Value Distribution for a Class of Small Functions in the Unit Disk

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 537478,24pages

doi:10.1155/2011/537478

Research Article

Value Distribution for a Class of Small Functions in

the Unit Disk

Paul A. Gunsul

Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA

Correspondence should be addressed to Paul A. Gunsul,[email protected]

Received 20 October 2010; Accepted 21 January 2011

Academic Editor: Brigitte Forster-Heinlein

Copyrightq2011 Paul A. Gunsul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Iffis a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic functionTr, f could be used to categorizef according to its rate of growth as|z| r → ∞. Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integerk,mr, fk_/f _o_T_{r, f}_as_r _{→ ∞}_{, possibly outside a set of finite measure}

wherem denotes the proximity function of Nevanlinna theory. Iff is a meromorphic function in the unit diskD {z : |z| < 1}, analogous results to the previous equation exist when lim sup_r_→₁−Tr, f/log1/1−r ∞. In this paper, we consider the class of meromorphic functions Pin D for which lim sup_r_→₁−Tr, f/log1/1−r < ∞, limr→1−_T_{r, f} ∞_, andmr, f/f oTr, fasr → 1. We explore characteristics of the class and some places where functions in the class behave in a significantly diﬀerent manner than those for which lim sup_r_→₁−Tr, f/log1/1−r ∞holds. We also explore connections between the class

Pand linear diﬀerential equations and values of diﬀerential polynomials and give an analogue to Nevanlinna’s five-value theorem.

1. Introduction

This paper uses notation from Nevanlinna theory which is summarized here for the reader’s convenience. We denote bynr, fthe number of poles off in|z| ≤ r <1, where each pole is counted according to its multiplicity. Also,nr, fcounts the number of distinct poles of

f in |z| ≤ r < 1 disregarding multiplicity. Ifx ≥ 0, then logx max0,logx. We define the proximity functionmr, f, the counting functionNr, f, and the Nevanlinna characteristic functionTr, fas follows:

mr, f 1

2π

_2π

0

logf

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Nr, f

r

0

nt, f−n0, f t dtn

0, flogr,

Tr, fmr, fNr, f.

1.1

Also, we have that

Nr, f

_r

0

nt, f−n0, f t dtn

0, flogr. 1.2

A meromorphic function f in the unit diskD {z : |z| < 1} can be categorized according to the rate of growth of its Nevanlinna characteristicTr, fas|z|rapproaches one. If

lim sup

r→1

Tr, f

−log1−r ∞, 1.3

many value distribution theorems analogous to those for transcendental meromorphic functions in the complex plane can be derived. In particular, results useful in studying solutions of linear diﬀerential equations which are analogous to theorems of H. Milloux can be shown—namely, for each positive integerk,

m

r,f

k

f o

Tr, f 1.4

asrapproaches one, possibly outside a set of finite measure wheremdenotes the proximity function of Nevanlinna theory. For meromorphic functions of lesser growth than 1.3, analogous theorems need not hold. IfFconsists of those meromorphic functionsfinDfor which

lim sup

r→1

Tr, f

−log1−r α

f<∞, 1.5

Shea and Sons1showed thatfis inFfor eachfinF, but also that there exist functionsf

inFwith unbounded characteristic for which

m

r,f

f

>log 1

1−r log log

1

1−r 1.6

on a sequence ofr approaching one. Thus all functions inFwith unbounded characteristic need not satisfy1.4fork1.

In this paper, we denote byPthose functionsfinFfor whichTr, fis unbounded as

rapproaches one and for which1.4does hold fork1. We derive striking properties of class

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equations defined inD. For functions inFwithαf>0, we develop an interesting theorem analogous to Nevanlinna’s five-value theorem for functions in the plane. Further, we prove a value distribution theorem for diﬀerential polynomials

Φ n

k0

akfk, 1.7

wherefis inP, theakare meromorphic functions inD, andTr, ak oTr, f, asr → 1.

Our paper proceeds as follows. InSection 2, we note examples of functions inPand properties of the class. InSection 3, we prove a uniqueness theorem for functions in classF

and hence in classP. InSection 4, we look at diﬀerential equations for which functions in

Pare either coeﬃcients or solutions, and inSection 5we consider diﬀerential polynomials. Much of the research reported here was part of the author’s Ph.D. dissertation written at Northern Illinois University2.

2. Properties and Examples of Functions in Class

P

First we note thatPis not empty. Forβ >0, the functionfdefined inDby

fz exp

_βi

1−z

2.1

is in classP, since

Tr, f β

2πlog

1r

1−r

2.2

by a calculation in Benbourenane3and by properties ofmand a lemma of Tsuji4, page 226,

m

r,f

f

O

log log 1 1−r

, r−→1. 2.3

Clearlyαfas defined in1.5isβ/2πfor this function.

The following proposition gives some simple closure properties ofP.

Proposition 2.1. Iffandgare inPandcis a nonzero complex number, we have

icfis inP;

ii1/fis inP;

iiifn_{is in}_P_{for each positive integer}_n_;

ivfgmay not be inP;

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The proof ofi,ii, andiiiinProposition 2.1follows by easy calculation. To seeiv, letg1/f, and to seev, letg−f.

The complicated nature of classPis demonstrated by the following theorem whereby some sums and products are inP.

Theorem 2.2. Letfbe a meromorphic function in classP.

iIfcis a nonzero complex number for which

lim inf

r→1

Nr,−c

Tr, f 0, 2.4

thenfcis inP.

iiIfgis a meromorphic function inDwhich is not identically zero and such thatTr, g

oTr, f,r → 1, andmr, g/g oTr, f,r → 1, thenfgis inP.

iiiThere exists a Blaschke productBsuch thatBfis not inP.

ivThere exists a Blaschke productBsuch thatBfis inP.

Remark 2.3. In Nevanlinna theory, the Valiron deficiency of a complex value c for a

meromorphic functionfinDis defined by

Δc, flim inf

r→1

Nr, c

Tr, f. 2.5

It is knowncf. Theorem 2.20 on page 210 in5that if

lim

r→1T

r, f ∞, _2.6

then

lim

r→1

Nr, a

Tr, f 1 2.7

except for at most a set ofa-values of vanishing inner capacity. This fact enables us to show that the functionFinPdefined by

Fz exp

i

1−z

2.8

hasFcinPfor all complex numbersc, because

Δ−c, F 0 2.9

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Remark 2.4. The following example illustrates partiiofTheorem 2.2.

Example 2.5. Letfz ei/1−zandgz e1z/1−z. Theng is not identically zero, and it is

well known thatTr, g O1. Therefore,Tr, g oTr, fasr → 1. Also we have that

m

r,g

g

≤T

r, 2

1−z2 O1 asr −→1, 2.10

since 2/1−z2is the quotient of two bounded, analytic functions in the unit disk. And so we have thatfg ei1z/1−z∈ P.

We turn to the proof ofTheorem 2.2.

Proof of Part (i). Letg fc. Thengf. We will show thatg ∈ P.

First, by calculation and properties of the Nevanlinna characteristic, note that

Tr, gTr, fO1 asr −→1. 2.11

Also, by calculation and properties of the proximity function, we get

m

r,g

g

m

r, f

fc

≤m

r,f

f

m

r, f fc

≤m

r,f

f

m

r,1 c

fc

≤m

r,f

f

m

r, 1 fc

O1 asr −→1.

2.12

Now, sincef∈ P,

m

r,f

f

oTr, f asr −→1,

oTr, g asr−→1.

2.13

Also, sinceΔ−c, f 0,

lim sup

r→1

mr,1/fc

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and so

m

r, 1 fc

oTr, f asr−→1,

oTr, g asr−→1.

2.15

Therefore,g ∈ Psincemr, g/g oTr, gasr → 1, andTr, gis unbounded as

r → 1.

Proof of Part (ii). First,Tr, fgis unbounded, since it can be shown that

Tr, fgTr, fO1 asr−→1, 2.16

sinceTr, fis unbounded.

Now note thatfg/fg g/gf/f. Therefore,

m

r,

fg fg ≤m

r,g

g

m

r,f

f

,

oTr, f asr−→1,

oTr, fg asr−→1.

2.17

Thusfg∈ P.

Proof of Part (iii). LetBbe the Blaschke product defined in6, Proposition 6.1, page 273. This

Blaschke product has the feature that for any >0 there exists an exceptional setE1 ⊂0,1

satisfying

E1

dr

1−r <∞, 2.18

such that

B_B_zzO

₁

1− |z|

_1.5

, |z|∈/E1, 2.19

and there exists a setF1∈0,1, satisfying

F1

dr

1−r ∞, 2.20

and a constantC >0, such that

B_B_xx≥ C

1−x1.5 log

1

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Letgz i/1−z. Thenegz ∈ P. Now defineqz Bzegz. We will now show thatqz∈ P/ .

First, it is easily shown thatTr, egz Tr, q O1asr → 1.

Note that sinceegzgegz, we haveg egz/egz. Therefore, sinceegz∈ P,

mr, goTr, egz asr −→1. 2.22

And so

m

r,B

B

m

r,q

q −g

_≤_m_r,q

q

mr,−glog 2. 2.23

Therefore,

m

r,q

q

≥m

r,B

B

−mr,−g−log 2. 2.24

Using2.21, we have on a small exceptional set with|z|rforr∈F1\E1,

B_B≥ C

1−r1.5log

1

1−r. 2.25

Taking the logof both sides, we get

logB

B

≥log

C

1−r1.5log

1 1−r

≥log C

1−r1.5

≥logC1.5 log 1 1−r.

2.26

So, calculating the following ratio yields

mr, q/q Tr, q ≥

mr, B/B Tr, egz −

mr, g Tr, egz −

log 2

Tr, egz −→3π >0 asr −→1. 2.27

Therefore,q /∈ P.

Proof of Part (iv). LetBbe a Blaschke product with zeros{zn}such that|zn|1−1/n5_{for all}

integersn≥2. Theorem B in Heittokangas6shows thatBis inHp_for_p_in₀_,₃_/₄_{, so}_B_/B

is of bounded characteristic. HenceBfis inPforfinP.

Remark 2.6. Since a Blaschke product is a bounded, analytic function, we see from partsiii

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Further study of examples in classP shows that the function f defined byfz

expi/1−zhas1.4holding for allk. However, there are alsof inPfor which1.4does not hold fork2. We have the following theorem.

Theorem 2.7. There exists an analytic functionhinPsuch thatmr, h/h/oTr, h, asr → 1.

Proof. First, we begin by constructing a function which has unbounded characteristic asr →

1, but its derivative is of bounded characteristic. This construction is from7, page 557. Letabe an integer greater than or equal to 2. Define form≥1,

Qm m

k1

ak a a−1a

m₋₁_,

tm1−

γ Qm,

2.28

whereγis a constant such that 0< γ <1. Now define

Fz

_z

0

fwdw, 2.29

where

fz

∞

m1

1

z tm

am

. 2.30

Shea showed in7thatfis analytic in the unit disk and satisfies

αlog

1 1−r

< Tr, f≤logMr, f< βlog

1 1−r

asr−→1, 2.31

whereαandβare constants such that

0< α < γ

a, β > γ

log 2

loga, 2.32

and Mr, f is the maximum modulus function for f. Choose γ and a such that γ

log 2/loga < 1, so we can take β < 1. This implies that F is bounded in the unit disk by the following argument: forzreiθ_,

|Fz|

_z

0

fwdw≤

_r

0

fρeiθdρ≤

_r

0

Mρ, fdρ. 2.33

By2.31, we have that

Mρ, f< 1

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Therefore, sinceβ <1, we have by a simple integration that

|Fz|<

_r

0

1

1−ρβ

dρO1 asr−→1. _2.35

LetKz ₀zFwdw. Define hz Kzegz wheregz i/1−z. Recall that

egz ∈ P. We now show thathz ∈ P. First, we see thatTr, K O1asr → 1, by the following:

Tr, K mr, K

≤ 1

2π

0

log

r

0

Fρeiθdρ dθ,

2.36

and sinceFis bounded, there exists a constantMsuch that

Tr, K≤ 1

2π

_2π

0

log

_r

0

Mdρ dθ≤logM. 2.37

Thus,Tr, h≤Tr, K Tr, egz≤Tr, egz O1, asr → 1. On the other hand,

Tr, egzT

r, h K

≤Tr, h O1 asr −→1.

2.38

Therefore,Tr, h∼Tr, egzasr → 1.

Now, we bound mr, h/h from above by using properties of the Nevanlinna characteristic:

m

r,h

h

m

r,Kg

_eg_K_eg

Keg

≤m

r,K

K

mr, glog 2

≤mr, gO1 asr−→1

oTr, egz asr −→1

oTr, h asr−→1.

2.39

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Now we show thatmr, h/h/oTr, hasr → 1. By a quick calculation, we have

m

r,K

K

≤m

r,h

h −2g

K

K −g

₋_g2

≤m

r,h

h

mr,2 mr, gm

r,K

K

mr, g2mr, glog 4

≤m

r,h h

oTr, h asr−→1.

2.40

Also we have from the above constructionKfso there exists anα >0 such thatmr, K> αlog1/1−r. And so,

m

r,K

K

≥mr, K−mr, K

≥αlog

1 1−r

O1 asr−→1.

2.41

Combining2.40and2.41, we have

mr, h/h

Tr, h ≥2πα /0 asr−→1. 2.42

Remark 2.8. If we defineAto be the set of functions inFsuch that1.4holds for all positive

integersk,Theorem 2.7showsAis properly contained inF. Further, we note that in the proof

ofTheorem 2.7abovehKeg_{can be replaced with}_h_Kp_where_p_∈_A_{. Also the idea of the}

proof ofTheorem 2.7can be used to show that fork >1 there exist functionshinFfor which

m

r,h

j

h oTr, h asr −→1 2.43

for all integers 1< j≤k, but

m

r,hk1

h /oTr, h asr −→1. 2.44

The functionhin the proof ofTheorem 2.7provides us with further information about

P.

Theorem 2.9. There exists a functionhinPsuch thathis not inP.

Proof. The function h Keg _of _{Theorem 2.7} _{is in} _P_{. Using the Nevanlinna calculus and}

properties ofK, one can show mr, h/h/oTr, h asr → 1. We omit the details here

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For functionsfin classF, we may callαfdefined in1.5the index off. In1, Shea and Sons showed forfinFthat

αf≤αf1kf1, 2.45

where

kflim sup

r→1

Nr, f

Tr, f1, 2.46

and this inequality is best possible. For analytic functions in classP, we get

Theorem 2.10. Iffis an analytic function in classP, then

iTr, f≤Tr, f oTr, fasr → 1;

iiαf≤αf;

iiiifNr,1/f olog 1/1−rasr → 1, thenαf αf.

Proof. For partisincefis analytic inP, we have

Tr, fmr, f≤m

r,f

f

mr, f

m

r,f

f

Tr, f

2.47

from which the result follows. Using the definition of the index offand off, partiicomes fromi.

To seeiii, we observe

Tr, f≤Tr, fT

r, f f

Tr, fT

r,f

f

O1 asr −→1. 2.48

Thus,

Tr, f≤Tr, fm

r,f

f

Nr, fN

r,1 f

O1 asr −→1. 2.49

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3. Connections between Class

P

and Differential Equations

We discuss some relationships between the coefficients of the linear differential equation and its solutions and how the coefficients and solutions relate to classP. We consider the complex linear differential equation

fnan−1zfn−1· · ·a0zf0 3.1

in the unit disk, with analytic coeﬃcients.

There has been a tremendous amount of recent research on the relationship between the growth of the solutions of3.1and the growth of the analytic coeﬃcients in the unit disk. Some recent papers include8–10. We now quote some of the important results that have a connection with classFand, therefore, classP. The theorems use the definitions of the weighted Hardy space and weighted Bergman space which are stated below for convenience.

Definition 3.1. We say that an analytic functionf in the unit disk is in the weighted Hardy

spaceHqpfor 0< p <∞and 0≤q <∞if

sup

0≤r<1

1−r2q

1 2π

2π

0

freiθpdθ

1/p

<∞. 3.2

We say thatfis inH_q∞if

sup

z∈D

1− |z|2q fz<∞. _3.3

Definition 3.2. We say that an analytic function f in the unit disk, D, is in the weighted

Bergman spaceApqif the area integral overDsatisfies

D

fzp1− |z|2qdσ

1/p

<∞ 3.4

for 0< p <∞and−1< q <∞.

The theorems below also mention the Nevanlinna classN, the meromorphic functions of bounded characteristic in D. If a function is in N, then it is not in P, since P only has functions of unbounded characteristic.

Theorem 3.3see10, page 320. Let f be a nontrivial solution of 3.1with analytic coeﬃcients

aj,j0, . . . , n−1, in the unit disk. Then we have that

iif−1< α <0 andaj∈H_a1n−j1/n−j for allj 0, . . . , n−1, thenf ∈N;

iiifaj∈A1/n−jfor allj 0, . . . , n−1, oraj∈A1_n−j−1for allj0, . . . , n−1, thenf∈N;

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Theorem 3.4see10, page 320. We have that

iif all nontrivial solutions f ∈ N, then the coeﬃcients aj ∈ _0<p<1/n−jAp for allj

0, . . . , n−1;

iiif all nontrivial solutionsf ∈ F, then the coeﬃcientsaj ∈

0<p<1/n−jH1/pp for allj

0, . . . , n−1.

We also have the following characterization, which uses the order of growth offin the unit disk defined as

ρflim sup

r→1−

logTr, f

−log1−r. 3.5

Theorem 3.5see9, page 44. All solutionsf of 3.1, where aj is analytic inD for all j

0, . . . , k−1, satisfyρf 0 if and only ifaj∈_0<p<1/n−jApfor allj0, . . . , n−1.

Whenn 1 in 3.1, we observe using Theorem 3.3, iff/f −a0 ∈ H_α11 with −1< α <0, thenf ∈Nand, therefore,f /∈ P. We can also conclude that iff/f −a0 ∈A1,

thenf ∈Nand sof /∈ P. Also, iff/f−a0 ∈H₁1, thenf∈ F, which meansf may be inP.

On the other hand, usingTheorem 3.4, we have iff∈ P, thenf/f−a0∈

0<p<1H

p

1/p.

Ifa0z −βi/1−z2, thenfz eβi/1−z is a solution of the diﬀerential equation. However, ifa0βi/1−zkfor an integerk≥3, thena0is of bounded characteristic, but the

solution

fCeβi/1−zk−1 3.6

has orderρ k−2 > 0 and, therefore,f /∈ F. This shows the delicate nature between the growth of the coeﬃcient and the solution; that is, a subtle change in growth of the coeﬃcient can result in a solution that is no longer considered slow growth.

Whenn2 in3.1, Theorems3.3and3.4have the following corollary.

Corollary 3.6. Letfbe a non-trivial solution of 3.1with analytic coeﬃcientsa0anda1in the unit

disk. Then

iifa1∈H_α11 anda0∈H_2α11/2 for−1< α <0, thenf∈Nandf /∈ P;

iiifa1∈A1anda0∈A1/2ora0anda1∈A1₁, thenf∈Nandf /∈ P;

iiiifa1∈H₁1anda0∈H₂1/2, thenf∈ Fand, therefore, could be inP;

ivif all non-trivial solutionsf∈ F, thena1∈

0<p<1H

p

1/panda0∈

0<p<1/2H

p

1/p.

The functionweβi/1−zis a solution to the equation

w− βi

1−z2w

₋ 2βi

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Sincewhas no zeros, another solution of the above second-order diﬀerential equation that is linearly independent ofwf1zis

f2z f1z

dz

f1z2 e

βi/1−z_e−2βi/1−z_dz.

3.8

Computation shows

f2z eβi/1−z

−1−z−2βilog

₋

2βi

1−z

∞

n1

−2βi

nn1!

₋

2βi

1−z

n

. 3.9

We know thatf1 ∈ P, but what can be said aboutf2? The above form off2makes it

diﬃcult to calculate the growth, but we do know that, byiiiinCorollary 3.6, ifa0 ∈ H₂1/2

anda1∈H₁1, thenf2∈ F.

Example 3.7. We show fora0 −2βi/1−z3anda1 −βi/1−z2,a0 ∈H₂1/2anda1 ∈H₁1.

To seea0 ∈H₂1/2, we first note

|a0|1/2

−2βi

1−z3

1/2

2β

|1−z|3/2. 3.10

We integrate and apply a lemma from Tsuji4, page 226which states that, in particular,

_2π

0

dθ

1−reiθ3/2 O

1

1−r1/2 , 3.11

and get that there exists a constantMsuch that

1 2π

2π

0

2β

|1−z|3/2dθ

2β

2π

0

1

|1−z|3/2dθ≤

2β

2π M

1−r1/2. 3.12

And so

1−r22

1 2π

_2π

0

|a0|1/2dθ

2

≤1r21−r2 2βM

2π1−r1/2

2

1r21−rC,

3.13

which goes to zero asr → 1. Therefore,a0∈H₂1/2.

A similar calculation shows thata1 ∈H₁1.

The question as to whetherf2∈ Pis not a trivial question as there exist examples, such

asExample 3.9below, where at least one solution is in classPand at least one solution is not

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Example 3.8. The functionhinTheorem 2.7is a solution of3.1withn2 whena0 −g− g2anda1−K2g K/KgK. Then since

h h −a1

h

h −a0, 3.14

we have that

m

r,h h

≤mr,−a1 m

r,h h

mr,−a0 log 2. 3.15

Now, recall from the proof ofTheorem 2.7that

αlog 1 1−r ≤m

r,h

h

oTr, h asr−→1, 3.16

and sinceh ∈ P, by3.15, we conclude that at least one ofa0 ora1 has index greater than

or equal toα. Therefore, as a consequence ofTheorem 2.7, we have a growth estimate for the coeﬃcients of this diﬀerential equation.

It can also be shown thata0 ∈H₂1/2. However,a1is not inH₁1, and thus the converse

ofCorollary 3.6iiiis not true.

For diﬀerential equations of the form3.1wheren≥3, we first quote two examples. The first example has some solutions of3.1inPand some not.

Example 3.9see9, Example 10, page 52. The functions

f1z ei1z/1−z−e−i1z/1−z, f2z ei1z/1−z, f3z 1z

1−z 3.17

are linearly independent solutions of

fa1zfa1zfa0zf 0, 3.18

where

a0z −8

1z1−z5,

a1z 4

1−z4 2

3z28z5

1z21−z2,

a2z −2 3z4

1z1−z.

3.19

It can be shown thatf2 ∈ P. However,f3is of bounded characteristic and, therefore,f3 ∈ P/ . It is known thatf1has order zero but unknown iff1 ∈ P.Also, according to9, we have

thata0∈0<p<1/3A

p

−1/3,a1∈0<p<1/2A

p

0, anda2 ∈

0<p<1A

p

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This next example is also from9.

Example 3.10see9, Example 11, page 53. The functions

f1z ei1z/1−z−e−i1z/1−z, f2z ei1z/1−z,

f3,4z

1z

1−z

×e±i1z/1−z

3.20

are linearly independent solutions of

f4a3zf3a2zfa1zfa0zf0, 3.21

where

a0z 16

1−z8,

a1z 16

1z2

1z1−z5 −

39z9z2₃_z3

1z31−z3 ,

a2z 8

1−z4 4

918z9z2

1z21−z2,

a3z −12 1z

1z1−z.

3.22

It is known thatf2,f3, andf4are all inP, and it is known thatf1has order zero. Also, it can

be shown thata0∈

0<p<1/4A

p

0,a1∈

0<p<1/3A

p

−1/3,a2 ∈

0<p<1/2A

p

0, anda3∈

0<p<1A10.

Proceeding as in the discussion ofExample 3.8, we have the following theorem.

Theorem 3.11. If a functionfinPsatisfies a diﬀerential equation of the form3.1such that

m

r,f

n

f /o

Tr, f asr −→1,

m

r,f

j

f o

Tr, f as r−→1

3.23

for all integersjsuch that 1≤j < n, then at least one of the analytic coeﬃcientsaj has index greater

than or equal toαasr → 1 for someα >0.

For nonhomogeneous diﬀerential equations of the form

(17)

whereaj is analytic in the disk for allj 0, . . . , n, we state a result from9that applies to

classF.

Theorem 3.12see9, Theorem 7, page 46. All solutionsfof3.24satisfyρf 0 if and only

ifρan 0 andaj ∈_0<p<1/k−jApfor allj0, . . . , n−1. Therefore, if all solutionsfof3.24are

inP, thenρan 0 andaj ∈

0<p<1/k−jApfor allj0, . . . , n−1.

Theorems3.3 and 3.4 do not tell the whole story regarding class F. Instead of the coeﬃcients being in a certain function class, what can we say about the solutions of3.1if we know the coeﬃcients have a certain index in classF? We show the following proposition.

Proposition 3.13. Letkbe a positive integer. Ifa0is an analytic function inDfor which the index of

a0isαa0> k, thenf /∈ Pforf∈ Fwhenfk−a0f0.

Proof. Note that sincefk−a0f 0, we havea0fk/f, and we want to show that

lim sup

r→1−

mr, f/f

Tr, f /0. 3.25

Sinceαa0> k, there exists a real numbers > ksuch thatTr, a0≥slog1/1−ron some

sequence ofr’s asr → 1. Also, sincef ∈ F, we haveTr, f ≤ tlog1/1−r. Now, since

f ∈ F,fk ∈ Fand sofi/fi−1 ∈ Ffor 1≤i≤k. By Shea and Sons1,mr, fi/fi−1 ≤

log1/1−r2o1log log1/1−rfor 1≤i≤kasr → 1. Now, we have the following:

m

r,f

k

f m

r, f

k

fk−1 fk−1

fk−2 · · ·

f f ≤m r, f k

fk−1 · · ·m

r,f f m r,f f ,

≤klog

1 1−r

k2o1log log

1 1−r

asr −→1.

3.26

So, sincea0is analytic, we have

m r,f f ≥m r,f k

f −m

r, f

k

fk−1 − · · · −m

r,f

f

≥slog

1 1−r

−klog

1 1−r

−k2o1log log

1 1−r

asr −→1

s−klog

1 1−r

−k2o1log log

1 1−r

asr −→1.

3.27

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With a similar argument as above,Proposition 3.13is also true ifa0 is meromorphic

andNr, a0 olog1/1−rasr → 1.

4. The Identical Function Theorem

For functions in classF, we have an analogue to the Nevanlinna five-value theorem which we quote as stated in11.

Theorem 4.1see11, page 48. Suppose thatf1andf2are meromorphic in the plane and letEja

be the set of pointszsuch thatfjz aj 1,2). Then ifE1a E2afor five distinct values ofa,

f1z≡f2zorf1andf2are both constant.

Our analogue and proof follow. The proof has a subtle diﬀerence from the direct analogue of the proof of Theorem 4.1 in11.

Theorem 4.2. Letf1zandf2zbe meromorphic functions in classFsuch thatαf1≥αf2>0

and letEjabe the set of pointszsuch thatfjz aforj 1,2. Then ifE1a E2aforq

distinct values ofasuch thatqis an integer andq >42/αf1, thenf1z≡f2z.

Proof. Suppose f1 and f2 are not identical and that {a1, a2, . . . , aq} are q distinct complex

numbers such thatE1aνandE2aνare identical forν1,2, . . . , qandqis an integer greater than or equal to 42/αf1. We write the following notations:

Nνr N

r, 1 f1z−aν

N

r, 1 f2z−aν

ν1,2, . . . , q. 4.1

Now using a reformulation of Nevanlinna’s inequality for functions in classF1, we have the following for allr <1:

q−2Tr, f1

≤ q

ν1

Nνr log 1 1−r O

log log 1 1−r

asr−→1, 4.2

q−2Tr, f2

≤ q

ν1

Nνr log 1 1−r O

log log 1 1−r

asr−→1. 4.3

Now assume that αf1 ≥ αf2 > 0. Then from the definition of index we have that for 0< < αf1, there exists a sequence{rm} → 1 such that

Trm, f1

>αf1

−log 1

1−rm ∀m−→ ∞,

or log 1 1−rm

< 1 αf1

−T

rm, f1

.

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So, combining4.2and4.3and using4.4, we get

q−2Trm, f1

Trm, f2

≤2

q

ν1

Nνrm 2o1log 1 1−rm

asm−→ ∞

<2

q

ν1

Nνrm 2o1 αf1

−T

rm, f1

asm−→ ∞,

4.5

which leads to

q−2− 2o1

αf1

−

Trm, f1

Trm, f2

≤2

q

ν1

Nνrm. 4.6

Sincef1andf2are not identical, we have

T

rm, 1

f1−f2

Trm, f1−f2

O1 asm−→ ∞

≤Trm, f1

Trm, f2

O1 asm−→ ∞

≤ 2

q−2−2o1/αf1

−

q

ν1

Nνrm O1 asm−→ ∞.

4.7

On the other hand, every common root of the equationsfνz ais a pole of 1/f1−f2, and

so we have

q

ν1

Nνrm≤N

rm,

1

f1−f2

O1

≤ 2

q−2−2o1/αf1

−

q

ν1

Nνrm O1 asr−→1,

4.8

which gives a contradiction sinceq >42/αf1implies 2/q−2−2o1/αf1−<1 asr → 1, unlessq_ν1Nνrm O1. This, however, cannot occur sinceTr, f1andTr, f2

are unbounded. Therefore, the result follows.

Remark 4.3. SinceP ⊂ F, Theorem 4.2 gives conditions for when two functions in P are

identical.

5. Values of Differential Polynomials

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Theorem 5.1see12, Theorem 4. Letfbe a meromorphic function inDwhich is in classFand for which

N

r,1 f

Nr, foTr, f, asr −→1. 5.1

Letnbe a positive integer, and fork0,1,2, . . . , nletakbe a meromorphic function inDfor which

Tr, ak oTr, f, asr −→1. 5.2

Ifψis defined inDby

ψ

n

k0

akfk 5.3

and ψ is nonconstant, then ψ assumes every complex number except possibly zero infinitely often

provided the index offisα >1nn1/2.

Theorem 5.2. Letfbe a meromorphic function inDwhich is in classPand for which

N

r,1 f

Nr, foTr, f, asr −→1. 5.4

Letnbe a positive integer, and fork0,1,2, . . . , n, letakbe a meromorphic function inDfor which

Tr, ak oTr, f, asr −→1. 5.5

Also, defineEto be the set{k:mr, fk/f oTr, fasr → 1}. Ifψis defined inDby

ψ

n

k0

akfk 5.6

and ψ is nonconstant, thenψ assumes every complex number except possibly zero infinitely often,

provided the index offisα >1nn1/2−E, whereEis the sum of the values ofE.

Proof. Since class Fis closed under diﬀerentiation, addition, and multiplication, we know

thatψ is in classF. Therefore, we can apply the reformulation of the Second Fundamental theorem for classF1toψ. Thus, using 0,∞, andc, a nonzero complex number, we get

Tr, ψ≤N

r, 1 ψ

N

r, 1 ψ−c

Nr, ψlog

1 1−r

O

log log

1 1−r

, asr −→1.

(21)

Since poles ofψcome from poles ofakorf, we have an upper bound forNr, ψ:

Nr, ψ≤Nr, f

n

k0

Nr, ak. 5.8

From the hypothesis, we then have

Nr, ψ≤Nr, foTr, f, asr−→1. 5.9

Therefore, using5.9and the First Fundamental theorem, we get

Tr, ψm

r, 1 ψ N r, 1 ψ

O1, r −→1

≤N r, 1 ψ N r, 1 ψ−c

Nr, ψlog

1 1−r

O log log 1 1−r

, asr −→1,

≤N r, 1 ψ N r, 1 ψ−c

Nr, flog

1 1−r

O log log 1 1−r

, asr −→1.

5.10

Now, solving formr,1/ψin the above calculation, we have the following inequality:

m r, 1 ψ ≤N r, 1 ψ −N r, 1 ψ N r, 1 ψ−c

Nr, f

log

1 1−r

oTr, fO

log log

1 1−r

asr−→1.

5.11

SinceNr,1/ψ≤Nr,1/ψ, theNr,1/ψterms cancel, and so we can say that

m r, 1 ψ ≤N r, 1 ψ−c

Nr, flog

1 1−r

oTr, f

O

loglog

1 1−r

asr−→1.

(22)

So, using the First Fundamental theorem, properties of the proximity function and5.12give us the following:

Tr, fm

r, 1 f N r,1 f

O1

≤m

1,1 ψ ψ f N r,1 f

O1

≤m r, 1 ψ m r,ψ f N r,1 f

O1

≤N

r, 1 ψ−c

Nr, flog

1 1−r

m r,ψ f N r,1 f

oTr, fO

log log

1 1−r

asr−→1.

5.13

Noticing the fact thatNr, f≤Nr, fand using the hypothesis that

Nr, fN

r, 1 f

oTr, f asr−→1, 5.14

we can say that

Tr, f≤N

r, 1 ψ−c

m r,ψ f log 1 1−r

oTr, f

O

log log

1 1−r

asr −→1.

5.15

We now estimatemr, ψ/f. By using properties of the proximity function, we get

m r,ψ f m r, _n

k0akfk

f m

r, n k0 ak fk f

≤n

k0

m

r, ak

fk

f logn1

≤n

k0

mr, ak

n k1 m r,f k

f logn1.

5.16

Recall the setE{k :mr, fk/f oTr, fasr → 1}. Notice that sincef ∈ P,Eis not empty. The setEalso allows us to split the following sum into two pieces. Indeed,

n k1 m r,f k f k∈E m r,f k f k/∈E m r,f k

(23)

But now we can say that

k∈E

m

r,f

k

f o

Tr, f asr−→1, 5.18

since this is true for eachk∈E. Therefore, using5.18and the hypothesis

Tr, ak oTr, f, asr−→1, 5.19

we can update5.16to say that

m

r,ψ f

≤

k/∈E

m

r,f

k

f o

Tr, f asr−→1. 5.20

We use3.26to say that

m

r,f

k

f ≤klog

1 1−r o

Tr, f asr −→1, 5.21

and, thus,5.20becomes

m

r,ψ f

≤

k/∈E

klog 1 1−r o

Tr, f asr−→1. 5.22

We can calculate_k/_∈Eklog1/1−rby noting that

k/∈E

klog 1

1−r log

1 1−r

k/∈E

k

nn1 2 −

E

log 1

1−r, 5.23

whereEis the sum of the elements inE. Note thatnn1/2−Eβ≥0. Thus,5.20 becomes

m

r,ψ f

≤

nn1 2 −

E

log 1 1−r o

Tr, f asr−→1. 5.24

Therefore, we can now update5.15to say

Tr, f ≤N

r, 1 ψ−c

nn1 2 −

E1

log 1 1−r

oTr, fO

log log

1 1−r

asr−→1.

(24)

Since the index offis equal toα > nn1/2−E1, we have that

γ Tr, f≤N

r, 1 ψ−c

oTr, f asr −→1, 5.26

whereγ > 0. SinceTr, fis unbounded, we have proved the claim thatψ assumes every complex number except possibly zero infinitely often.

References

1 D. F. Shea and L. R. Sons, “Value distribution theory for meromorphic functions of slow growth in the disk,” Houston Journal of Mathematics, vol. 12, no. 2, pp. 249–266, 1986.

2 P. Gunsul, A Class of Small Functions in the Unit Disk, ProQuest LLC, Ann Arbor, Miss, USA, 2009. 3 D. Benbourenane, Value Distribution for Solutions of Complex Diﬀerential Equations in the Unit Disk,

ProQuest LLC, Ann Arbor, Miss, USA, 2001.

4 M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing, New York, NY, USA, 1975. 5 L. R. Sons, “Unbounded functions in the unit disc,” International Journal of Mathematics and

Mathematical Sciences, vol. 6, no. 2, pp. 201–242, 1983.

6 J. Heittokangas, “Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class,” Kodai Mathematical Journal, vol. 30, no. 2, pp. 263–279, 2007.

7 D. F. Shea, “Functions analytic in a finite disk and having asymptotically prescribed characteristic,”

Pacific Journal of Mathematics, vol. 17, pp. 549–560, 1966.

8 J. Heittokangas, R. Korhonen, and J. Rättyä, “Linear differential equations with coefficients in weighted Bergman and Hardy spaces,” Transactions of the American Mathematical Society, vol. 360, no. 2, pp. 1035–1055, 2008.

9 R. Korhonen and J. Rättyä, “Finite order solutions of linear differential equations in the unit disc,”

Journal of Mathematical Analysis and Applications, vol. 349, no. 1, pp. 43–54, 2009.

10 I. Laine, Handbook of Diﬀerential Equations: Ordinary Diﬀerential Equations, vol. 4, chapter 3, Elsevier,

New York, NY, USA, 2008.

11 W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.

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