ and deduce some equivalent conditions for the concavity (Section 3). For a hyperbolic subregion Ω of a simply connected hyperbolic region X, we consider the ratio ρ_{Ω,X}(w) of the hyperbolic metric λ_{Ω}(w)|dw| of Ω to the hyperbolic metric λ_{X}(w)|dw| of X:

ρ_{Ω,X}(w) = \frac{λ_{Ω}(w)|dw|}{λ_{X}(w)|dw|}.

We call ρ_{Ω,X}(w) the relative hyperbolic density of the hyperbolic metric of Ω to the hyperbolic metric of X or simply, the relative hyperbolic density of Ω to X. We write ρ_{Ω} for ρ_{Ω,X} when it is clear in the context. Note that ρ_{Ω}(w) ≥ 1 holds for all w ∈ Ω and equality holds at some point precisely when Ω = X. We will say that Ω(⊂ X) is hyperbolically convex in X if for any two points A, B in Ω, the geodesic arc joining A and B in X lies entirely in Ω. We will give characterizing properties of hyperbolically convex regions of X in terms of the relative hyperbolic density ρ_{Ω}(w) in Section 4. To this end, we use the natural differential operators ∂_{X}, ∂_{X}^{2} and Δ_{X}, called hyperbolic differential operators (see Section 2 for precise definitions and fundamental properties of them). Moreover, by using the notion of hyperbolic concavity (Section 4), we give some sufficient conditions for subregions of X to be hyperbolically convex, which are expected to be equivalent ones. Meanwhile, the notion of hyperbolic concavity may be of independent interest.

2. Hyperbolic calculus

In the complex plane, the partial differential operators

∂ = \frac{∂}{∂w} = \frac{1}{2} (\frac{∂}{∂u} − i\frac{∂}{∂v} ) ,
¯∂ = \frac{∂}{∂w̄} = \frac{1}{2} (\frac{∂}{∂u} + i\frac{∂}{∂v} )

where w = u + iv and the Laplacian Δ = 4∂ ¯∂ are of fundamental importance. However, the following differential operators are more natural for the regions X which is a hyperbolic region in ℙ or the Riemann sphere ℙ when we are concerned with intrinsic geometry of those regions (cf. [4]):

∂_{X} = \frac{1}{λ_{X}} ∂,
∂_{X}^{2} = \frac{1}{λ_{X}^{2}} [∂^{2} − 2(∂_{X}λ_{X})∂] = \frac{1}{λ_{X}^{2}} [∂^{2} − 2(∂ log λ_{X})∂] ,
Δ_{X} = \frac{1}{λ_{X}^{2}} Δ.

We note that ∂_{X}^{2} is not equal to ∂_{X}∂_{X} unlike the euclidean case: ∂^{2} = ∂∂. For instance,

∂_{𝔻}^{2}r(z) = ∂_{𝔻}∂_{𝔻}r(z) − z̄∂_{𝔻}r(z)

for the unit disk 𝔻. When X = ℙ, the above-defined operators are called spherical differential operators. When X is a hyperbolic region in ℙ, we call them hyperbolic differential operators relative to X. We now observe a behavior of these operators under a holomorphic covering projection p. (The special case when X = 𝔻 and p is an analytic automorphism of 𝔻 was given in Section 3.2 of [4].)

Lemma 1. Let X and Y be hyperbolic regions in ℂ and suppose that p : Y → X is a holomorphic (unbranched) covering projection of Y onto X. Let r be a function of class C^{2} defined in a neighborhood of a point a ∈ X and let b ∈ p^{−1}(a). Then the following formulae hold near the point b :

∂_{Y} (r ∘ p) = \frac{p′}{|p′|}[(∂_{X}r) ∘ p],

∂_{Y}^{2} (r ∘ p) = (\frac{p′}{|p′|})^{2} [(∂_{X}^{2}r) ∘ p],
Δ_{Y} (r ∘ p) = (Δ_{X}r) ∘ p.

Proof. A straightforward computation together with the fundamental relation (λ_{X} ∘ p)|p′| = λ_{Y} yields these formulae.

In particular, the quantities |∂_{X}r|, |∂_{X}^{2}r| and Δ_{X}r are invariant under analytic automorphisms of X. This fact will be crucial in the proof of our main theorems. We also remark that the hyperbolic differential operators ∂_{X} and ∂_{X}^{2} cannot be defined at the point at infinity although the absolute value of them is well defined there.

We now recall another kind of invariant differential operators [7]. For a holomorphic map f : 𝔻 → 𝔻, we define

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

and

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{2z̄(1 − |z|^{2})f′(z)}{1 − |f(z)|^{2}}.

It is sometimes convenient to use the notation

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

In terms of these operators, Ma and Minda (Theorems 3 and 5 in [7], Corollary 3.2 and Theorem 5.1 in [9]) established the following characterizations of hyperbolically convex regions in 𝔻.

Theorem C. Let f be a holomorphic universal covering projection of the unit disk 𝔻 onto a subregion Ω of 𝔻. Then, the following are equivalent:

(i) Ω is hyperbolically convex in 𝔻. (ii) |D_{h2}f(z)/(2D_{h1}f(z))| = |Q_{f}(z)/2| < 1 for z ∈ 𝔻. (iii) |D_{h2}f(z)/(2D_{h1}f(z))| = |Q_{f}(z)/2| ≤ 1 − |D_{h1}f(z)|^{2} for z ∈ 𝔻.

(iv) (1 − |z|^{2})^{2}|S_{f}(z)| + \frac{3}{4}|Q_{f}(z)|^{2} ≤ 3 for z ∈ 𝔻.

In the above theorem, equalities hold when Ω is a hyperbolic half plane of 𝔻 in (iii) and (iv), and S_{f} denotes the Schwarzian derivative of f :

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

It is also easy to verify the following chain rules (cf. Section 3.5 in [4]).

Lemma 2. Let Ω be a subregion of the unit disk 𝔻 and r : Ω → ℝ be a function of class C^{2}. For a holomorphic map f : 𝔻 → Ω, the following formulae hold:

∂_{𝔻}(r ∘ f) = (∂_{𝔻}r) ∘ f · D_{h1}f,
∂_{𝔻}^{2}(r ∘ f) = (∂_{𝔻}^{2}r) ∘ f · (D_{h1}f)^{2} + (∂_{𝔻}r) ∘ f · D_{h2}f.

We also have the following analog of formulae (4) and (5) in [4].

Lemma 3. Let f : 𝔻 → 𝔻 be a locally injective holomorphic map. Then

∂_{𝔻}|D_{h1}f| = \frac{1}{2}|D_{h1}f|Q_{f},
∂_{𝔻}^{2}|D_{h1}f| = \frac{1}{2}|D_{h1}f| (λ_{𝔻}^{−2}S_{f} + Q_{f}^{2}) .

Proof. It is easy to check the first formula. In order to show the second, we observe that

∂_{𝔻}^{2}|D_{h1}f| = ∂_{𝔻}(∂_{𝔻}|D_{h1}f|) − z̄∂_{𝔻}|D_{h1}f|

= ∂_{𝔻} (\frac{1}{2}|D_{h1}f|Q_{f}) − \frac{z̄}{2}|D_{h1}f|Q_{f}

= \frac{1}{2}Q_{f} ∂_{𝔻}|D_{h1}f| + \frac{1}{2}|D_{h1}f|∂_{𝔻}Q_{f} − \frac{z̄}{2}|D_{h1}f|Q_{f}

= \frac{1}{2}|D_{h1}f| (∂_{𝔻}Q_{f} + \frac{1}{2}Q_{f}^{2} − z̄Q_{f}) .

A straightforward computation yields the relation,

(∂_{𝔻}Q_{f})(z) = (1 − |z|^{2})^{2}S_{f}(z) + \frac{1}{2}Q_{f}(z)^{2} + z̄Q_{f}(z),

and hence, we obtain the desired identity.

3. Hyperbolically concave functions

Concavity (or convexity) for real-valued functions is a fundamental notion in real analysis. Beckenbach [1] developed a theory of generalized convexity for real-valued functions on (one-dimensional) intervals while some people used the notion of geodesically convex functions on a Riemannian manifold which are convex along each geodesic arc with constant speed. We somewhat combine these notions to give a definition of hyperbolically concave functions in this article.

Recall that a real-valued function r defined on a plane region Ω ⊂ ℂ is said to be concave in Ω if the inequality

r((1 − t)w_{0} + tw_{1}) ≥ (1 − t)r(w_{0}) + tr(w_{1})

holds for every t ∈ [0, 1] whenever the line segment [w_{0}, w_{1}] joining two points w_{0} and w_{1} in Ω is contained in Ω (cf. [3]). Note that we do not require Ω to be convex.

Similarly, we can define hyperbolic concavity. Let Ω be a subregion of a simply connected hyperbolic region X in ℂ. A real-valued function r on Ω is said to be hyperbolically concave relative to X if the inequality

(3.1) r(w_{t}) ≥ \frac{sinh[2(1 − t)d]r(w_{0}) + sinh[2td]r(w_{1})}{sinh[2d]}

holds for each t ∈ [0, 1], where d = d_{X}(w_{0}, w_{1}) and w_{t} is the unique point in X such that d_{X}(w_{0}, w_{t}) = td and that d_{X}(w_{t}, w_{1}) = (1 − t)d for w_{0}, w_{1} ∈ Ω, whenever the geodesic arc γ joining w_{0} and w_{1} in X lies entirely in Ω. We remark that the point w_{t} lies in γ necessarily. For brevity, we will write w_{t} = P_{X}(w_{0}, w_{1}, t).

First we show a continuity of this function for completeness though we do not use it in this article. In the proof of the next lemma, set D_{X}(w_{0}, d) = {w ∈ X : d_{X}(w_{0}, w) ≤ d} for a point w_{0} ∈ Ω.

Lemma 4. A hyperbolically concave function r : Ω → ℝ relative to X is continuous.

Proof. Let w_{0} be an arbitrary point in Ω. First we assume that r ≥ −M in the neighborhood V = D_{X}(w_{0}, d_{0}) ⊂ Ω, where M and d_{0} are positive constants. Note that for t ∈ [0, 1], D_{X}(w_{0}, td_{0}) = {P_{X}(w_{0}, w, s) : 0 ≤ s ≤ t, w ∈ ∂V }. We take an arbitrary point w_{1} on ∂V and set w_{t} = P_{X}(w_{0}, w_{1}, t). By (3.1), for each t where 0 ≤ t ≤ 1, we get the inequality

(3.2) r(w_{t}) − r(w_{0}) ≥ (\frac{sinh[2(1 − t)d_{0}]}{sinh[2d_{0}]} − 1) r(w_{0}) − \frac{sinh[2td_{0}]}{sinh[2d_{0}]}M

In order to get an upper bound of [r(w_{t}) − r(w_{0})], we choose a point w_{−1} ∈ ∂V so that w_{0} is the hyperbolic midpoint of w_{1} and w_{−1} in X. Since w_{0} = P_{X}(w_{−1}, w_{t}, 1/(1 + t)), we obtain the inequality

r(w_{0}) ≥ \frac{sinh[2td_{0}]r(w_{−1}) + sinh[2d_{0}]r(w_{t})}{sinh[2(1 + t)d_{0}]}

by (3.1), and hence,

(3.3) r(w_{t}) − r(w_{0}) ≤ (\frac{sinh[2(1 + t)d_{0}]}{sinh[2d_{0}]} − 1) r(w_{0}) + \frac{sinh[2td_{0}]}{sinh[2d_{0}]}M.

Both of the right-hand sides in (3.2) and (3.3) tend to 0 as t → 0. Hence we have shown that r is continuous at w_{0} if it is locally bounded below.

On the other hand, the local lower boundedness of r can be easily shown. Indeed, consider a compact hyperbolic triangle T in Ω relative to X. Then, by (3.1), the function r is bounded below on each side of T. Applying (3.1) again, we can deduce that r is bounded below on T. Since Ω is covered by such triangles, the local lower boundedness follows.

Now, we give characterizations of hyperbolic concavity when r is of class C^{2}. First, we need a kind of minimum principle for solutions to a boundary value problem for an ordinary differential equation. See [8] for a proof of the following result or apply the Strong Minimum Principle to the function v − u (see [12], p. 260).

Lemma 5. Let u and v be real-valued functions of class C^{2} on the interval [a, b] and suppose that v′′ ≤ 4v and u′′ = 4u there. If u(a) = v(b) and u(b) = v(b), then either v = u on [a, b] or v > u on (a, b).

We are now ready to state our important result for hyperbolic concavity.

Theorem 1. Let Ω be a subregion of a simply connected hyperbolic region X in ℂ and r be a real-valued function of class C^{2} on Ω. Then the following are equivalent:

(i) r is hyperbolically concave on Ω relative to X. (ii) Whenever the geodesic arc joining w_{0} and w_{1} in X is contained in Ω, the midpoint

m of it satisfies the inequality,

r(m) ≥ \frac{r(w_{0}) + r(w_{1})}{2 cosh d_{X}(w_{0}, w_{1})}. (3.4)

(iii) Whenever the geodesic arc w(s) in X parametrized by its hyperbolic arclength is contained in Ω, the function v(s) = r(w(s)) satisfies the differential inequality v′′(s)− 4v(s) ≤ 0.

(iv) The inequality

|∂_{X}^{2}r(w)| + \frac{1}{4}Δ_{X}r(w) ≤ 2r(w) (3.5)

holds on Ω.

Proof.

(i)⇒(ii): Just put t = 1/2 in the inequality (3.1).

(ii)⇒(iii): Let w(s) be a geodesic arc in Ω parametrized by its hyperbolic arclength and set s = s_{0}. For w_{0} = w(s_{0} − δ) and w_{1} = w(s_{0} + δ), we obtain the inequality

v(s_{0}) ≥ \frac{v(s_{0} − δ) + v(s_{0} + δ)}{2 cosh[2δ]}

by (3.4) or, equivalently,

\frac{v(s_{0} − δ) + v(s_{0} + δ) − 2v(s_{0})}{δ^{2}} ≤ 2 \frac{cosh[2δ] − 1}{δ^{2}} v(s_{0}).

Letting δ → 0, we obtain the inequality v′′(s_{0}) ≤ 4v(s_{0}).

(iii)⇒(i): Let w_{0}, w_{1} ∈ Ω and suppose that the geodesic arc joining w_{0} and w_{1} in X is contained in Ω. Set w(s) = w_{s/d} = P (w_{0}, w_{1}, s/d) for s ∈ [0, d], where d = d_{X}(w_{0}, w_{1}) and u(s) = (sinh[2(d − s)]r(w_{0}) + sinh[2s]r(w_{1}))/ sinh[2d]. Then u satisfies the differential equation u′′ − 4u = 0 and the boundary conditions u(0) = r(w_{0}), u(d) = r(w_{1}). Applying Lemma 5 to the function v(s) = r(w(s)) yields the inequality r(w_{s/d}) ≥ u(s) for s ∈ [0, d], which is same as (3.1).

(iii)⇔(iv): Since the left-hand side of (3.5) is conformally invariant by Lemma 1, we may assume that X = 𝔻.

Let γ : s ↦ w(s) be a smooth arc parametrized by hyperbolic arclength in 𝔻 and suppose that γ lies in Ω. (For a while, we do not assume γ to be a geodesic arc.) Letting θ(s) = arg w′(s), we may write w′(s) = |w′(s)|e^{iθ(s)} = (1 − |w(s)|^{2})e^{iθ(s)}.

Since r is real-valued, the derivative of v(s) = r(w(s)) can be obtained by

v′(s) = ∂r(w(s))w′(s) + ¯∂r(w(s))\overline{w′(s)}
= 2 Re {∂r(w(s))w′(s)}
= 2 Re {(1 − |w(s)|^{2})∂r(w(s))e^{iθ(s)}}
= 2 Re {e^{iθ(s)}∂_{𝔻}r(w(s))}.

Let κ_{𝔻}(w(s), γ) and κ_{ℂ}(w(s), γ) be the hyperbolic curvature and the euclidean curvature of γ at w(s), respectively (cf. [5]). That is,

κ_{ℂ}(w(s), γ) = \frac{Im \{\frac{w′′(s)}{w′(s)}\}}{|w′(s)|} = \frac{Im \{\frac{w′′(s)}{w′(s)}\}}{1 − |w(s)|^{2}}

and

κ_{𝔻}(w(s), γ) = (1 − |w(s)|^{2})κ_{ℂ}(w(s), γ) + 2 Im \{\overline{w(s)}e^{iθ(s)}\}.

Then

Im \{\frac{w′′(s)}{w′(s)}\} = κ_{𝔻}(w(s), γ) − 2 Im \{\overline{w(s)}e^{iθ(s)}\}.

By the relation w′(s) = (1 − |w(s)|^{2})e^{iθ(s)}, we get

Re \{\frac{w′′(s)}{w′(s)}\} = −2 Re \{\overline{w(s)}e^{iθ(s)}\},

and hence,

w′′(s) = −2 \overline{w(s)}(1 − |w(s)|^{2})e^{2iθ(s)} + i(1 − |w(s)|^{2})e^{iθ(s)}κ_{𝔻}(w(s), γ).

By using the previous equality, we have

v′′(s) = 2 Re {∂^{2}r(w(s))w′(s)^{2}} + 2 Re {∂r(w(s))w′′(s)} + 2 ∂ ¯∂r(w(s))|w′(s)|^{2}
= 2 Re \{[(1 − |w(s)|^{2})^{2}∂^{2}r(w(s)) − 2 \overline{w(s)}(1 − |w(s)|^{2})∂r(w(s))] e^{2iθ(s)}\}
+2 (1 − |w(s)|^{2})^{2}∂ ¯∂r(w(s)) − 2 κ_{𝔻}(w(s), γ)Im {(1 − |w(s)|^{2})∂r(w(s))e^{iθ(s)}} .

Hence,

v′′(s) = 2 Re \{e^{2iθ(s)} · ∂_{𝔻}^{2}r(w(s))\} + \frac{1}{2}Δ_{𝔻}r(w(s))
−2 κ_{𝔻}(w(s), γ) · Im {e^{iθ(s)}∂_{𝔻}r(w(s))}.

From now on, we assume γ to be a geodesic arc. So, κ_{𝔻}(w(s), γ) = 0, and therefore,

v′′(s) = 2 Re \{e^{2iθ(s)} · ∂_{𝔻}^{2}r(w(s))\} + \frac{1}{2}Δ_{𝔻}r(w(s)).

Since the argument θ(s) can be chosen arbitrarily for a given point w = w(s) by taking a suitable geodesic arc passing through w, the inequality (3.5) holds at the point w. The converse is obvious by the last formula.

Remark. We can similarly define spherical concavity for a real-valued function defined on a hyperbolic region in ℙ by replacing d_{X}(w_{0}, w_{1}) and sinh by d_{ℙ}(w_{0}, w_{1}) and sin, respectively in the inequality (3.1). When r is of class C^{2}, our definition is equivalent to that of Kim and Minda [4]: A real-valued function r of class C^{2} is called spherically concave in Ω if the function v given by v(s) = r(w(s)) satisfies the inequality v′′(s) + 4v(s) ≤ 0 for a spherical geodesic arc w(s) in Ω which is parametrized by its spherical arclength.

The composition of a hyperbolically concave function with a certain function can also be hyperbolically concave. We show this in the next lemma.

Lemma 6. Suppose that a function h : (0, M) → ℝ satisfies the following three conditions on (0, M): (a) h is non-decreasing, (b) h(x)/x is non-increasing, and (c) h is concave. Let Ω be a subregion of a hyperbolic region X in ℂ. If a hyperbolically concave function r : Ω → ℝ relative to X takes its values in (0, M), then the composed function h ∘ r is also hyperbolically concave on Ω relative to X.

Proof. We need to show the inequality (3.1) for h ∘ r. First, we put

c = \frac{sinh[2(1 − t)d] + sinh[2td]}{sinh[2d]} and s = \frac{sinh[2td]}{sinh[2(1 − t)d] + sinh[2td]}.

Note that 0 < c ≤ 1 because sinh x is super-additive: sinh(x + y) > sinh x + sinh y for x, y > 0. Now the hyperbolic concavity of r gives

r(w_{t}) ≥ c[(1 − s)r(w_{0}) + sr(w_{1})].

By the condition (b), h(cx) ≥ ch(x) for x ∈ (0, M). This together with (a) and (c) implies

h(r(w_{t})) ≥ h(c[(1 − s)r(w_{0}) + sr(w_{1})])
≥ ch([(1 − s)r(w_{0}) + sr(w_{1})])
≥ c[(1 − s)h(r(w_{0})) + sh(r(w_{1}))],

which is the desired inequality.

4. Main theorems

In this section, we give characterizations of hyperbolically convex regions Ω in a simply connected hyperbolic region X in ℂ in terms of the relative hyperbolic density ρ_{Ω} of Ω to X.

Theorem 2. Let Ω be a subregion of a simply connected hyperbolic region X in ℂ and let ρ_{Ω} be the relative hyperbolic density of Ω to X. Then the following are equivalent:

(i) Ω is hyperbolically convex in X.

(ii) |∂_{X} \frac{1}{ρ_{Ω}} | ≤ 1.

(iii) |∂_{X} \frac{1}{ρ_{Ω}} | ≤ 1 − \frac{1}{ρ_{Ω}^{2}} .

(iv) Δ_{X} \frac{1}{ρ_{Ω}} ≤ \frac{4}{ρ_{Ω}} .

(v) 1/ρ_{Ω} is superharmonic in Ω.

(vi) Δ_{X} \frac{1}{ρ_{Ω}} ≤ −\frac{4}{ρ_{Ω}} (1 − \frac{1}{ρ_{Ω}^{2}} ).

(vii) \frac{1}{ρ_{Ω}} |∂_{X}^{2} \frac{1}{ρ_{Ω}} | ≤ \frac{3}{2} (1 − |∂_{X} \frac{1}{ρ_{Ω}} |^{2}) .

Equalities hold in (iii), (vi) and (vii) when Ω is a hyperbolic half plane of X. Note that we do not assume Ω to be simply connected in the theorem. In euclidean convex regions, a condition analogous to (iii) cannot appear as a characterizing property of euclidean convex regions since equality holds for a half plane in (ii) of Theorem A.

We now give sufficient conditions for hyperbolic convexity in terms of hyperbolic concavity of some functions involving the relative hyperbolic density.

Theorem 3. Let Ω be a subregion of a simply connected hyperbolic region X in ℂ and let ρ_{Ω} be the relative hyperbolic density of Ω to X. Then, concerning the following conditions, the implication relations (i) ⇔ (ii) ⇒ (iii) ⇒ (iv) hold:

(i) \frac{1}{ρ_{Ω}} |∂_{X}^{2} \frac{1}{ρ_{Ω}} | + |∂_{X} \frac{1}{ρ_{Ω}} |^{2} ≤ 1 + \frac{1}{ρ_{Ω}^{2}} .

(ii) 1/ρ_{Ω} is hyperbolically concave on Ω relative to X. (iii) tanh(1/ρ_{Ω}) is hyperbolically concave on Ω relative to X. (iv) Ω is hyperbolically convex in X.

In view of the equivalence of the condition (iv) in Theorem A or B to the corresponding convexity of the subregion, it is expected that the conditions in Theorem 3 are all equivalent. The authors, however, did not succeed in showing it.

Next, we establish characterizations of hyperbolically convex regions in X in terms of two-point distortion. Theses are hyperbolic analogs of the results in [15] for the euclidean case and in [4] for the spherical case. Theorem 4. Let Ω be a subregion of a hyperbolic region X in ℂ and denote by ρ_{Ω} the relative hyperbolic density of Ω to X. Then the following are equivalent:

(i) Ω is hyperbolically convex in X.

(ii) |\frac{1}{ρ_{Ω}(A)} − \frac{1}{ρ_{Ω}(B)}| ≤ 2 d_{X}(A, B) for all A, B ∈ Ω.

(iii) | arctanh (\frac{1}{ρ_{Ω}(A)}) − arctanh (\frac{1}{ρ_{Ω}(B)})| ≤ 2 d_{X}(A, B) for all A, B ∈ Ω.

To prove the above theorems, we first establish a formula related to the hyperbolic Laplacian of ρ_{Ω}. This is an analogous result to [4, Lemma 1 (8)] for the spherical case. Lemma 7. Let Ω be a subregion of a hyperbolic region X in ℂ. Then the relative hyperbolic density ρ_{Ω} of Ω to X satisfies the relation

\frac{1}{4 ρ_{Ω}} Δ_{X} (\frac{1}{ρ_{Ω}} ) = |∂_{X} (\frac{1}{ρ_{Ω}} )|^{2} − (1 − \frac{1}{ρ_{Ω}^{2}} ) . (4.1)

Proof. By the curvature equation for the hyperbolic metric, we have

Δ_{X} log ρ_{Ω} = \frac{1}{λ_{X}^{2}} Δ log ρ_{Ω}
= \frac{1}{λ_{X}^{2}} (Δ log λ_{Ω} − Δ log λ_{X})
= \frac{1}{λ_{X}^{2}} (4λ_{Ω}^{2} − 4λ_{X}^{2})
= 4 (ρ_{Ω}^{2} − 1).

On the other hand, by definition of Δ_{X}, we have

Δ_{X} log ρ_{Ω} = −\frac{1}{λ_{X}^{2}} Δ log \frac{1}{ρ_{Ω}} = −\frac{4}{λ_{X}^{2}} ∂ ¯∂log \frac{1}{ρ_{Ω}}
= −\frac{4}{λ_{X}^{2}} ∂[ρ_{Ω} ¯∂(\frac{1}{ρ_{Ω}} )]
= −\frac{4ρ_{Ω}}{λ_{X}^{2}} ∂ ¯∂(\frac{1}{ρ_{Ω}} ) − \frac{4}{λ_{X}^{2}} ∂ρ_{Ω} ¯∂(\frac{1}{ρ_{Ω}} )
= −ρ_{Ω} Δ_{X} (\frac{1}{ρ_{Ω}} ) + \frac{4ρ_{Ω}^{2}}{λ_{X}^{2}} ∂ (\frac{1}{ρ_{Ω}} ) · ¯∂(\frac{1}{ρ_{Ω}} )

= −ρ_{Ω} Δ_{X} (\frac{1}{ρ_{Ω}} ) + 4ρ_{Ω}^{2} |∂_{X} (\frac{1}{ρ_{Ω}} )|^{2} .

In view of these two representations of Δ_{X} log ρ_{Ω}, we obtain

ρ_{Ω} Δ_{X} (\frac{1}{ρ_{Ω}} ) = 4ρ_{Ω}^{2} |∂_{X} (\frac{1}{ρ_{Ω}} )|^{2} − 4(ρ_{Ω}^{2} − 1).

Let Ω be a subregion of a hyperbolic region X in ℂ and g : X → Y be a conformal homeomorphism. Then, we have the obvious relation ρ_{Ω,X} = ρ_{Ω′,Y} ∘ g on Ω where Ω′ = g(Ω). Therefore, the conditions which appear in Theorem 2 are invariant under conformal mappings and hence, it is enough to prove the theorem for X = 𝔻.

By combining the formulae in the following lemma with conditions in Theorem C, we can obtain some of our conditions in Theorem 2.

Lemma 8. Let f be a holomorphic universal covering projection of the unit disk 𝔻 onto a subregion Ω of 𝔻. Then the relative hyperbolic density ρ_{Ω} of Ω to 𝔻 satisfies the following relations:

(4.2)  (\frac{1}{ρ_{Ω}} ) ∘ f = |D_{h1}f|,
(4.3)  (∂_{𝔻} \frac{1}{ρ_{Ω}} ) ∘ f = \frac{|f′|}{f′} · \frac{D_{h2}f}{2 D_{h1}f} = \frac{|f′|}{f′} · \frac{Q_{f}}{2} ,
(4.4)  (∂_{𝔻}^{2} \frac{1}{ρ_{Ω}} ) ∘ f = \frac{|f′|}{f′} · \frac{λ_{𝔻}^{−2}S_{f}}{2 D_{h1}f} .

Proof. By using λ_{Ω}(f(z))|f′(z)| = λ_{𝔻}(z) = 1/(1 − |z|^{2}), we obtain (4.2) immediately. The second relation is deduced by Lemmas 2 and 3. Now, we show the third one. By Lemma 2, we have

(∂_{𝔻}^{2} \frac{1}{ρ_{Ω}} ) ∘ f = (D_{h1}f)^{−2} \{∂_{𝔻}^{2} (\frac{1}{ρ_{Ω}} ∘ f) − ∂_{𝔻} (\frac{1}{ρ_{Ω}} ∘ f) \frac{D_{h1}f}{D_{h2}f} \}
= (D_{h1}f)^{−2} {∂_{𝔻}^{2}|D_{h1}f| − ∂_{𝔻}|D_{h1}f|Q_{f}}

We then apply Lemma 3 to get (4.4).

We are now ready to prove our main theorems by making use of the previous lemmas.

Proof of Theorem 2. In view of Lemma 8, the equivalence of (i), (ii), (iii) and (vii) in Theorem 2 is just a restatement of Theorem C. Furthermore, by (4.1) in Lemma 7, we have the relations

ρ_{Ω} Δ_{X} (\frac{1}{ρ_{Ω}} ) = 4ρ_{Ω}^{2} \{|∂_{X} (\frac{1}{ρ_{Ω}} )|^{2} − 1\} + 4

= 4ρ_{Ω}^{2} \{|∂_{X} (\frac{1}{ρ_{Ω}} )|^{2} − (1 − \frac{1}{ρ_{Ω}^{2}} )^{2}\} − 4 (1 − \frac{1}{ρ_{Ω}^{2}} ) ,

which show the equivalence of (ii) and (iv) and the equivalence of (iii) and (vi). Hence, (iv) and (vi) are equivalent. We recall that 1/ρ_{Ω} is superharmonic if and only if Δ(1/ρ_{Ω}) ≤ 0, equivalently, Δ_{X}(1/ρ_{Ω}) ≤ 0. Clearly, (vi) implies (v) and (v) implies (iv). Therefore, we have shown that (i) through (vii) are all equivalent.

Proof of Theorem 3. Set r = 1/ρ_{Ω} for simplicity. Then, (4.1) in Lemma 7 yields the relation

r (|∂_{X}^{2}r| + \frac{1}{4}Δ_{X}r − 2r) = r|∂_{X}^{2}r| + |∂_{X}r|^{2} − 1 − r^{2}.

By Theorem 1, we see that the conditions (i) and (ii) are equivalent.

In order to show (ii)⇒(iii), we use Lemma 6. For h(x) = tanh x, it is sufficient to check that the function h(x) satisfies conditions (a), (b) and (c) in the lemma. We omit the proof for this since it is quite elementary.

Finally, we show that (iii) implies (iv). Since these conditions are conformally invariant, we may assume that X = 𝔻. Suppose that the condition (iii) is satisfied but Ω is not hyperbolically convex in 𝔻. Then, there are distinct points a and b in Ω such that the geodesic arc joining a and b in 𝔻 is not contained in Ω. Without loss of generality, we may further assume that a = 0. Note that any geodesic arc in 𝔻 passing through 0 is a line segment. Since Ω is connected, 0 and b can be connected by a smooth simple arc γ : s ↦ w(s), 0 ≤ s ≤ 1, with w(0) = 0 and w(1) = b in Ω. Let σ_{s} be the closed line segment joining 0 and w(s). Since 0 is an interior point of Ω, the segment σ_{s} is contained in Ω for sufficiently small s > 0. Let s_{0} be the smallest number such that σ_{s_{0}} is not contained in Ω. It is clear that 0 < s_{0} ≤ 1. Choose a point c in σ_{s_{0}} ∩ ∂Ω and put d_{0} = d_{𝔻}(0, w(s_{0})) and t_{0} = d_{𝔻}(0, c)/d_{0}. For any number 0 < s < s_{0}, let c(s) be the point lying in σ_{s} such that t_{0} = d_{𝔻}(0, c(s))/d(s), where d(s) = d_{𝔻}(0, w(s)). It is evident that c(s) → c as s → s_{0} − . Since the hyperbolic metric is complete, it follows that λ_{Ω}(c(s)) → +∞, and hence, ρ_{Ω}(c(s)) → +∞ as s → s_{0} − . On the other hand, by (3.1) for the function r = tanh(1/ρ_{Ω}), we have

tanh \frac{1}{ρ_{Ω}(c(s))} ≥ \frac{sinh[2(1 − t_{0})d(s)] tanh(1/ρ_{Ω}(0)) + sinh[2t_{0}d(s)] tanh(1/ρ_{Ω}(w(s)))}{sinh[2d(s)]}.

Clearly, the right-hand side of the above inequality tends to a positive number as s → s_{0}−, which contradicts the divergence of ρ_{Ω}(c(s)). Thus, Ω must be hyperbolically convex in 𝔻.

Proof of Theorem 4. Suppose that Ω is a hyperbolically convex region in 𝔻. Fix A, B ∈ Ω. Let γ : w = w(s), 0 ≤ s ≤ L, be the hyperbolic geodesic arc from A to B parametrized by hyperbolic arclength. Then γ ⊂ Ω and L = d_{𝔻}(A, B). If we set v(s) = 1/ρ_{Ω}(w(s)), then

|v′(s)| = 2 Re \{∂ \frac{1}{ρ_{Ω}} (w(s))w′(s)\} ≤ 2 |∂_{𝔻} \frac{1}{ρ_{Ω}} (w(s))| .

By using the condition (iii) of Theorem 2, we obtain

|v′(s)| ≤ 2 (1 − \frac{1}{ρ_{Ω}^{2}(w(s))}) = 2(1 − v^{2}(s))

or, equivalently,

−2 ≤ \frac{v′(s)}{1 − v^{2}(s)} ≤ 2.

By integrating these inequalities over [0, L], we obtain

| arctanh(v(L)) − arctanh(v(0))| ≤ 2L,

namely,

(4.5)  | arctanh (\frac{1}{ρ_{Ω}(B)}) − arctanh (\frac{1}{ρ_{Ω}(A)})| ≤ 2d_{𝔻}(A, B).

Next, we prove that if the preceding inequality (4.5) holds for all A, B ∈ Ω, then Ω is hyperbolically convex in 𝔻. We show that the inequality (4.5) implies

|∂_{𝔻} \frac{1}{ρ_{Ω}} | ≤ 1 − \frac{1}{ρ_{Ω}^{2}}

and so Ω is hyperbolically convex by Theorem 2. Fix w_{0} ∈ Ω. Let γ : w = w(s) be a hyperbolic geodesic arc in Ω parametrized by hyperbolic arclength on some interval containing 0 with w_{0} = w(0). Let v(s) = 1/ρ_{Ω}(w(s)), then we have

| arctanh(v(s)) − arctanh(v(0))| ≤ 2 d_{𝔻}(w(s), w(0)) = 2s

for all s sufficiently small. We now divide the both sides by s and let s tend to 0 to obtain

\frac{|v′(0)|}{1 − v^{2}(0)} ≤ 2.

Since

v′(0) = 2 Re \{(∂_{𝔻} \frac{1}{ρ_{Ω}} )(w_{0}) e^{iθ}\} ,

where θ = arg w′(0), and we can find a geodesic arc containing w_{0} in any direction there, we conclude that

|(∂_{𝔻} \frac{1}{ρ_{Ω}} )(w_{0})| ≤ 1 − \frac{1}{ρ_{Ω}^{2}(w_{0})}

holds by choosing a suitable θ. Thus, we have shown that (i) and (iii) are equivalent. In a similar way (even more easily), we can also show that (i) and (ii) are equivalent.

Acknowledgments.

The first author would like to thank Professor Fumio Maitani for his hospitality when she visited Kyoto in February of 1999. This work was begun with the second author during the visit.

References

[1] E. F. Beckenbach, Generalized convex functions, Bull. Amer. Math Soc. 43 (1937), 363–371.

[2] R. Harmelin, Hyperbolic metric, curvature of geodesics and hyperbolic discs in hyperbolic domains, Israel J. Math. 70 (1990), 111–126.

[3] S. Kim and D. Minda, The hyperbolic and quasihyperbolic metrics in convex regions, J. Analysis 1 (1993), 109–118.

[4] S. Kim and D. Minda, The hyperbolic metric and spherically convex regions, J. Math. Kyoto Univ. 41 (2001), 297–314.

[5] D. Minda, Hyperbolic curvature on Riemann surfaces, Complex Variables 12 (1989), 1–8.

[6] W. Ma and D. Minda, Spherical linear invariance and uniform local spherical convexity, Current Topics in Analytic Function Theory (Singapore) (H. M. Srivastava and S. Owa, eds.), World Scientific, 1992, pp. 148–170.

[7] W. Ma and D. Minda, Hyperbolically convex functions, Ann. Polon. Math. 60 (1994), 81–100.

[8] W. Ma and D. Minda, Two-point distortion theorems for strongly close-to-convex functions, Complex Variables Theory Appl. 33 (1997), 185–205.

[9] W. Ma and D. Minda, Hyperbolically convex functions II, Ann. Polon. Math. 71 (1999), 273–285.

[10] D. Mejía and Ch. Pommerenke, On hyperbolically convex functions, J. Geom. Anal. 10 (2000), 365– 378.

[11] D. Minda and D. Wright, Univalence criteria and the hyperbolic metric, Rocky Mt. J. Math. 12 (1982), 471–479.

[12] W. Walter, Ordinary Differential Equations, Springer, 1998.

[13] S. Yamashita, Convexité et inégalités différentielles concernant la densité hyperbolique, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 781–783.

[14] S. Yamashita, Univalent analytic functions and the Poincaré metric, Kodai Math. J. 13 (1990), 164–175.

[15] S. Yamashita, The Poincaré density, Lipschitz continuity and superharmonicity, Math. Japonica 38 (1993), 487–496.

Department of Mathematics, Woosuk University, Wanju-gun, Cheonbuk 565-701, South Korea

  E-mail address: sakim@core.woosuk.ac.kr

Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan

  E-mail address: sugawa@math.sci.hiroshima-u.ac.jp