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Indagationes Mathematicae xx (xxxx) xxx–xxxwww.elsevier.com/locate/indag

On harmonic functions and the hyperbolic metric

Q1 Marijan Markovic∗

Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b. 81000 Podgorica,Montenegro

Received 2 October 2013; received in revised form 7 March 2014; accepted 10 March 2014

Communicated by E.P. van den Ban

Dedicated to the memory of my grandmother Olga

12

Abstract 3

Motivated by some recent results of Kalaj and Vuorinen (2012), we prove that positive harmonic 4

functions defined in the upper half-plane are contractions w.r.t. hyperbolic metrics of half-plane and positive 5

part of the real line, respectively. 6

c⃝ 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 7

Keywords: Positive harmonic functions 8

9

1. Introduction and the main result 10

Denote by U = z ∈ C : |z| < 1 the unit disc of the complex plane C and by H = z ∈ C : 11

Im z > 0 the upper half-plane. R is the whole real axis, and the positive real axis is denoted by 12

R+= x ∈ R : x > 0. 13

Let dh stand for the hyperbolic distance on the disc U. With the same letter we denote the 14

hyperbolic distance on H and R+, since we believe that misunderstanding will not occur. We 15

have 16

dh(z, w) = infγ

γ

|dω|

1 − |ω|2= 2 tanh−1

z − w

1 − zw

17

∗ Tel.: +382 67577979.E-mail address: marijanmmarkovic@gmail.com.

http://dx.doi.org/10.1016/j.indag.2014.03.0020019-3577/ c⃝ 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG).

2 M. Markovic / Indagationes Mathematicae xx (xxxx) xxx–xxx

where γ ⊆ U is any regular curve connecting z ∈ U and w ∈ U. On the other hand, the hyper-1

bolic distance between z ∈ H and w ∈ H is2

dh(z, w) = infγ

γ

|dω|

Im ω= 2 tanh−1

z − w

z − w

,3

where now γ ⊆ H. In particular, the hyperbolic distance between x ∈ R+ and y ∈ R+, where4

x ≤ y is5

dh(x, y) = dh(i x, iy) =

y

x

dt

t= log

y

x.6

Recall the classical Schwarz–Pick lemma. An analytic function f of the unit disc into itself7

satisfies8 f (z) − f (w)

1 − f (z) f (w)

≤

z − w

1 − zw

9

for all z, w ∈ U. The equality sign occurs if and only if f is a Mobius transform of U onto itself.10

The previous result has a counterpart for analytic functions f : H → H. Using the Cayley11

transform one easily finds that the Schwarz–Pick inequality in this settings says12 f (z) − f (w)

f (z) − f (w)

≤

z − w

z − w

(1)13

for every z, w ∈ H. Letting z → w in (1), we obtain14

| f ′(z)|

Im f (z)≤

1Im z

. (2)15

Regarding the expression for the hyperbolic distance in the upper half-plane, (1) may be rewritten16

as17

dh( f (z), f (w)) ≤ dh(z, w), (3)18

which means that f is a contraction in the hyperbolic metric of H. It is well known that the19

equality sign attains in (1), (2), and (3) (for some z or for some distinct z and w, and therefore20

for all such points) if and only if21

f = a Mobius transform of H onto itself.22

During the past decade, harmonic mappings and functions have been extensively studied and23

many results from the theory of analytic functions have been extended for them.24

Quite recently Kalaj and Vuorinen [3] proved that a harmonic function f : U → (−1, 1) is a25

Lipschitz function in the hyperbolic metric, i.e., for every z, w ∈ U they obtained that26

dh(u(z), u(w)) ≤4π

dh(z, w). (4)27

Actually, using the classical results they firstly established28

|∇u(z)| ≤4π

1 − |u(z)|2

1 − |z|2(5)29

for z ∈ U. Both inequalities are sharp.30

We refer to [1] for a related result.31

M. Markovic / Indagationes Mathematicae xx (xxxx) xxx–xxx 3

We are interested here in the positive harmonic functions defined in H. As we have said in the 1

abstract, our main aim is to prove 2

Theorem 1.1. Let u : H → R+ be harmonic. Then 3

dh(u(z), u(w)) ≤ dh(z, w) (6) 4

for all z, w ∈ H. In other words, a positive harmonic function is a contractible function in the 5

hyperbolic metric. 6

Moreover, if the equality sign holds in (6) for some pair of distinct points z and w, then the 7

function u must be of the following form 8

u(z) = Im(a Mobius transform of H onto H). 9

Remark 1.2. The group of all conformal mappings of H onto itself is given by 10az + b

cz + d: a, b, c, d ∈ R, ad − bc > 0

. 11

Thus, a positive harmonic function u(z) is extremal for the inequality (6) if and only if it has the 12

form 13

u(z) = k · Im z or u(z) = k · P(z, t), 14

where k > 0 and t ∈ R; here 15

P(z, t) =1π

y

(x − t)2 + y2 , 16

z = x + iy ∈ H, t ∈ R is the Poisson kernel for the upper half-plane. 17

2. Proof of the result 18

Let Ω be any domain in C (or in R). A metric density ρ is any continuous function in Ω with 19

nonnegative values everywhere in Ω . The ρ-length (or just a length) of a curve γ in Ω is given 20

by 21γ

ρ(z) |dz|. (7) 22

The ρ-distance (or just a distance) between z ∈ Ω and w ∈ Ω is 23

dρ(z, w) = infγ

γ

ρ(z)|dz|, (8) 24

where γ is a regular curve in Ω connecting points z and w. Of course, if Ω is an interval in R, 25

then we need not the infimum sign in the preceding expression for the distance function. 26

For a regular curve γ we denote by tγ (ω) the unit tangent vector at a point ω ∈ γ consistent 27

with the orientation of γ . 28

We will prove the following lemma of somewhat general character. 29

Lemma 2.1. Let Ω ⊆ C be a domain, I ⊆ R an open interval, and u : Ω → I a smooth 30

function, i.e., u ∈ C1(Ω). Let ρ be a metric density in Ω and let ρ be a metric density in the Q2

4 M. Markovic / Indagationes Mathematicae xx (xxxx) xxx–xxx

interval I . If1

ρ(u(ω))|∇u(ω)| ≤ ρ(ω) (9)2

for every ω ∈ Ω , then we have3

dρ(u(z), u(w)) ≤ dρ(z, w) (10)4

for z, w ∈ Ω .5

If the equality sign is attained in (10) for some pair of distinct points z and w, then equality6

holds in (9) for some ω ∈ Ω .7

Proof. Let z and w be distinct fixed points in Ω . Without loss of generality, we may assume that8

u(z) ≤ u(w). Let γ be a regular curve connecting z and w. Orient it from z to w. Denote9

I0 =ω ∈ γ : there exist λ(ω) > 0 such that ∇u(ω) = λ(ω)tγ (ω)

.10

This set may be decomposed as11

I0 =

∞n=1

In,12

where In are intervals in γ (if the union is finite, we assume that In = ∅, starting from an integerQ313

n0). Let J1 = I1,14

J2 = I2 \ ω ∈ γ : u(ω) ∈ u(J1) ,15

and by induction for n > 1, let16

Jn+1 =

ω ∈ In : u(ω) ∈ u

n

k=1

Jk

.17

Denote18

J =

∞k=1

Jk .19

Then [u(z), u(w)] ⊆ u(γ ) and |[u(z), u(w)]| = |u(J )|. Moreover, u is injective in J . Therefore,20

by using inequality (9) we obtain21

dρ(u(z), u(w)) =

u(w)

u(z)ρ(ω) |dω| =

J

ρ(u(ω)) |∇u(ω)| |dω|22

≤

J

ρ(ω) |dω| ≤

γ

ρ(ω) |dω|.23

Since γ is any curve, we have24

dρ(u(z), u(w)) ≤ dρ(z, w),25

what we have to prove.26

The second part of this lemma follows immediately. 27

As an application of the preceding lemma and the Schwarz–Pick lemma we prove our main28

result here.29

M. Markovic / Indagationes Mathematicae xx (xxxx) xxx–xxx 5

Proof of Theorem 1.1. Let u : H → R+ be a harmonic function. Denote by f an analytic 1

mapping in the upper half-plane such that 2

Im f (z) = u(z) 3

for z ∈ H. Then f maps H into H. Since | f ′(z)| = |∇u(z)|, z ∈ H, applying the version of 4

Schwarz’s lemma for analytic function in the upper half-plane, i.e. (2), we obtain 5

|∇u(z)|

u(z)=

| f ′(z)|

Im f (z)≤

1Im z

(11) 6

for z ∈ H. The equality sign appears if and only if f is a Mobius transform of H onto H. This is 7

a counterpart of the sharp estimate (5) for harmonic functions in the unit disc. 8

Thus u satisfies the condition of Lemma 2.1 with the hyperbolic metric density on each sides. 9

Thus, according to this lemma we obtain 10

dh(u(z), u(w)) ≤ dh(z, w) (12) 11

for all z, w ∈ H. 12

If the equality sign holds for a pair of distinct points z, w ∈ H, then we must have the equality 13

sign in (11) for some z. As we know, this means that u must be of the form 14

u = Im(a Mobius transform of H onto itself), 15

what proves the second part of our theorem. 16

Remark 2.2. In order to prove that the inequality (5) is sharp the authors of [3] have found a 17

function for which the inequality (5) reduces to the equality. However, following the proof in [3], 18

one can deduce that the equality sign holds in (5) (for some z, and therefore for all z) if and only 19

if u(z) : U → (−1, 1) has the form 20

u(z) = Re

2i

πlog

1 + b(z)

1 − b(z)

= −

2π

arg1 + b(z)

1 − b(z), 21

where b(z) is a Mobius transform of the unit disc onto itself. 22

Using Lemma 2.1 and the sharp estimate (5) for harmonic functions u : U → (−1, 1) one 23

can derive (4) (what is the main result of [3]), along with all extremal functions. 24

Uncited references 25

Q4

[2]. 26

References 27

[1] H. Chen, The Schwarz–Pick lemma and Julia lemma for real planar harmonic mappings, Sci. China Math. 56 (2013)2327–2334.

28

[2] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. 29

[3] D. Kalaj, M. Vuorinen, On harmonic functions and the Schwarz lemma, Proc. Amer. Math. Soc. 140 (2012) 161–165. 30