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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2011, Article IDΒ 537478, 24 pages
http://dx.doi.org/10.1155/2011/537478
Research Article

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

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

Received 20 October 2010; Accepted 21 January 2011

Academic Editor: BrigitteΒ Forster-Heinlein

Copyright Β© 2011 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.

Abstract

If is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function could be used to categorize according to its rate of growth as . Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer , as , possibly outside a set of finite measure where denotes the proximity function of Nevanlinna theory. If is a meromorphic function in the unit disk , analogous results to the previous equation exist when . In this paper, we consider the class of meromorphic functions in for which , , and as . We explore characteristics of the class and some places where functions in the class behave in a significantly different manner than those for which holds. We also explore connections between the class and 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 by the number of poles of in , where each pole is counted according to its multiplicity. Also, counts the number of distinct poles of in disregarding multiplicity. If , then . We define the proximity function , the counting function , and the Nevanlinna characteristic function as follows: Also, we have that

A meromorphic function in the unit disk can be categorized according to the rate of growth of its Nevanlinna characteristic as approaches one. If 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 shownβ€”namely, for each positive integer , as approaches one, possibly outside a set of finite measure where denotes the proximity function of Nevanlinna theory. For meromorphic functions of lesser growth than (1.3), analogous theorems need not hold. If consists of those meromorphic functions in for which Shea and Sons [1] showed that is in for each in , but also that there exist functions in with unbounded characteristic for which on a sequence of approaching one. Thus all functions in with unbounded characteristic need not satisfy (1.4) for .

In this paper, we denote by those functions in for which is unbounded as approaches one and for which (1.4) does hold for . We derive striking properties of class and make some connections between functions in class and solutions of linear differential equations defined in . For functions in with , 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 where is in , the are meromorphic functions in , and , as .

Our paper proceeds as follows. In Section 2, we note examples of functions in and properties of the class. In Section 3, we prove a uniqueness theorem for functions in class (and hence in class ). In Section 4, we look at differential equations for which functions in are either coefficients or solutions, and in Section 5 we consider differential polynomials.

Much of the research reported here was part of the author's Ph.D. dissertation written at Northern Illinois University [2].

2. Properties and Examples of Functions in Class

First we note that is not empty. For , the function defined in by is in class , since by a calculation in Benbourenane [3] and by properties of and a lemma of Tsuji ([4, page 226]), Clearly as defined in (1.5) is for this function.

The following proposition gives some simple closure properties of .

Proposition 2.1. If and are in and is a nonzero complex number, we have (i) is in ;(ii) is in ;(iii) is in for each positive integer ;(iv) may not be in ;(v) may not be in .

The proof of (i), (ii), and (iii) in Proposition 2.1 follows by easy calculation. To see (iv), let , and to see (v), let .

The complicated nature of class is demonstrated by the following theorem whereby some sums and products are in .

Theorem 2.2. Let be a meromorphic function in class .(i)If is a nonzero complex number for which then is in .(ii)If is a meromorphic function in which is not identically zero and such that , , and , , then is in .(iii)There exists a Blaschke product such that is not in .(iv)There exists a Blaschke product such that is in .

Remark 2.3. In Nevanlinna theory, the Valiron deficiency of a complex value for a meromorphic function in is defined by It is known (cf. Theorem 2.20 on page 210 in [5]) that if then except for at most a set of -values of vanishing inner capacity. This fact enables us to show that the function in defined by has in for all complex numbers , because for all .

Remark 2.4. The following example illustrates part (ii) of Theorem 2.2.

Example 2.5. Let and . Then is not identically zero, and it is well known that . Therefore, as . Also we have that since is the quotient of two bounded, analytic functions in the unit disk. And so we have that .

We turn to the proof of Theorem 2.2.

Proof of Part (i). Let . Then . We will show that .
First, by calculation and properties of the Nevanlinna characteristic, note that
Also, by calculation and properties of the proximity function, we get Now, since , Also, since , and so
Therefore, since as , and is unbounded as .

Proof of Part (ii). First, is unbounded, since it can be shown that since is unbounded.
Now note that . Therefore, Thus .

Proof of Part (iii). Let be the Blaschke product defined in [6, Proposition 6.1, page 273]. This Blaschke product has the feature that for any there exists an exceptional set satisfying such that and there exists a set , satisfying and a constant , such that
Let . Then . Now define . We will now show that .
First, it is easily shown that as .
Note that since , we have . Therefore, since , And so Therefore, Using (2.21), we have on a small exceptional set with for , Taking the of both sides, we get So, calculating the following ratio yields
Therefore, .

Proof of Part (iv). Let be a Blaschke product with zeros such that for all integers . Theorem B in Heittokangas [6] shows that is in for in , so is of bounded characteristic. Hence is in for in .

Remark 2.6. Since a Blaschke product is a bounded, analytic function, we see from parts (iii) and (iv) above that multiplication by such functions may or may not yield a function in .

Further study of examples in class shows that the function defined by has (1.4) holding for all . However, there are also in for which (1.4) does not hold for . We have the following theorem.

Theorem 2.7. There exists an analytic function in such that , as .

Proof. First, we begin by constructing a function which has unbounded characteristic as , but its derivative is of bounded characteristic. This construction is from [7, page 557].
Let be an integer greater than or equal to 2. Define for , where is a constant such that . Now define where Shea showed in [7] that is analytic in the unit disk and satisfies where and are constants such that and is the maximum modulus function for . Choose and such that , so we can take . This implies that is bounded in the unit disk by the following argument: for , By (2.31), we have that Therefore, since , we have by a simple integration that
Let . Define where . Recall that . We now show that . First, we see that as , by the following: and since is bounded, there exists a constant such that Thus, , as . On the other hand, Therefore, as .
Now, we bound from above by using properties of the Nevanlinna characteristic: And so we have as and thus .
Now we show that as . By a quick calculation, we have Also we have from the above construction so there exists an such that . And so, Combining (2.40) and (2.41), we have

Remark 2.8. If we define to be the set of functions in such that (1.4) holds for all positive integers , Theorem 2.7 shows is properly contained in . Further, we note that in the proof of Theorem 2.7 above can be replaced with where . Also the idea of the proof of Theorem 2.7 can be used to show that for there exist functions in for which for all integers , but
The function in the proof of Theorem 2.7 provides us with further information about .

Theorem 2.9. There exists a function in such that is not in .

Proof. The function of Theorem 2.7 is in . Using the Nevanlinna calculus and properties of , one can show as . We omit the details here (cf. [2]).

For functions in class , we may call defined in (1.5) the index of . In [1], Shea and Sons showed for in that where and this inequality is best possible. For analytic functions in class , we get

Theorem 2.10. If is an analytic function in class , then (i) as ;(ii);(iii)if as , then .

Proof. For part (i) since is analytic in , we have from which the result follows. Using the definition of the index of and of , part (ii) comes from (i).
To see (iii), we observe Thus, Dividing both sides of (2.49) by and taking the limit superior as approaches one, we get .

3. Connections between Class 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 class . We consider the complex linear differential equation 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 of (3.1) and the growth of the analytic coefficients in the unit disk. Some recent papers include [8–10]. We now quote some of the important results that have a connection with class and, therefore, class . 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 function in the unit disk is in the weighted Hardy space for and if We say that is in if

Definition 3.2. We say that an analytic function in the unit disk, , is in the weighted Bergman space if the area integral over satisfies for and .

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

Theorem 3.3 (see [10, page 320]). Let be a nontrivial solution of (3.1) with analytic coefficients , , in the unit disk. Then we have that(i)if and for all , then ;(ii)if for all , or for all , then ;(iii)if for all , then .

Theorem 3.4 (see [10, page 320]). We have that(i)if all nontrivial solutions , then the coefficients for all ;(ii)if all nontrivial solutions , then the coefficients for all .

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

Theorem 3.5 (see [9, page 44]). All solutions of (3.1), where is analytic in for all , satisfy if and only if for all .

When in (3.1), we observe using Theorem 3.3, if with , then and, therefore, . We can also conclude that if , then and so . Also, if , then , which means may be in . On the other hand, using Theorem 3.4, we have if , then .

If , then is a solution of the differential equation. However, if for an integer , then is of bounded characteristic, but the solution has order and, therefore, . 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.

When in (3.1), Theorems 3.3 and 3.4 have the following corollary.

Corollary 3.6. Let be a non-trivial solution of (3.1) with analytic coefficients and in the unit disk. Then(i)if and for , then and ;(ii)if and or and , then and ;(iii)if and , then and, therefore, could be in ;(iv)if all non-trivial solutions , then and .

The function is a solution to the equation

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

Computation shows

We know that , but what can be said about ? The above form of makes it difficult to calculate the growth, but we do know that, by (iii) in Corollary 3.6, if and , then .

Example 3.7. We show for and , and . To see , we first note We integrate and apply a lemma from Tsuji [4, page 226] which states that, in particular, and get that there exists a constant such that And so which goes to zero as . Therefore, .
A similar calculation shows that .

The question as to whether is not a trivial question as there exist examples, such as Example 3.9 below, where at least one solution is in class and at least one solution is not in class .

Example 3.8. The function in Theorem 2.7 is a solution of (3.1) with when and . Then since we have that
Now, recall from the proof of Theorem 2.7 that and since , by (3.15), we conclude that at least one of or has index greater than or equal to . Therefore, as a consequence of Theorem 2.7, we have a growth estimate for the coefficients of this differential equation.
It can also be shown that . However, is not in , and thus the converse of Corollary 3.6(iii) is not true.

For differential equations of the form (3.1) where , we first quote two examples.

The first example has some solutions of (3.1) in and some not.

Example 3.9 (see [9, Example 10, page 52]). The functions are linearly independent solutions of where It can be shown that . However, is of bounded characteristic and, therefore, . (It is known that has order zero but unknown if .) Also, according to [9], we have that , , and .

This next example is also from [9].

Example 3.10 (see [9, Example 11, page 53]). The functions are linearly independent solutions of where It is known that , , and are all in , and it is known that has order zero. Also, it can be shown that , , , and .

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

Theorem 3.11. If a function in satisfies a differential equation of the form (3.1) such that for all integers such that , then at least one of the analytic coefficients has index greater than or equal to as for some .

For nonhomogeneous differential equations of the form where is analytic in the disk for all , we state a result from [9] that applies to class .

Theorem 3.12 (see [9, Theorem 7, page 46]). All solutions of (3.24) satisfy if and only if and for all . Therefore, if all solutions of (3.24) are in , then and for all .

Theorems 3.3 and 3.4 do not tell the whole story regarding class . Instead of the coefficients being in a certain function class, what can we say about the solutions of (3.1) if we know the coefficients have a certain index in class ? We show the following proposition.

Proposition 3.13. Let be a positive integer. If is an analytic function in for which the index of is , then for when .

Proof. Note that since , we have , and we want to show that Since , there exists a real number such that on some sequence of 's as . Also, since , we have . Now, since , and so for . By Shea and Sons [1], for as . Now, we have the following: So, since is analytic, we have Therefore, as .

With a similar argument as above, Proposition 3.13 is also true if is meromorphic and as .

4. The Identical Function Theorem

For functions in class , we have an analogue to the Nevanlinna five-value theorem which we quote as stated in [11].

Theorem 4.1 (see [11, page 48]). Suppose that and are meromorphic in the plane and let be the set of points such that ). Then if for five distinct values of , or and are both constant.

Our analogue and proof follow. The proof has a subtle difference from the direct analogue of the proof of Theorem 4.1 in [11].

Theorem 4.2. Let and be meromorphic functions in class such that and let be the set of points such that for . Then if for distinct values of such that is an integer and , then .

Proof. Suppose and are not identical and that are distinct complex numbers such that and are identical for and is an integer greater than or equal to . We write the following notations: Now using a reformulation of Nevanlinna's inequality for functions in class [1], we have the following for all : Now assume that . Then from the definition of index we have that for , there exists a sequence such that So, combining (4.2) and (4.3) and using (4.4), we get which leads to
Since and are not identical, we have On the other hand, every common root of the equations is a pole of , and so we have which gives a contradiction since implies as , unless . This, however, cannot occur since and are unbounded. Therefore, the result follows.

Remark 4.3. Since , Theorem 4.2 gives conditions for when two functions in 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 class . In a preliminary report by Sons [12], the author explores various results for functions satisfying (1.3) in the disk and their analogues for functions in class . Some of these results for class can be refined further if we restrict the functions to class . We state a theorem from Sons (without proof) and follow it with a refinement for class .

Theorem 5.1 (see [12, Theorem 4]). Let be a meromorphic function in which is in class and for which Let be a positive integer, and for let be a meromorphic function in for which If is defined in by and is nonconstant, then assumes every complex number except possibly zero infinitely often provided the index of is .

Theorem 5.2. Let be a meromorphic function in which is in class and for which Let be a positive integer, and for , let be a meromorphic function in for which Also, define to be the set . If is defined in by and is nonconstant, then assumes every complex number except possibly zero infinitely often, provided the index of is , where is the sum of the values of .

Proof. Since class is closed under differentiation, addition, and multiplication, we know that is in class . Therefore, we can apply the reformulation of the Second Fundamental theorem for class [1] to . Thus, using 0, , and , a nonzero complex number, we get Since poles of come from poles of or , we have an upper bound for : From the hypothesis, we then have Therefore, using (5.9) and the First Fundamental theorem, we get Now, solving for in the above calculation, we have the following inequality: Since , the terms cancel, and so we can say that So, using the First Fundamental theorem, properties of the proximity function and (5.12) give us the following: