An Inequality for Algebroid Functions and Some Applications

An Inequality for Algebroid Functions and Some

Applications

Ángel J. Alonso Gómez

Abstract In this paper we present a version for algebroid functions of an inequality due to Barsegian (Math URSS Sbornik 42(2):155, 1982) and we get two consequences: first a kind of deficiency relation for curves (preimages of an algebroid function, the so-called T-lines) which is an extension of the well-known relation of Barsegian (T-lines. On the geometry of real and complex functions. Taylor & Francis, London, 2002) and secondly a proximity principle for the preimages of an algebroid function, also an extension of the principle for meromorphic functions.

Keywords Algebroid functions · Gamma-lines · Deficiency relations · Proximity principles

Mathematics Subject Classification 30C99

1 Algebroid Functions

An algebroid function is a multivaluated one w = w (z) with a finite number of branches in C and whose only singularities are poles and algebraic branch points [10,11]. These functions satisfy an equation like

with Ai, i = 1, 2 , . . . , k meromorphic functions. k is the order of the algebroid function.

But we can also consider algebroid functions as holomorphic functions defined on a Riemann surface with k sheets [4]: we have the Riemann surface

S = {(z,wi (z)), z e C, i = {(z, wi (z)), z e C, i = 1, 2 , . . . , k]

where w1, w2, .. .wk are the branches of the algebroid function. The canonical projection

p : S -+ C (z,wi (z)) ^ z

is an holomorphic covering of C and

f : S -+ C (z,w i (z)) -^- wi (z)

is also holomorphic. It is the function associated to the algebroid function.

2 The Inequality We can consider the relation

f S -+ C

between the function associated to the algebroid function and its branches. S is now a Riemann surface of k sheets and p is locally the identity.

Then

wi oo p = f p =

but p' = 1 and p" = 0 ==> f" = w" o p.

From now on S [r] is the part of the Riemann surface S which projects by means of p onto D [r], the closed disk of C centered in (0, 0), with radius r.

p:E^C

p(ζ) = z

and

is the pull-back of the element of surface on the plane to the surface S . Proposition 1

f

"(ζ)\ A

w

'(z)|

f (ζ) i 1 D [ r ] w (z)

$ < k i () rdrd

n f' l f l w i

a [ r ] v> / i = 1 D [ r ] i

with z = rei .

Proof Around any point of D [r] there is a neighbourhood Ui with p-1 (Ui n D [r]) c

S [r] and a wi defined in Ui n D [r].

p is locally an homeomorphism (it is the identity indeed) and applies every neigh­ bourhood in S [r] onto a neighbourhood in D [r].

Then, if around every point of D [r], z = p (ζ) there is a neighbourhood Ui where

a branch wi of f is defined, w1, w2,... ,wk are defined in U1 n D [r], U 2nD [r],

…, in Uk n D [r], in such a way that

U

i = 1

then

f

"(ζ)\

k

|

w

( z )

|

U (Ui n D [r]) = D [r]

i

I

but

— ^ =

—-i = 1 p -1 (Ui n D [r]) f (ζ) i = 1 Ui n D [r] i ( )

hence

|

f

(ζ )

|*<

1

f

(ζ )

1*

i f'(ζ ) " - 1nD f'(ζ ) A [r] i = 1 p (Ui n [r])

f

"

( ζ )

L ,

k w

(z)l

$ << rdrd

n[ r ] f'(ζ) i 1 Ui r> D [ r ] w i (z)

But this argument only is valid with only one sheet in the surface S , p being a homeomorphism only locally because it projects points ζ in the different sheets of S

onto the same point z e C, hence we must consider the totality of the sheets and we get

I "

(ζ )

1

k w

(z)

|

f— $ < k j rdrd

finally, we observe that

\wi (ζ)_{, Irdrd} \ \ _{< \—} wi (z)

_ w, (ζ) w,' (z)

Ui n D [r] i D [ r ] i

and we finally get

f"(ζ)I k |w i(z)|

" f'(ζ) ~

A [r] i = 1 D [r]

f

(ζ) w

;

(z)

V i = 1 D [ r ] i

Observation: arguing in the same way for the integral

1 |f'(ζ)|2

A(r) = — <t>

π S[r] 1 + lf'(ζ)l2

which is the density of the well-known Ahlfors-Shimizu characteristic, we get

k w i ( z ) 2

A (r) < k 2rdrd

i = 1 D [ r ] 1 + |w ij (z)|

and we shall call, from now on

Ai (r) = i 2wi (z) rdrd

D [r] 1 + |w i' (z)|2

A (r) is the Ahlfors-Shimizu density of the Riemannian image of S [r] under f and Ai (r) those of D [r] under the branches of the algebroid function.

3 The Defect Relation

Let Tν, ν = 1, 2 , . . . , q be q Jordan, smooth, disjoint, bounded curves, with ν (Tν) <

oo, where ν (T) = Varζerαr (ζ), being αp (ζ) the angle between the tangent to p (T)

in p (ζ) and the OX axis. We add the hypothesis that f~1 (Tν) does not contain poles

nor branch points of H.

Def.- A (r) == k r, ) , where A (r, T) is the length of the preimages

r —> 0 0 r i=1 Ai (r)

of r in S [r], under the algebroid function f.

A (r)is the deficiency of T and we shall call T a deficient curve if and only if A (O ^ 0.

Theorem 1 Let Tν, ν = 1, 2 , . . . , q be a collection of curves as above and let A (Tν)

be the deficiency of every Tν, then:

q

A (rν) < Ck

ν = 1

where C is an absolute constant and k is the order of the algebroid function.

Proof We begin with

A (r, rν) < K <t> + hπrkA (r) + 2w2kπr

ν = 1 S [r] f (ζ)

Kand h being independent of the function f. [1]

Then

q | /; |

A (r, Tν) < K k i rdrd + hπrkA (r) + 2\J2kπr

w,' (z)

ν = 1 D [ r ] i

but

w(z)

—, rdrd < KrAi (r)

D[ r ] w' (z)

lim inf

for r e Ei c t such that 1 Ein0,) dlnt = Ci e (0, 1) , i. e. Ei being a

set of lower logarithmic density Ci. (Branches wi are meromorphic functions) [3].

Then

q k

A (r, rν) < Kkr Ai (r) + hπrkA (r) + 2w2kπr

ν = 1 i=1

but

kA (r) < kA (r) < k Ai (r) i = 1

and we get

A (r, rν) < Kkr Ai (r) + hπrk Ai (r) + 2w2kπr

and finally

q k |

A (r, rv) < kr \ C1 Ai (r) + C2

v = 1 i = 1

with r e E as above.

Then, with the definition of A (I"1) we get the proposition we were looking for.

Now, if we have q -> 00 we get

00

J2 A (rv) < Ck V = 1

and it remains bounded, so that we must have A (T) = 0 for almost all curves and only a numerable quantity of curves are such that A (T) > 0. Those are the deficient curves.

4 The Proximity Principle

We are now speaking about the problem of the distribution of preimages of a function (an algebroid one, in our case) and about an estimation of their proximity. If we call a-points corresponding to z those that satisfies f (z) = a , we will try to progress in the direction of the so-called Littlewood Property [6] which this author has asserted for entire functions and says, roughly speaking, that almost all a-points of those functions, for almost all values of a , belong to a subset of C of small area. Littlewood has proved that the property is true under the hypothesis that the bound

I / I

|p,( z )|2 rdrdy<cn 1 2

-is fulfilled by a polinomial p of degree less or equal than n and a > 0. This is the conjecture of Littlewood, proved by Lewis and Wu [7].

In this direction, we extend a result of Barsegian [Bar1] for meromorphic functions to algebroid functions.

Let av, bv, v = 1, 2 . . . , q be values in C , f an algebroid function, fi (z) are the

preimages of z, i. e. f (fi (z)) = z.n (r, z) is the number of preimages of z in S [r] .

Let Tv be Jordan, bounded, smooth, disjoint, bounded curves, with v (Tv) < 00,

and the extreme points of Tv are av and bv.

f~1 (Tv) n S [r] contains n (r, av) curves yi, i = 1 , 2 , . . . ,n (r, av) with only

ordinary points, i. e. they do not pass through singular points.

We assign to the preimages of the extreme points av and bv the same index, fi (av), ?i (bv) respectively. The other points from n (r, av) to n (r, bv) remains

So, we define

n (r, a)

B (r, a, b) = dist (fi (a), fi (b))

i = 1

where

dist (fi (a), fi (b)) = inf ds

y (fi (a), fi (b))

being ds the metric on the surface S and y (a, /S) those regular curves joining the points a and /S and

n (r, b)

A(r,a,b) = |zi (b)| i = n (r, a) + 1

with zi (b) = p (fi (b)).

The proximity principle is given by the next

Theorem 2 There exist K, and Ce (0, 1) absolute constants such that for every r e E c [0, oo) of lower logarithmic density C

q k | | A (r, av.bv) + B (r, av, bv)| < Kr A(r) + Ai (r) + O (rA (r)) v = 1 i = 1

where A (r) and Ai (r) are as above.

Proof The extreme points of yi area-points and b-points, fi (a)andfi (b) respectively,

then there are n (r, av) pairs (fi (av) are summed in an arbitrary manner.

then there are n (r, av) pairs (fi (av), t,i (bv)), i = 1,2,... ,n(r, av) and the remain

Let Fr be the riemmanian image of S [r] under f. There are n0 (r , T) simple islands

of Fr over T [5] and n (r, T) multiple islands (any of them counted only once).

Now, if n (r, T) is the sum of the multiplicities of all islands over T, we have

n (r, T) = n0 (r, r) + n1 (r, V) + n (r, V)

where n1 (r, T) is the sum of the orders of all islands over I\ Then

and so

A(r) — n (r, V) + n1 (r) = A(r) — n0 (r, T) — n (r, V)

but, by the second fundamental theorem of Ahlfors [9]

n (r, rν) — n1 (r, rν) > (q — 2) A (r) — hL (r)

ν=1 ν=1

where h is a constant depending on the curves and L (r) is the length of the relative boundary of Fr and consequently

hL (r) = o (A (r))

then

q

(A (r) — n0 (r , Tν) — n (r, Tν)) < 2A (r) + hL (r)

ν = 1

but, from the first fundamental theorem of Ahlfors [9] we can deduce

n (r, rν) < qA (r) + hL (r)

ν = 1

then

q q q

n0 (r, rν) + n1 (r, rν) + n (r, rν) < qA (r) + hL (r)

ν = 1 ν = 1 ν = 1

and consequently

q q q n (r, rν) < qA (r) — n0 (r, Tν) — n1 (r, Tν) + hL (r)

ν= 1 ν= 1 ν= 1

butn < n1,hence

q q q n (r, rν) < qA (r) — n0 (r , rν) — n (r, Vν) + hL (r)

and

q q

qA (r) — n (r, Tν) — n0 (r, rν) < 2A (r) + hL (r)

ν=1 ν = 1

so that

n (r, rν) < 2A (r) + hL (r)

ν = 1

with different constant h and, by

q q

(A (r) — n (r, rν)) = (A (r) — n0 (r, Tν) — n1 (r, Tν) — n (r, rν))

ν = 1 ν = 1

we get

(A (r) — n0 (r , Tν)) < 4A (r) + hL (r)

ν = 1

with different h again and

q q

(n1 (r, rν) + n (r, rν)) < 4A (r) — (A (r) — n (r, Tν)) + hL (r)

ν=1 ν=1

but Miles [8] has proved that

|A (r) — n (r, aν)| < KA (r), r -> oo

ν = 1

with r e E , a subset of [0, oo) with lower logarithmic density C e (0, 1) and every aν belongs to a rν so that

|A (r) — n (r, Vν)| < KA (r), r -> oo r e E

ν= 1

and the same is true for n (r, bν), hence

(n1 (r, aν) + n (r, aν)) < 4A (r) — (A (r) — n (r, aν)) + hL (r)

q q £ (m (r, bv) + h (r, bv)) < 4A (r) - £ (A (r) - n (r, bv)) + hL (r)

v=1 v = 1

and we get

£ («i (r, AV) + «! (r, bv) + n (r, av) + h (r, bv)) < 8A (r) + 2KA (r) + hL (r)

v= 1

always with r -> oo, r e E .

Over Tv there are n (r, rv) simple islands of Fr. Their preimages are y* (v), i = 1, 2 , . . . , «o 0", rv) and «o (/, Tv) < n (r, av).

The extreme points of y* (v) c S [r] are not multiple points, but ordinary points, namely f; (av) and f; (fev) and there are «o (/, rv) pairs of points of this kind and

«o (r, Tv)

£ / e « ^ ( y « ( v ) ) < A ( r , rv) i = 1

where A (r, Tv) is the sum of the lengths of preimages of / . And so

«o (r, Tv)

£ dirt (&(flv), &(&„))< A ( r , rv) i = 1

It is obvious that

n (r, av)

£ dist (& (AV) , & (fcv)) + A (r, av, bv) <

i = n0 (r, Fv) + 1

< r {n (r, av) - n0 (r, rv) + n (r, bv) - n0 (r, rv) ) hence

n (r, av)

£ dist (& (ov), & (&„)) + A (r, av, 6V) < 2r (n (r, Tv) - n0 (r, Tv)) i = n0 (r, Fv) + 1

and

A(r,av,bv) + B(r,av,bv)= £ dist (^ (av), ^ (bv))

n (r, rv)

+ E dist(b(av),b(bv)) + A(r,av,bv)

no (r, rv) + 1

q q

=» E \A(r,av,bv) + B(r,av,bv)\ < E A(r,Vv)+(8+ 2K)rA(r)hrL(r)

v=1 v = 1

but we saw previously that

q

E A ( r , rv) < £ r v = 1

Ci E AKr) + C2

i = 1

and Anally

q

E l(r, av,bv) + B(r, av,bv)\

v = 1

< £ r | C i E Ai{r) + C2\+{% + 2K)rA{r) + hrL{r)

i = 1

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