Bounds on the Ratio $n(r, a)/S(r)$ for Meromorphic Functions

Bounds on the Ratio $n(r, a)/S(r)$ for Meromorphic FunctionsAuthor(s): Joseph MilesSource: Transactions of the American Mathematical Society, Vol. 162 (Dec., 1971), pp. 383-393Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1995760 .

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 162, December 1971

BOUNDS ON THE RATIO n(r, a)/S(r) FOR MEROMORPHIC FUNCTIONS(')

BY

JOSEPH MILES(2)

Abstract. Let f be a meromorphic function in the plane. We prove the existence of an absolute constant K such that if a1, a2, . . ., aq are distinct elements of the Riemann sphere then lim inf, <. (Z;q=' In(r, a1)/S(r)- 1i) < K. We show by example that in general no such bound exists for the corresponding upper limit. These results involving the unintegrated functionals of Nevanlinna theory are related to previous work of Ahlfors, Hayman and Stewart, and the author.

Introduction. If f is a nonconstant meromorphic function in lzl <??, we let

n(r, a) = n(r, a,f) denote the number of roots counting multiplicity of the equation

f(z) = a in lzI ? r and let S(r) = S(r,f) denote the mean covering number of the map f of lzl <r into the Riemann sphere. Our principal result is that there exists an absolute constant K such that if a1, a2,..., aq are distinct elements of the Riemann sphere then

lim inf ( n(r, aj)_ ) < K.

The lower bound for :q= 1 (n(r, aj)/S(r) - 1) contained in this result is not new; it is weaker than the bound given in Ahlfors' " unintegrated " second fundamental theorem [1] for meromorphic functions. However, Ahlfors' methods give no upper bound for D = 1 (n(r, aj)/S(r) -1). We give an example of a meromorphic function for which lim SUpr o0 n(r, O)/S(r) = oo; thus our result cannot be improved to yield

lim SUpr, 0 (Ij= 1 In(r, aj)/S(r) - II) <K. Finally we use our main result to show for any nonconstant meromorphic function and any Hausdorff measure A that there exists r oo such that n(rn, a)/S(rn) -* 1 for all a 0 A where A(A) = 0. This

is a strengthening of Theorem 2 in [6].

1. Terminology and notation. We assume familiarity with the basic notation and definitions of Nevanlinna theory. If I denotes the Riemann sphere and m

Received by the editors April 29, 1970 and, in revised form, May 21, 1970 and April 7, 1971. AMS 1969 subject classifications. Primary 3058, 3061; Secondary 3055, 3060. Key words and phrases. Meromorphic function, Nevanlinna theory, counting function,

mean covering number. (1) This research was conducted while the author held a Postdoctoral Research Associate-

ship sponsored by the National Research Council and the Office of Naval Research. (2) The author is indebted to Professor W. K. Hayman for several comments concerning

the presentation of these results and is particularly grateful for the suggestion of the present form of Theorem 1, which is a strengthening of the author's original statement.

Copyright (? 1972, American Mathematical Society

383



384 JOSEPH MILES [December

denotes normalized area measure on 1, then the spherical characteristic T(r) =T(r,f) of a functionf meromorphic in lzl < oo is defined by

(1.1) T(r) d JS(t)

where

(1.2) S(r) = n(r, a) dm(a).

The difference between the spherical and Nevanlinna characteristics off is bounded as r -*0oo. S(r) is called the mean covering number of the map f: IzI < r -* E. If y is an analytic arc on E, s is arc length measure on y, and L(y) is the length of y, then the mean covering number of f: IzI < r -* E with respect to y is

(1.3) S(r, y) = LI f n(r, a) ds(a).

L(r) denotes the length on E of the curve f(rei)-), -r < 0 < 7r. If we must specify the function under consideration, we use the notation S(r, y,f) and L(r,f).

If a E E and b E E, the distance between a and b is defined to be the length of the shorter great circle arc on E joining a and b. This distance is denoted by 8(a, b). Where there is no danger of confusion we do not distinguish between a complex number and its stereographic projection on E.

If Ec [1, oo), we denote the logarithmic measure of E by ml(E) = fE dt/t. If E, =E r [1, r], by the upper (lower) logarithmic density of E we mean

lim sup (inf) ml(E,)/log r. t-+ X

Suppose h(r) is a continuous, strictly increasing function such that h(O) = 0. The Hausdorff measure A associated with h is defined as follows. For any set A in the plane and any p >0, we consider all coverings of A by countable collections of disks Di with radius ri < p and let H(p) be the infimum over all such coverings of

Ji h(ri). Evidently H(p) increases as p decreases. We define A(A) to be lim,0 H(p). Many of our inequalities are valid only for sufficiently large values of the

variable, denoted by r > rO or t > to. It is not intended that ro and to have the same value each time they occur.

Our proofs rely heavily on the geometry of the curvef(reie). We recall in particular that the derivative with respect to 0 of argf(reie) is Re (reief'(rei9)/f(reie)) if f(rei9) #0 , oo and that the derivative of the argument of the vector tangent to the curve

f(reie) is Re (1 + (reief"(rei9)/f'(rei9))) iff'(reie) # 0, oo.

2. Statement and discussion of results. We state our principal result.

THEOREM 1. There exist absolute constants K< oo and C E (0, 1) such that iff is any nonconstant meromorphic function in lzl < oo then there exists Es [1, oo) having



1971] BOUNDS ON THE RATIO n(r, a)/S(r) 385

lower logarithmic density at least C with the property that, if E> 0, there exists ro=ro(E) such that if a1, a2, ... aq are elements of 2 with 8(ai, aj)> s for i1$j, then

(2.1) 2 (a1) < K ,=1 S(r)

for all r E, r > ro(E).

Our proof will in fact show that K may be chosen to be less than 800. A careful examination of the reasoning in ?4 shows this estimate for K could be considerably reduced.

Theorem 1 is related to the following result of Ahlfors [1, p. 189], which may be regarded as an unintegrated second fundamental theorem.

THEOREM (AHLFORS). If f is a nonconstant meromorphic function in IzI < oo and

a,, a2,..., aq are distinct elements of E, then there exists h> 0 depending on a1, a2, ..., aq such that

(2.2) :(n S(r )a-)I > -2-hS(r)

for all r > 0.

Since, for 71 >0, there exists E'c [1, oo) having finite logarithmic measure such that L(r)<S(r)1"2 + for all r e [1, oo)-E', we see that the lower bound for

=1 (n(r, aj)/S(r) -1) contained in Theorem 1 is weaker than the corresponding bound in Ahlfors' theorem. However, Theorem 1 also provides an upper bound for,

D = 1 (n(r, aj)/S(r)-1). The principal result concerning upper bounds for n(r, a)/S(r) is due to Hayman

and Stewart [5]. It states that if a> e, then there exists EaC [1, oo) having positive lower logarithmic density such that supaec n(r, a) < aS(r) for all r c Ea. The Hayman-Stewart result and the upper bound for :^ = 1 (n(r, aj)/S(r) - 1) contained in Theorem 1 are independent; this is a consequence of the fact that K does not depend on q in Theorem 1. Thus the two results complement one another.

In ?5 we give an example of a meromorphic function with order zero for which lim sup,,, n(r, O)/S(r) = oo. This example shows that the exceptional set of r- values is in general unbounded both in Theorem 1 and in the Hayman-Stewart result. (A second example showing that the exceptional set of Theorem 1 is unbounded is Example 2 of [6].)

We obtain Theorem 2 as a corollary of Theorem. 1.

THEOREM 2. If f is a nonconstant meromorphic function in lzl < oo and A is a Hausdorff measure, then there exists rp -- oo and there exists a set A of complex numbers with A(A) = 0 such that n(rp, a)/S(rp) -* 1 for all a 0 A.

The conclusion of Theorem 2 cannot in general hold as r -* oo through all values. This is a consequence of Example 2 in [6].




In [6] it is shown for any nonconstant meromorphic function f that there exists Ec [1, oo) having logarithmic density zero and there exists a set A' in the plane having inner logarithmic capacity zero such that lim,co;r?E n(r, a)/S(r) = 1 for all a 0 A'. We note that Theorem 2 has a smaller exceptional set of a-values than does this result, but has a larger exceptional set of r-values.

3. A lemma. The following lemma is used in the proof of Theorem 1. It is similar to a result of Fuchs [2]. Our proof is considerably less involved than is the proof of Lemma 1 in [2] because we are not concerned with

Im (reief"(reie)/f'(reie)).

LEMMA. There exist absolute constants K1 < oo and C E (0, 1) such that if f is a nonconstant meromorphic function in lzl < oo then there exists El F [1, oo) having lower logarithmic density at least C such that

(3.1) 1 Re reief"(reie) dO < K1S(r,f) 27T jf '(reie)

for all sufficiently large r E E1.

Proof. It follows from elementary considerations that if f is rational, then (3.1) holds for all sufficiently large values of r with K1 = 3. Thus we restrict our attention to transcendental functions.

We consider any r > 0 and associate with r the positive quantity h = hr = T(r,f)/3S(r,f). We let pl= reh, p2 = re2h, and p3 = rea3. By the differentiated Poisson-Jensen formula [4, p. 22] applied to f'(z) we have that if Iz =r and f'(z)

009, oo, then

zf"(Z) _1[log jf'(pleiQ0)j 2zp 1e' d(p (3.2) f '(Z) 2= r j (ple'_ Z)2

+ (z + a, _ 2 + 2b,z) +

~~~+ 2z - z _ _

where {an} are the zeros off'(z) and {bn} are the poles off'(z). The function w=z/(z-a) maps Iz =r> lal onto a circle in Re w>; it maps

I zi =r< IaI onto a circle in Re w< 2. Furthermore

1 on reie (33) 2 j -_ Re reie dO = 1 if r > lal,

= 0 if r < lal.

Combining (3.3) with the above observations about the function z/(z-a) we conclude for all complex a that

(3.4) Re r d < 1. 27T ~ree - a




From (3.2) and (3.4) we deduce

I f j Re reief"(reie) d6 -r f'(reie)

2Pi 2

_ _ r_ _ _ d2 (3.5) -< Ilog jf(pjei0)j I dcI)(2!p r2 pir d) 2,ff 27r ~~~(2 r2) p~loe'w- re 012

+ 2n(pl, 0, f ')+ 2n(p1, oo, f).

Standard estimates yield that

I J Re reief'(rel) d6

(3.6) 2 f '(reie)

2_ r2 {m(p1,f') + m(p1, I /f')} + T {N(p2, O,f') + N(p2, oo,f')}f

Certainly

(3.7) 2rpl/(p2-r2) = 2eh/(e2h_1) < 1/h.

Thus

(3.8) 21 Re reiof"(reie) d6 _ 2

{3T(P2,f) + 0(21) 27T , f'(reie) =y 3(2f)+o1}

We have

(3.9) T(p2,f') < 2T(p2, f)+m(p2, f/f).

Suppose s> 0. The lemma on the logarithmic derivative [4, p. 36] in combination with (3.8) and (3.9) implies that for r > ro(s)

(3.10) 1 Re reef'(rele) df < 12+ T(P3,f)+3 log + I 27r f '(re"e) = h h

We apply the following growth lemma, due to Hayman [3, Lemma 1 and Lemma 6].

LEMMA (HAYMAN). Suppose that p(x) and p'(x) are positive and nondecreasing for x > xo and that s > 0. For x > xo let h'= h= p(x)/p'(x). Then p(x + h') < h'(e + 6)p9(X) holds for all x in a set having lower density at least C(s), where C(s) is positive and depends only on s.

We apply this lemma to p(x) = T(ex, f) and note that h' = T(ex)IS(ex). Thus

(3.11) T(ex+h') < h'(e+s)S(ex)

holds on a set of values of x having lower density at least C(s). Introducing the change of variables r = ex, we note h'= T(r)/S(r) = 3h. Thus

(3.12) T(p3) < 3h(e + s)S(r)

holds on a set E2 of r-values having lower logarithmic density at least C(s).




Let E3 = {r > 1 : S(r) > T(r)2}. Then

(3.13) 3 t dt < | tT(t)2 dt T(1) c

Finally let E1 = E2 - E3. Then for r E E1

1 3S(r) (3.14) log, h = log, T(r) = o(T(r)) (r-*oo).

By (3.13), the lower logarithmic density of E1 is at least C(s). From (3.10), (3.12), and (3.14) we conclude

(3.15) 1 -.I Re reief'(reie) d < 3(12 + 2s)(e+ s)S(r)

27T j, f '(reie) =

for r>r0(s), r e E1. Setting K1=3(12+2s)(e+s) and C= C(s) we have the desired conclusion. We note that K1 may thus be chosen to be less than 98.

4. Theorems 1 and 2. We now use the lemma to prove Theorem 1. Since the case of f rational clearly causes no difficulty, we assume f is transcendental. Thus S(r) -* oo as r -* oo. We let K1 and C be the constants of the lemma and let El* be the set of all r> 1 for which (3.1) holds and for which f'(z) #0, oo for any z of modulus r. We let E'={r: L(r)?> S(r)213} and E=E*-E'. Thus E is an open set having lower logarithmic density at least C such that (3.1) holds for each r E E.

We are given E > 0 and elements a1, a2, . . ., a, of I such that 8(ai, aj) _ E if i #j. This condition implies the existence of k(s) < oo such that q <k(e). Without loss of generality we may assume aj 1 oo for 1 _j < q; this follows from the fact [4, p. 13] that transformations of the form w = (exp (i6O))(I + dz)/(z - a) correspond to rota- tions of E and hence preserve distances and areas on E.

For any r E E we partition {1, 2,..., q} into sets Hf, H2, and H3r by the following definitions:

H ={j: 1 <?jq and S(rj)1i < l/q

(4.1) H2= {j:1 < j< q and n(r, aj)/S(r) ? 1 + l/q}, and

H'r = {j: 1 < j < q and n(r, aj)/S(r) < 1-l/q}.

Clearly we have

(4.2) n n(r,aj) i 1.

je-Hr S(r)

For 1 <j<q we let yj be a closed line segment in the complex plane which has a1 as one endpoint, which lies on a ray from 0 to oo, and which has a stereographic projection on E of length s/4. Thus the distance on E between yi and yj is at least e/2 if i--]j.




We consider a particular j. It follows from Theorem 2 of [1, p. 165] that corresponding to yj there exists a constant h > 0 independent off such that, for all r > 0,

(4.3) JS(r,f)-S(r,,yj,f)J < hL(r,f).

The constant h in fact depends only on the length s/4 of yj and is independent of the position of yj. This can be deduced either from the proof of Theorem 2 in [1] or from the following elementary argument. If y is an arc of a great circle on E and has length e/4 then there is a rotation of E which maps y onto yp. Hence for an appropriate choice of a and 60, if we let F= (exp (i60))(1 + af)/(f- a), we then have S(r, y, f) = S(r, yj, F), S(r, f) = S(r, F), and L(r, f) =L(r, F). Applying (4.3) to the function Fwe conclude IS(r,f)-S(r, ,yf)I <hL(r,f). Thus h=h(s)>0 is such that for all r>0

(4.4) |S(r)-S(r, yj)J < hL(r)

for 1 <j? q. We now consider any j E H2. Since L(r) < S(r)213 for r E E, we conclude from

(4.4) that there exists rJ(e) > 0 such that r > rO and r E E together imply for j E H2

(4.5) S(r, yj) < S(r)(1 + 1/2k(s)).

For r > ro(s), r E E, and j E H2, the fact that q < k(s) combined with (4.5) implies the existence of a point z1,r E yC satisfying

(4.6) n(r, zj,r) < S(r)(1 + 1/2q).

Since E is open, for all r > ro(s) there exists r' > r independent of j E H2r such that

(i) r' E E,

(ii) n(r', aj) = n(r, aj), (iii) n(r', Z1,r) = n(r, Z1,r), and

(iv) S(r') < 2S(r).

Thus IzI = r' implies f(z) # aj and f(z) #Z,r for] e H2. From the argument principle we have for continuous determinations of the arguments involved and for all j E H2

(4.8) n(r, a1) - n(r, Z1,r) = (1/27T)[arg (f(r'e") - aj) - arg (f(r'e - ill) - aj)]

-(1/27T)[arg (f(r'e1")- z1, )- arg (f(r'e - i")_-Z,

Let [aj, z1,r] be the closed line segment from a1 to Zj,r. The transformation g(z)

=(z-aj)/(z-z1,r) maps [aj, Z1,r] onto a ray from 0 to oo. From (4.8) we have

(4.9) n(r, aj) - n(r, Z1,r) = (1/27T)[arg g(f(r'eil)) - arg g(f(r'e -))].

Elementary properties of the argument, applied to the function g(f), enable us to conclude from (4.9) that there exist nj > n(r, aj) - n(r, Zj,r) -1 disjoint open intervals




Ili, . . ., IV in (-7T, 7r) with Ij,=(sji, t,j) such that for 1 ?k<nn

(i) f(r' exp (iskj)) c- [aj, Zj,r] CYi5

(ii) f(r' exp (it i)) e [aj,, Zjr] yj, and

... [arg (f(r' exp (itt)) - aj) - arg (f(r' exp (is i)) - aj)]

- [arg (f(r' exp (it)) - Zj, )- arg (f(r' exp (iSj))-Z.,r)] > 27.

For notational convenience we do not indicate the dependence of nj or the intervals Ik on r. From (4.10) we conclude that the curvef(r'e 0), restricted to sJ0 < 0 < tj, has a tangent parallel to yj at some point in (sji, tj) and has a tangent perpendicular toyj at some point in (si, t,). Hence for 1 < k <nj,

t 'eief"1(r'ie) ? O T 1 (4.11) =Re ft i) +1 dO >

RefjI(rie) =2(27T) 4

For r e E, r > r0(8), and j e H2r, we let

U {Iki : I-< k < nj} and (4.12) u1={,1kn} ad

Vj = {Ik, there exists 0 e (sk, t,) such that f(r'eie) e yi for some i j j}.

We do not indicate the dependence of Uj and Vj on r. From the definition of V1, (4.10)(i), and (4.10)(ii) it follows that UjeHr U r UjeH2 Vj is a disjoint collection of intervals. We note that Ik, e Vj implies that the length on E of the curve f(r'ei0). sJ < 0 < t,, is at least ? since the distance between yi and yj is at least 8/2 if ioj. From (4.7) it follows that for each r e E we have L(r') < S(r')213 < 2S(r)213. The last two observations together imply for each j e H2r that the number of elements of Vj is at most L(r')/8 < 2S(r)2 3/8. For r> ro(8) and r c E we thus have that UjEH2 U -U1EH2r Vj is a disjoint collection of at least

f {n(r, aj)-n(r, Zj,r)1 -2S(r)213/8} jeH2

intervals, for each of which (4.11) holds. Thus

I n iRe r+e

t eio) +1 dO f '(r'te10)

(4.13) >1 ( (- > {n(r, aj) - n(r, Zi,r) -1-2S(r)213/8}) 4

eH"2

From (4.6) and (4.13) we deduce for r>ro(8), r c E,

1 'a R r'e1Of"(r'ei0) dO

i+yJ Re7 f '(r'ei0)

(4.14) ~~~~~~> I

(-q +~ fn(r, aj) -S(r) -S(r)12q -2S(r )213/e}) jeH2

f ~n(r, aj) -S(r)}) -q/4 -S(r)/8 -qS(r)21/8




From (4.7)(iv), the lemma of ?3, and the fact that q < k(s) we conclude

(4.15) 4I ( 2 {n(r, aj)- S(r)}) < k(s)/4 + S(r)/8 + k(s)S(r)213/2? + I + 2K1 S(r).

Thus for r > ro(s) and r c E,

(4.16) E {n(r, aj)-S(r)} ? (8K1 + I)S(r). jiH2

A similar discussion applies to the set H3. The required inequality in this case also follows from Ahlfors' theorem mentioned in ?2 if one checks that the constant h of that theorem depends only on mini0j 8(ai, aj). The proof is completed upon combining (4.2), (4.16), and an inequality for H3r similar to (4.16).

We now use Theorem 1 to prove Theorem 2. Let h be the strictly increasing function associated with the Hausdorff measure A. Let p be a positive integer and

pp > O be such that h(pp) < I /p3. There exists cp < oo such that Ia-bI/8(a, b) < cp if

jai <p, Ibi <p and a=Ab. Applying Theorem 1 with s=pp/cp we conclude there exists rp > 2P such that {IaI < p: In(rp, a)/S(rp) - 1 ? l/p} can be covered by a set of [Kp] disks in the plane each of radius pp where [ ] is the greatest integer function. We denote the union of these disks by Dp. Thus n(rp, a)/S(rp) -- 1 for all a 0 A where A = nm=i UP= m Dp. From the definition of Hausdorff measure we see that A(A) does not exceed Z1po= m (Kp)h(pp) for any value of m. Hence A(A) = 0.

5. An example. We now exhibit a meromorphic function for which

lim sup n(r, O)/S(r) = oo. r -o

We define a sequence Nn=- 224n and verify directly that

(5.1) 4Nn E Nj2N, = o(1), i = n +

n - 1

(5.2) ZNj2<Nnl/2, n-2,3,4,..., i=1

and

(5.3) 2Nn + 1 > 2(4Nn), n=,2 1, 2, 3,

In (5.1) and throughout ?5 all quantities indicated to be o(1) are o(1) as n oo. The required function is

(5.4) f(z) = f1 ( n ? )N)/fI ( zNn

We adopt the notationfn(z) =gn(z)/hn(z) where

(5.5) gn(z) = (1 + Z/4Nn)N2 and hn(z) = (1 - z/2Nn)Nn.




For n ?2 we let rn = 4N- and select a?n (0, -] in a manner to be specified later. Letting Pn=rn(l +?an) we have if IZi =Pn that

nl-1 n-1 3(N.) AN2

(5.6) log 171 gj(z) ? log 3(4 n-1

= 2 NN2(log 3 + Nn log 4-N1 log 4). i= 1

Likewise we find for ZI =Pn

n -1 n -1 Nn NJ

log 11hj(z) ? log jji (_4)

(5.7) =1 j=h(Z -1 22>

(57)= Nj(Nn log 4-Nj log 2-log 2). j=1

Combining (5.2), (5.6), and (5.7) we see that on IZI =Pn

n-i

(5.8) log+ j7 {f(z) = o(Nn)

Similarly if IZi =Pn then by (5.1)

(5.9) log <I g1(z) - N; 4NnNi = o(1) j=n+l j=n+l

and by (5.1) and (5.3)

co co

(5.10) log 11 hj(z) ? -3 Nj4 Nn2Nj= o(1). i=n+i i=n+i

If IZ =Pn we have

(5. 1 1 ) log 12 Nhn(Z)I = log I2Nn_z/4NnINn

>N log |10 +1an-2 Nni _ -2Nn2 -Nn = o(1).

Thus on IZI=Pn

(5.12) log Ihn(Z)I _ Nn2 log 2-o(1).

From (5.8), (5.9), (5.10), and (5.12) we conclude there exists 81(n) 0 as n oo

such that for any choice of a?n (0, 4-] we have everywhere on jzj = Pn rn( + an)

(5.13) log lf(z)j -' Nn2{log I '(I +z/rn)I? +s(n)}.

An elementary calculation shows that log I (1 + pneiO/r)>0 if and only if cos 0 2-(4?c?ac)/(2+2an). For 0< an-? we have

cos 4a" /2 < 1 _ 42ocn/2! + 44a2/4! (5.14)

n n

< I1-2an < 1-(4an+ n)/(2+2an).




We deduce that if log 11(l + pne10/rn)I >0 and 0 & [-, r] then 101 ? 4an 2. Combin- ing this fact with (5.13) we have

(5.15) M(Pn f) < N2{i (n)+ 812 log (I +n)

=< Nn2{-- (n) +?n }

Letting 82(n)=Nn-1 we have

(5.16) N(pn,f)-N(rn,f) = n(rn,f) log (1 + an) ? 2Nn 1og (1 +an) _ Nn = E2(n)Nn2.

Because the difference between the spherical characteristic and the Nevanlinna characteristic is bounded, we see from (5.15), (5.16) and the definition of 82(n) that for n> nO

S r)<T(pn, f) - T(rn, f) S(r ) < log (1 + an)

< m(Pn, f) + N(pn f) - N(rn f) + 0(1)

(5.17) log2(1 ?1 + ) < Nn {e1(n) + e2(n) + cnl2} + 0(1)

CCn/2

< 2Nn2{sj(n) + 2e2(n) + acx32}

(Xn

We choose an =min (-, [s1(n) + 2s2(n)]2/3) and note that a?n 0 as n oo. Thus for

n>no

(5.18) S(rn) < 4N(2 61(n)+2e2(n))2/3 n

Sincef has Nn2 zeros on Izi =rn, we conclude that lim supr,, n(r, O)/S(r)= oo.

REFERENCES 1. L. V. Ahlfors, Zur Theorie der Uberlagerungsfldchen, Acta Math. 65 (1935), 157-194. 2. W. H. J. Fuchs, A theorem on the Nevanlinna deficiencies of meromorphic functions of

finite order, Ann. of Math. (2) 68 (1958), 203-209. MR 20 #7106. 3. W. K. Hayman, An inequality for real positive functions, Proc. Cambridge Philos. Soc. 48

(1952), 93-105. MR 13, 631. 4. , Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press,

Oxford, 1964. MR 29 #1337. 5. W. K. Hayman and F. M. Stewart, Real inequalities with applications to function theory,

Proc. Cambridge Philos. Soc. 50 (1954), 250-260. MR 15, 857. 6. J. Miles, On the counting function for the a-values of a meromorphic function, Trans. Amer.

Math. Soc. 147 (1970), 203-222. MR 40 #7454.

UNIVERSITY OF ILLINOIS, URBANA, ILLINOIS 61801



Bounds on the Ratio $n(r, a)/S(r)$ for Meromorphic Functions

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