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

(1)

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,paul.gunsul@gmail.com 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, f^k/f oTr, fasr → ∞, 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⁻Tr, 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:

m r, f

1 2π

_2π

0

logf

re^iθdθ,

(2)

N r, f

_r

0

n t, f

−n 0, f

t dtn

0, f logr, T

r, f m

r, f N

r, f .

1.1

Also, we have that N

r, f

_r

0

n t, f

−n 0, f

t dtn

0, f

logr. 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

T r, 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

T r, 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

T r, 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 Pand make some connections between functions in classPand solutions of linear diﬀerential

(3)

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

akf^k, 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

T r, 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;

iiifⁿis inPfor each positive integern;

ivfgmay not be inP;

vfgmay not be inP.

(4)

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 T

r, 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

Δ c, f

lim inf

r→1

Nr, c T

r, f. 2.5

It is knowncf. Theorem 2.20 on page 210 in5that if

rlim→1T r, f

∞, 2.6

then

rlim→1

Nr, a T

r, 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 for allc /0.

(5)

Remark 2.4. The following example illustrates partiiofTheorem 2.2.

Example 2.5. Letfz e^i/1−zandgz e^1z/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 e^i1z/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 T

r, g T

r, f

O1 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

o

T r, f

asr −→1, o

T r, g

asr−→1.

2.13

Also, sinceΔ−c, f 0,

lim sup

r→1

m r,1/

fc T

r, f 0, 2.14

(6)

and so

m

r, 1 fc

o

T r, f

asr−→1, o

T r, 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 T

r, fg T

r, f

O1 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

, o

T r, f

asr−→1, o

T r, 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

Bz Bz

O 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 Bx Bx

≥ C

1−x^1.5 log 1

1−x, x∈F1\E1. 2.21

(7)

Letgz i/1−z. Thene^gz ∈ P. Now defineqz Bze^gz. We will now show thatqz∈ P./

First, it is easily shown thatTr, e^gz Tr, q O1asr → 1.

Note that sincee^gzge^gz, we haveg e^gz/e^gz. Therefore, sincee^gz∈ P, m

r, g o

T

r, e^gz

asr −→1. 2.22

And so

m

r,B B

m

r,q

q −g

≤m

r,q q

m

r,−g

log 2. 2.23

Therefore,

m

r,q q

≥m

r,B B

−m r,−g

−log 2. 2.24

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

B

≥ C

1−r^1.5log 1

1−r. 2.25

Taking the logof both sides, we get

log B

B

≥log C

1−r^1.5log 1 1−r

≥log C 1−r^1.5

≥logC1.5 log 1 1−r.

2.26

So, calculating the following ratio yields m

r, q/q T

r, q ≥ mr, B/B T

r, e^gz − m r, g T

r, e^gz − log 2 T

r, e^gz −→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/n⁵for all integersn≥2. Theorem B in Heittokangas6shows thatBis inH^pforpin0,3/4, soB/B is of bounded characteristic. HenceBfis inPforfinP.

Remark 2.6. Since a Blaschke product is a bounded, analytic function, we see from partsiii andivabove that multiplication by such functions may or may not yield a function inP.

(8)

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, Q_m^m

k1

a^k 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 t_m

_a^m

. 2.30

Shea showed in7thatfis analytic in the unit disk and satisfies αlog

1 1−r

< T r, f

≤logM r, 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: forzre^iθ,

|Fz|

_z

0

fwdw

≤ _r

0

f

ρe^iθdρ≤ _r

0

M ρ, f

dρ. 2.33

By2.31, we have that

M ρ, f

< 1

1−ρ_β. 2.34

(9)

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

|Fz|<

_r

0

1

1−ρ_βdρO1 asr−→1. 2.35

LetKz _z

0Fwdw. Define hz Kze^gz wheregz i/1−z. Recall that e^gz ∈ P. We now show thathz ∈ P. First, we see thatTr, K O1asr → 1, by the following:

Tr, K mr, K

≤ 1 2π

_2π

0

log _r

0

F

ρe^iθdρ 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, e^gz≤Tr, e^gz O1, asr → 1. On the other hand,

T r, e^gz

T

r, h K

≤Tr, h O1 asr −→1.

2.38

Therefore,Tr, h∼Tr, e^gzasr → 1.

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

m

r,h h

m

r,Kge^gKe^g Ke^g

≤m

r,K K

m

r, g log 2

≤m r, g

O1 asr−→1 o

T

r, e^gz

asr −→1 oTr, h asr−→1.

2.39

And so we havemr, h/h oTr, hasr → 1 and thush∈ P.

(10)

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

r,K

K

≤m

r,h

h −2gK

K −g− g₂

≤m

r,h h

mr,2 m r, g

m

r,K K

m

r, g 2m

r, g log 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

≥m r, 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 2.7abovehKe^gcan be replaced withhKpwherep∈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,h^k1

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 Ke^g 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 cf.2.

(11)

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

α f

≤α f

1k f

1, 2.45

where

k f

lim sup

r→1

N r, f T

r, f

1, 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

T r, f

m r, f

≤m

r,f f

m

r, f m

r,f

f

T

r, f 2.47

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

To seeiii, we observe

T r, f

≤T r, f

T

r, f f

T

r, f T

r,f

f

O1 asr −→1. 2.48

Thus,

T r, f

≤T r, f

m

r,f f

N

r, f N

r,1

f

O1 asr −→1. 2.49

Dividing both sides of2.49by log 1/1−rand taking the limit superior asr approaches one, we getαf≤αf.

(12)

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

fⁿa_n−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 spaceH_q^pfor 0< p <∞and 0≤q <∞if

sup

0≤r<1

1−r²_q 1 2π

_2π

0

f

re^iθ^pdθ

1/p

<∞. 3.2

We say thatfis inH_q^∞if

sup

z∈D

1− |z|²_q fz<∞. 3.3

Definition 3.2. We say that an analytic function f in the unit disk, D, is in the weighted Bergman spaceA^p_qif the area integral overDsatisfies

D

fz^p

1− |z|²_q dσ

_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 anda_j∈H_a1n−j^1/n−j for allj 0, . . . , n−1, thenf ∈N;

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

iiiifaj∈H_n−j^1/n−jfor allj0, . . . , n−1, thenf∈ F.

(13)

Theorem 3.4see10, page 320. We have that

iif all nontrivial solutions f ∈ N, then the coeﬃcients aj ∈

0<p<1/n−jA^p for allj 0, . . . , n−1;

iiif all nontrivial solutionsf ∈ F, then the coeﬃcientsa_j ∈

0<p<1/n−jH_1/p^p for allj 0, . . . , n−1.

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

ρ f

lim sup

r→1⁻

logT r, f

−log1−r. 3.5

Theorem 3.5see9, page 44. All solutionsf of 3.1, where a_j is analytic inD for all j 0, . . . , k−1, satisfyρf 0 if and only ifaj∈

0<p<1/n−jA^pfor 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_1/p^p . Ifa0z −βi/1−z², thenfz e^βi/1−z is a solution of the diﬀerential equation.

However, ifa₀βi/1−z^kfor an integerk≥3, thena₀is of bounded characteristic, but the solution

fCe^βi/1−z^k−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_2α1^1/2 for−1< α <0, thenf∈Nandf /∈ P;

iiifa₁∈A¹anda₀∈A^1/2ora₀anda₁∈A¹₁, thenf∈Nandf /∈ P;

iiiifa1∈H₁¹anda0∈H₂^1/2, thenf∈ Fand, therefore, could be inP;

ivif all non-trivial solutionsf∈ F, thena₁∈

0<p<1H_1/p^p anda₀∈

0<p<1/2H_1/p^p . The functionwe^βi/1−zis a solution to the equation

w− βi

1−z²w− 2βi

1−z³w0. 3.7

(14)

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 e^βi/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 ∈ H₂^1/2 anda1∈H₁¹, thenf2∈ F.

Example 3.7. We show fora0 −2βi/1−z³anda1 −βi/1−z²,a0 ∈H₂^1/2anda1 ∈H₁¹. To seea0 ∈H₂^1/2, 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

1−dθre^iθ^3/2 O 1

1−r^1/2 , 3.11

and get that there exists a constantMsuch that 1

2π _2π

0

2β

|1−z|^3/2dθ 2β

2π _2π

0

1

|1−z|^3/2dθ≤ 2β

2π M

1−r^1/2. 3.12 And so

1−r²₂ 1 2π

_2π

0

|a0|^1/2dθ

2

≤1r²1−r² 2βM 2π1−r^1/2

2

1r²1−rC, 3.13 which goes to zero asr → 1. Therefore,a0∈H₂^1/2.

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 in classP.

(15)

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 ∈H₂^1/2. 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 e^i1z/1−z−e^−i1z/1−z, f₂z e^i1z/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^p₀, anda2 ∈

0<p<1A^p₀.

(16)

This next example is also from9.

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

f1z e^i1z/1−z−e^−i1z/1−z, f2z e^i1z/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 −39z9z²3z³ 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^p₀,a₁∈

0<p<1/3A^p_−1/3,a₂ ∈

0<p<1/2A^p₀, 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,fⁿ f /o

T r, f

asr −→1,

m

r,f^j

f o

T r, 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

fⁿa_n−1zfⁿ⁻¹· · ·a1zfa0zfanz, 3.24

(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−jA^pfor allj0, . . . , n−1. Therefore, if all solutionsfof3.24are inP, thenρan 0 andaj ∈

0<p<1/k−jA^pfor 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∈ Fwhenf^k−a0f0.

Proof. Note that sincef^k−a0f 0, we havea0f^k/f, and we want to show that

lim sup

r→1⁻

m

r, f/f T

r, 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,f^k ∈ 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

f^k−1 f^k−1 f^k−2 · · ·f

f

≤m

r, f^k

f^k−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

f^k−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 Therefore,mr, f/f/Tr, f≥s−k/t >0 asr → 1.

(18)

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 letE_ja 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 letE_jabe the set of pointszsuch thatf_jz 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, . . . , a_q} 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:

N_νr N

r, 1

f₁z−a_ν

N

r, 1

f₂z−a_ν 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−2 T

r, f1

≤ q ν1

Nνr log 1 1−r O

log log 1 1−r

asr−→1, 4.2 q−2

T r, f₂

≤ q ν1

N_νr 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

T rm, f1

>

α f1

−

log 1

1−rm ∀m−→ ∞, or log 1

1−rm

< 1 α

f1

−T r_m, f₁

.

4.4

(19)

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

T r_m, f₁

T

r_m, f₂

≤2 q ν1

N_νrm 2o1log 1 1−rm

asm−→ ∞

<2 q ν1

Nνrm 2o1 α

f₁

−T rm, f1

asm−→ ∞,

4.5

which leads to

q−2− 2o1 α

f1

− T

r_m, f₁ T

r_m, f₂

≤2 q ν1

N_νrm. 4.6

Sincef1andf2are not identical, we have

T

rm, 1

f₁−f₂ T

rm, f1−f2

O1 asm−→ ∞

≤T rm, f1

T rm, f2

O1 asm−→ ∞

≤ 2

q−2−2o1/

α f1

− q ν1

N_νrm 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/

α f1

− q ν1

N_νrm 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

We now turn our focus on determining values for diﬀerential polynomials in the disk as it relates to classP. In a preliminary report by Sons12, the author explores various results for functions satisfying1.3in the disk and their analogues for functions in classF. Some of these results for classFcan be refined further if we restrict the functions to classP. We state a theorem from Sonswithout proofand follow it with a refinement for classP.

(20)

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

N

r,1 f

N

r, f o

T r, f

, asr −→1. 5.1

Letnbe a positive integer, and fork0,1,2, . . . , nleta_kbe a meromorphic function inDfor which Tr, ak o

T r, f

, asr −→1. 5.2

Ifψis defined inDby

ψⁿ

k0

akf^k 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

N

r, f o

T r, f

, asr −→1. 5.4

Letnbe a positive integer, and fork0,1,2, . . . , n, leta_kbe a meromorphic function inDfor which Tr, ak o

T r, f

, asr −→1. 5.5

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

ψⁿ

k0

akf^k 5.6

and ψ is nonconstant, thenψ assumes every complex number except possibly zero infinitely often, provided the index offisα >1nn1/2−

E, where

Eis 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

T r, ψ

≤N

r, 1 ψ

N

r, 1

ψ−c

N r, ψ

log 1

1−r

O

log log 1

1−r

, asr −→1.

5.7

(21)

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

N r, ψ

≤N r, f

ⁿ

k0

Nr, ak. 5.8

From the hypothesis, we then have

N r, ψ

≤N r, f

o T

r, f

, asr−→1. 5.9

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

T r, ψ

m

r, 1 ψ

N

r, 1

ψ

O1, r −→1

≤N

r, 1 ψ

N

r, 1

ψ−c

N r, ψ

log 1

1−r

O

log log 1

1−r

, asr −→1,

≤N

r, 1 ψ

N

r, 1

ψ−c

N r, f

log 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

N r, f log

1 1−r

o

T r, f

O

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

N

r, f log

1 1−r

o

T r, f O

loglog

1 1−r

asr−→1.

5.12

(22)

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

T r, f

m

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

N

r, f log

1 1−r

m

r,ψ

f

N

r,1 f

o

T r, f

O

log log 1

1−r

asr−→1.

5.13

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

r, f N

r, 1

f

o T

r, f

asr−→1, 5.14

we can say that T

r, f

≤N

r, 1 ψ−c

m

r,ψ

f

log 1

1−r

o T

r, 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

k0akf^k

f m

r,

n k0

ak

f^k f

≤ⁿ

k0

m

r, a_kf^k

f logn1

≤ⁿ

k0

mr, ak

n k1

m

r,f^k

f logn1.

5.16

Recall the setE{k :mr, f^k/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

f . 5.17

(23)

But now we can say that

k∈E

m

r,f^k

f o

T r, f

asr−→1, 5.18

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

T r, f

, asr−→1, 5.19

we can update5.16to say that

m

r,ψ f

≤

k/∈E

m

r,f^k

f o

T r, f

asr−→1. 5.20

We use3.26to say that

m

r,f^k

f ≤klog 1 1−r o

T r, f

asr −→1, 5.21

and, thus,5.20becomes

m

r,ψ f

≤

k/∈E

klog 1 1−r o

T r, 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

where

Eis 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

T r, f

asr−→1. 5.24

Therefore, we can now update5.15to say

T r, f

≤N

r, 1 ψ−c

nn1

2 −

E1

log 1 1−r o

T r, f

O

log log 1

1−r

asr−→1.

5.25

(24)

Since the index offis equal toα > nn1/2−

E1, we have that γ T

r, f

≤N

r, 1 ψ−c

o

T r, 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.

12 L. R. Sons, “Values for diﬀerential polynomials in the disk,” Abstracts of the AMS, vol. 28, p. 640, 2007.

(25)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Probability and Statistics

Journal of

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Operations Research

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Algebra

Discrete Dynamics in Nature and Society

Decision Sciences

Discrete Mathematics

^{Journal of}

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

Research Article

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

Paul A. Gunsul

1. Introduction

2. Properties and Examples of Functions in Class P

3. Connections between Class P and Differential Equations

4. The Identical Function Theorem

5. Values of Differential Polynomials

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Algebra

Decision Sciences

Discrete Mathematics

Stochastic Analysis