Abstract. Nevanlinna-type equalities are given for bounded analytic functions in the unit disk D.

VALUE DISTRIBUTION OF BOUNDED ANALYTIC FUNCTIONS

Shamil Makhmutov

(Received March 2006)

Abstract. Nevanlinna-type equalities are given for bounded analytic functions in the unit disk D.

1. Introduction

Let D be the open unit disk in the complex plane and let ϕ : D → D be an analytic self-map of D. Define the hyperbolic characteristic function T ∗ (r, ϕ), 0 < r < 1, of ϕ by

T _∗ (r, ϕ) = 1 π

Z Z

|z|≤r

(ϕ ^∗ (z)) ² ln r

|z| dxdy

where ϕ ^∗ (z) = |ϕ ⁰ (z)|(1 − |ϕ(z)| ² ) ⁻¹ is the hyperbolic derivative of ϕ.

For a ∈ D, we define the hyperbolic proximity function

m ∗ (r, ϕ, a) = − 1 2π

2π

Z

0

ln s

   

1 − aϕ(re ^iθ ) ϕ(re ^iθ ) − a

   

2

− 1dθ + ln s

   

1 − aϕ(0) ϕ(0) − a

   

2

− 1,

if ϕ(0) 6= a and

m _∗ (r, ϕ, a) = − 1 2π

2π

Z

0

ln s 

  

1 − aϕ(re ^iθ ) ϕ(re ^iθ ) − a

   

2

− 1dθ − ln |c _k (a)|

if ϕ(0) = a and where c k (a) = lim

w→0

ϕ(w) − a w ^k 6= 0.

The counting function N (r, ϕ, a), a ∈ D, is the same as in the Nevanlinna theory of meromorphic functions [3]

N (r, ϕ, a) = X

0<|a

j

|≤r

ln r

|a j | + n(0, a) ln r,

where {a j } are roots of equation ϕ(z) = a and the sum is counted according to multiplicity.

The First Main Theorem in the Theory of Meromorphic Functions in the geo- metric interpretation says ([3, p. 175], [4, p. 24])

For any function f meromorphic in the disk |z| < R ≤ ∞ the equality m # (r, f, a) + N (r, f, a) = T # (r, f ) + m # (0, f, a)

2000 Mathematics Subject Classification Primary 30D35; Secondary 30D50, 30D45.

holds for any a ∈ C ∪ {∞}.

In Nevanlinna’s theorem the characteristic function T # (r, f ) involves the spher- ical derivative of f and m # (r, f, a) is the spherical proximity function to the point a ∈ C S{∞}.

2. The First Fundamental Theorem Involving the Hyperbolic Derivative

Theorem 1. Let ϕ be an analytic self-map of D. For any a ∈ D and 0 < r < 1 m _∗ (r, ϕ, a) − N (r, ϕ, a) = T _∗ (r, ϕ) . (1) In the proof of Theorem 1 we shall need Theorem A and Lemmas 1 and 2.

Theorem A. ([2, p. 13]) Let G be a simply connected domain and its boundary γ be a piecewise smooth curve. Let g(z, w) be the Green’s function of G. Suppose that u is twice continuously differentiable in G and once on γ except a finite set of points c 1 , . . . , c m in G. If u can be represented in a neighborhood of c j , j = 1, . . . , m, in the form

u(z) = d j ln |z − c j | + u j (z)

where d j is a constant and u j (z) is twice continuously differentiable in a neighbor- hood of c j , then for every w ∈ G, w 6= c 1 , . . . , c m

u(w) + 1 2π

Z

G

∆u(z)g(z, w)dxdy = 1 2π

Z

γ

u(z) ∂g

∂n ds − X

c

_j

∈G

d _j g(c _j , w) .

Here ds is the arc length element on Γ, ∂g

∂n is taken with respect to the inner normal on Γ and ∆ stands for the Laplace operator ∂ ²

∂x ² + ∂ ²

∂y ² = 4 ∂ ²

∂z∂z . Lemma 1. Let ϕ be an analytic self-map of D. For any a ∈ D,

the function u _a (z) = − ln s

   

1 − aϕ(z) ϕ(z) − a

   

2

− 1 , satisfies the condition

∆u _a (z) = 2(ϕ ^∗ (z)) ² , for all z ∈ D, except for the a-points of ϕ.

Proof. For every a ∈ D    

1 − aϕ(z) ϕ(z) − a

   

2

− 1 = (1 − |a| ² )(1 − |ϕ(z)| ² )

|ϕ(z) − a| ² . T hen ∆u a (z) = −∆(ln p

1 − |ϕ(z)| ² ) + ∆(ln p|ϕ(z) − a| ² )

where ∆(ln p|ϕ(z) − a| ² ) = 0 for every a ∈ D and except for the a-points of ϕ.

We have to prove that

−∆(ln p1 − |ϕ(z)| ² ) = 2(ϕ ^∗ (z)) ² .

The last follows from the direct calculation of ∆ 

ln p1 − |ϕ(z)| ² 

. 

Lemma 2. Let ϕ be a holomorphic function in D R = {|z| ≤ R}, 0 < R < ∞, and

|ϕ(z)| < 1 in D R . If {b n } is the set of zeros of ϕ, then 1

π Z Z

|z|≤R

(ϕ ^∗ (z)) ² ln R

|z| dxdy

= − 1 2π

2π

Z

0

ln s

1

|ϕ(Re ^iθ )| ² − 1dθ + ln s

1

|ϕ(0)| ² − 1 − X

|b

n

|<R

ln R

|b n | if ϕ(0) 6= 0 and

1 π

Z Z

|z|≤R

(ϕ ^∗ (z)) ² ln R

|z| dxdy (2)

= − 1 2π

2π

Z

0

ln

s 1

|ϕ(Re ^iθ )| ² − 1dθ − ln |c k | − k ln R − X

0<|b

n

|<R

ln R

|b _n | if ϕ(0) = 0 and lim

z→0 ϕ(z)

z

^k

= c k 6= 0, k ≥ 1. The sums are counted according to multiplicity.

Proof. Let u(z) = u 0 (z). According to Lemma 1, ∆u(z) = 2(ϕ ^∗ (z)) ² for every z ∈ D R \ {b n }. By Theorem A, we have

− ln

s 1

|ϕ(w)| ² − 1 + 1 π

Z Z

|z|≤R

(ϕ ^∗ (z)) ² ln    

R ² − wz R(z − w)

   

dxdy

= − 1 2π

2π

Z

0

   

R ² − wz R(z − w)

   

ln

s 1

|ϕ(Re ^iθ )| ² − 1dθ − X

|b

_n

|<R

ln    

R ² − b _n w R(b n − w)

   

(3)

If ϕ(0) 6= 0, then we set w = 0 in (3) to obtain

− ln s

1

|ϕ(0)| ² − 1 + 1 π

Z Z

|z|≤R

(ϕ ^∗ (z)) ² ln R

|z| dxdy

= − 1 2π

2π

Z

0

ln s

1

|ϕ(Re ^iθ )| ² − 1dθ − X

0<|b

_n

|<R

ln R

|b n | . (4)

If lim

w→0

ϕ(w)

w ^k = c _k 6= 0, k ≥ 1, then

− ln

s 1

|ϕ(w)| ² − 1 = k ln |w| + ln |c _k | + o(1) as w → 0 (5) and

X

|b

n

|<R

ln    

R ² − b n w R(b n − w)

   

= k ln R

|w| + X

0<|b

_n

|<R

ln R

|b n | + o(1) as w → 0 . (6)

Passing to the limit as w → 0 in (4) and taking in account (5) and (6), we obtain

(2). 

Proof of Theorem 1. Equality (1) for a = 0 was proved in Lemma 2.

Let S _a (z) = a − z

1 − az , a ∈ D.

T hen m ∗ (r, S a ◦ ϕ, 0) − N (r, S a ◦ ϕ, 0) = T ∗ (r, S a ◦ ϕ) . According to Lemma 1

T ∗ (r, S a ◦ ϕ) = T ∗ (r, ϕ) for any a ∈ D and also

m ∗ (r, S a ◦ ϕ, 0) = m ∗ (r, ϕ, a) N (r, S a ◦ ϕ, 0) = N (r, ϕ, a).

Thus (1) holds for any a ∈ D. 

We write now the hyperbolic characteristic function T _∗ (r, ϕ) in another form T _∗ (r, ϕ) =

r

Z

0

A _∗ (t, ϕ) t dt where

A _∗ (r, ϕ) = 1 π

Z Z

|z|≤r

(ϕ ^∗ (z)) ² dxdy.

Let ϕ map the disk D _r = {|z| ≤ r} on a Riemann surface F _r and µ be unit mass distribution spread over the unit disk, dµ(z) = 1

π dx dy. Then for the hyperbolic mass distribution λ with dλ(a) = dµ(a)

(1 − |a| ² ) ² it holds A _∗ (r, ϕ) = 1

π Z Z

D

r

(ϕ ^∗ (z)) ² dA(z) = Z Z

F

r

dλ(z) = Z Z

|a|<1

n(r, ϕ, a)dλ(a)

Theorem 2. Let ϕ be an analytic self-map of D. Then T _∗ (r, ϕ) =

Z Z

|a|<1

N (r, ϕ, a)dλ(a).

Proof. Since the function n(t, ϕ, a) is a measurable function with respect to arguments a, |a| < 1, and t, 0 < t ≤ 1, we get

T _∗ (r, ϕ) =

r

Z

0

A _∗ (t, ϕ) t dt =

r

Z

0

dt t

Z Z

|a|<1

n(t, ϕ, a)dλ(a)

= Z Z

|a|<1

dλ(a)

r

Z

0

n(t, ϕ, a) t dt =

Z Z

|a|<1

N (r, ϕ, a)dλ(a)



Finally, we remind the reader that if ϕ is a bounded analytic function in D, the counting function N (r, ϕ, a) is bounded for every a ∈ D. By the Littlewood inequality,

N (1, ϕ, a) ≤ log    

1 − ϕ(0)a ϕ(0) − a

   

and if ϕ is an inner function then this inequality becomes equality for all a in D except for a set of zero logarithmic capacity in D.

As a function of argument a N (1, ϕ, a) = O

 log 1

|a|



as |a| → 1 . (See [5, p. 187-197].)

3. Analytic Form of the First Fundamental Theorem

In Theorem 1 we obtained the first fundamental theorem of the value distribution in hyperbolic form. In the case of meromorphic functions in the disk D R = {|z| <

R}, 0 < R ≤ ∞, we have the classical (spherical) result of R. Nevanlinna [3].

Let f be an analytic function in the disk D R = {|z| < R}, 0 < R ≤ ∞, and define a characteristic function e T (r, f ), 0 < r < R, by

T (r, f ) = e 2 π

Z Z

|z|≤r

|f ⁰ (z)| ² ln r

|z| dxdy.

Let m(r, f, a) = e 1 2π

2π

Z

0

|f (re ^iθ ) − a| ² dθ − |f (0) − a| ² , a ∈ C be the analytic proximity function of f to the point a ∈ C.

Theorem 3. Let f be an analytic function in the disk D R , 0 < R ≤ ∞. Then for each r, 0 < r < R,

T (r, f ) = e m(r, f, a), e a ∈ C (7) T (r, f ) = e

Z Z

|a|<∞

N (r, f, a)dµ(a) . (8)

Equality (7) follows from the Littlewood-Paley identity [5, p. 178] applied to the function u(z) = f (z) − a, a ∈ C.

In order to prove (8) we should consider e T (r, f ) in the form

T (r, f ) = e

r

Z

0

A(t, f ) t dt

A(r, f ) = 1 π

Z Z

|z|≤r

|f ⁰ (z)| ² dx dy.

Then the proof of (8) proceeds in the same way as the proof of Theorem 2.

Consider the Hardy space H ² of analytic functions f in the unit disk D. Ac- cording to definition of analytic proximity function m(r, f, a) and Theorem 3 the e boundness of e T (r, f ) is the necessary and sufficient condition for f to be in H ² (see [1]).

Corollary 1. Let f be an analytic function f in D. Then f ∈ H ² if and only if Z Z

|a|<∞

N (r, f, a)dµ(a) < ∞.

Acknowledgements. The author wishes to express his sincere gratitude to the referee for helpful comments. This work was partially supported by Sultan Qaboos University, project IG/SCI/DOMS/04/04.

References

[1] P.L. Duren, Theory of H ^p Spaces, Academic Press, New York, 1970.

[2] A. Goldberg and I. Ostrowski, Value distribution of meromorphic functions, Nauka, Moscow, 1972. (Russian). English translation by Mikhail Ostrovskii, Translations of Mathematical Monographs, 236. American Mathematical Soci- ety, Providence, RI, 2008.

[3] R. Nevanlinna, Analytic Functions, Springer-Verlag, Berlin, 2nd ed., 1970.

[4] J.L. Schiff, Normal Families, Springer-Verlag, New York, 1993.

[5] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer- Verlag, New York, 1993.

Shamil Makhmutov

Department of Mathematics and Statistics Sultan Qaboos University

P.O. Box 36, Al Khodh 123 Oman

[email protected]