Abstract
We will prove the assertions which give necessary and sufficient conditions for a normal meromorphic function on the open unit disk to have an angular limit. The results obtained show that the conditions from the classical Lindelöf theorem, as well as the theorems of Lehto and Virtanen and Bagemihl and Seidel, concerning angular limit values of meromorphic functions, can be weakened.
1. Introduction and Preliminaries
Let be the open unit disk in the complex plane , with the boundary , and let be the Riemann sphere. For any and a function , let , where () is a Möbius map. Further, the pseudohyperbolic distance on is given by The function defined on as is the hyperbolic metric on . The chordal metric on the Riemann sphere is defined as
All the convergence in this paper will be considered with respect to some of the aforementioned metrics. Since the convergence with respect to the hyperbolic and pseudohyperbolic metrics on the disk is equivalent, in our proofs we will use one of these metrics that “simplifies” the related proof.
For a fixed , the set defined as is called the pseudohyperbolic disk with the pseudohyperbolic center and the pseudohyperbolic radius . Notice that for , , . Similarly, for a fixed , the set defined as is called the hyperbolic disk with the center and the hyperbolic radius .
For simplicity, here as always in the sequel, for an arbitrary nonempty set we write if a sequence of complex functions defined on the disk tends uniformly on to the function with respect to the pseudohyperbolic (or hyperbolic) metric (or ) and the chordal metric of the Riemann sphere .
Given a set so that ( is the closure of ) and the function , denote by the cluster set of the function at the point with respect to the set . Namely, is the set of all points for which there exists a sequence in so that and as . It is known that . If is a Stolz angle of the disk with the vertex at the point , then the cluster set is the limit value of the function along the angle . If for each with we have for some , then is said to be a Fatou point of the function and is its angular limit value.
Here, as always in the sequel, will denote a Jordan arc that ends at a point . If for some then is said to be an asymptotic value of the function at the point along the curve . The classical Lindelöf theorem on boundary values of holomorphic functions asserts that if a bounded analytic function on the disk has an asymptotic value , , at a point , then is its angular limit value and is a Fatou point of the function [1].
Seidel [2] and Seidel and Walsh [3] investigated the boundary properties of bounded analytic functions and analytic functions that omit two complex values via the convergence of their Heine’s sequences. They proved that the results of Lindelöf’s theorem are also valid for univalent analytic functions.
In [4], Lehto and Virtanen extended Lindelöf’s result to the class of normal meromorphic functions in .
A function meromorphic on the disk is said to be normal in if the family is normal in the disk in the sense of Montel, which means that each sequence in contains a subsequence which converges uniformly on each compact subset of . Namely, Lehto and Virtanen proved that if a normal meromorphic function in the disk has an asymptotic value , , at a point , then is its angular limit value and is a Fatou point of the function [4, Theorem 2].
In the same paper, in terms of the estimate of growth of the spherical derivative, Lehto and Virtanen established a necessary and sufficient conditions for a function meromorphic on hyperbolic domains of to be a normal function. For more information on the properties and applications of normal meromorphic functions in the theory of functions see [5].
In [6], Bagemihl and Seidel examined the existence of angular limits of a function meromorphic on at a point , , via the existence of its Heine’s sequences , so that as .
In further considerations, the notion of the range will be significant, where is a function defined in . The range is defined as the set of all points so that there exists a sequence for which and for all .
Bagemihl and Seidel [6, Theorem 1] showed that if for a normal meromorphic function on there exists a sequence such that , and for some with , then is a Fatou point of the function and is its angular limit value. In [6, Examples 1 and 2], the authors gave examples of analytic functions that show that the condition of normality for the function and the condition related to the sequence with cannot be omitted. Example 4 in [6] of a Blaschke product shows the fact that the condition is necessary in that assertion.
Further boundary properties of meromorphic functions via their behaviour along sequences with were investigated by Gavrilov (see, e.g., [5, 7, 8]) and Gauthier ([9, 10]).
In this paper, we prove Theorem 6 of Section 3 and Theorems 7–10 of Section 4, which give necessary and sufficient conditions for a normal meromorphic function in at a point to have an angular limit in terms of a sequence with and under the condition . Here it is not supposed that . The omission of the condition and the retention of the condition related to terms of the sequence require “good” boundary behaviour of the function along sufficiently “thick” sets whose elements lie within a “small hyperbolic distance” from the points , , with and . Namely, in Theorem 7 these sets are sufficiently large disks whose pseudohyperbolic (hyperbolic) centers are the terms of the sequence (see Figure 1); in Theorem 8, these sets are sufficiently large parts of Jordan’s arcs, and in Theorem 10 they are many sequences of points. These theorems give new criteria for the existence of angular limit values of meromorphic functions on at points of the unit circle .
In Section 2, we prove Lemmas 1, 3, and 4. These lemmas present auxiliary results for proofs of other assertions in this paper.
2. Auxiliary Lemmas
In this section, we consider the uniform convergence of function sequences to constant functions. Related classical results, for example, can be found in [11–13], while the uniform convergence of function sequences via properties of boundary sets of functions were investigated in [14, 15]; also see [16].
Lemma 1. Let be an arbitrary function, let be a sequence such that , and let be an arbitrary nonempty subset of with , . Then the following conditions are equivalent:(i) with and , ;(ii).
Proof. From the condition it follows that
(i) (ii). It follows from (i) that for each there exists a positive integer such that for all and . This yields that for all and , and thus,
that is,
Let be a sequence contained in such that . We will show that . For each there is a such that . From it follows that , and so, for there is a such that for every . As , by the previous argument it follows that for all . This together with (11) gives
Hence, for each there exists , such that for each . Therefore, and .
(ii) (i). Suppose that (i) does not hold. Then we will show that (ii) also is not satisfied. The assumption that (i) is not true implies that there exists such that for every there exist and satisfying . Take , . Then for each , and from it follows that , and so, we have . Therefore, we obtain , whence it follows that there exists a sequence contained in such that , but the condition is not satisfied. Hence, the condition is not satisfied, and the proof is completed.
Lemma 2 (Schiff [13, pp. 72–75]). Let be a sequence of meromorphic functions , where is a domain in complex plane . If forms a normal family of functions on and there exists a set with which has at least one accumulation point in , such that a sequence converges to the limit function for each , then on each compact subset of .
Lemma 3. Let be a sequence of meromorphic functions , where is a domain in complex plane . If forms a normal family of functions on and there exists a set with which has at least one accumulation point in , such that , with some , for each , then on each compact subset of and on .
Proof. By Lemma 2, it follows that on each compact subset of , where is a meromorphic function in the domain . From the conditions of Lemma 3, it follows that for each . As the set has an accumulation point in , by the uniqueness theorem for meromorphic functions, it follows that on , and thus, on each compact subset of .
Lemma 4. Let be a normal meromorphic function in the disk , let be a sequence such that , and let be an arbitrary subset of with , , which has at least one accumulation point. Then the following assertions are equivalent:(i) with ;(ii)For each , .
Proof. (i) (ii). By (i) and Lemmas 1 and 3 it follows that on each compact subset of . Taking , with , then applying Lemma 1, we find that . Since , we have for each with , and therefore, , for all with and . Since for each with and , it holds with , we have for each and .
For any given , there exists with for which . Then
whence we see that for all . This shows that
as desired.
(ii) (i). Suppose that and . Then
and hence, for all and . Since a sequence defined as forms a normal family of functions on , it follows by Lemma 3 that , with , for each compact subset of . Therefore, for each compact subset of which has at least one accumulation point, we have . Then by Lemma 1, , as required.
Lemma 5 (Schiff [13, p. 98]). Let be a normal family of meromorphic functions , where is a domain of the complex plane , and suppose that , , for all and . Then for every meromorphic function for which there exists a sequence such that on each compact subset of either there holds for each , or on .
3. The Main Result
Theorem 6. Let be a normal meromorphic function in . Then the following assertions are equivalent.(i)There exists a sequence such that , and let be an arbitrary subset of with , , which has at least one accumulation point for which , ;(ii)A point is a Fatou point of a function , and is its angular limit value at .
Proof. (i) (ii). From (i) and Lemma 4 it follows that for any ,
Suppose that . Then is a simply connected domain. If denotes the arc of geodesic curve in hyperbolic metric that joins points and , then is a Jordan curve that lies in and ends at the point . Therefore, , whence it follows that is an asymptotic value of a function at point . Then by [4, Theorem 2], it follows that is a Fatou point of a function , and is its angular limit.
(ii) (i). Let be the radius of the disk that joins the center 0 of and a point on the unit circle . Then the set , , is a domain bounded by parts of two hypercircles and which join points and and by arc of the circle . Hypercircles and form the angle with the radius at the point . Therefore, a domain lies in a Stolz angle . Then assuming that the terms of a sequence are points of the radius satisfying and (16) and taking the disk with as the set , we have
Since is a Fatou point of a function and is its angular limit, we conclude that , as desired. This completes the proof.
4. Applications
Theorem 7. Let be a normal meromorphic function in the disk , and let be a sequence contained in satisfying and (16). If , , for some , then is a Fatou point of a function , and is its angular limit at .
Proof. The set is presented in Figure 1. As , with , that is, for all , we have and taking in Theorem 6, from this theorem it follows that is a Fatou point of a function , and is its angular limit.
Theorem 8. Let be a normal meromorphic function in the disk , and let be a Jordan arc contained in such that . Further, let be a sequence satisfying and (16), and let with and . If the equality holds for some and for some , then is a Fatou point of the function , and is its angular limit at .
Proof. From the conditions of the theorem it follows that a sequence forms a normal family of functions meromorphic on the disk . This means that contains a subsequence that converges uniformly on compact subsets of to a function meromorphic on . For simplicity, in the sequel we will write instead of , and instead of , . The proof will be deduced using the pseudohyperbolic metric. Then for any fixed with . Set and , . For any fixed and each , let , with . Points may be chosen so that when or . In such a way for any fixed , we find a sequence . It follows that a sequence can be chosen so that because of . We will prove that for each .
Let be an arbitrary positive number. Then
Since is a function meromorphic on , then using a standard “” estimate of each term on the right hand side of (20), we find that for any given , and thus, for each . Because and since a sequence has an accumulation point in , by using the uniqueness theorem for meromorphic functions it follows that on .
From the previous part of the proof we see that every subsequence of a sequence which uniformly converges on compact subsets of the disk converges to a constant . Suppose that there exists a subsequence of the sequence that does not converge uniformly on compact subsets of the disk to a constant . Then there exists such that for each there is a and a point satisfying
As a sequence forms a normal family of meromorphic functions, it follows that a sequence has a subsequence that converges uniformly on compacts of the disk . As was previously proved, it follows that this subsequence converges uniformly on compacts of the disk to a constant . This shows that (21) does not hold for all . This contradiction proves our assertion.
Hence, as proved above, a sequence satisfies . Since
this together with Lemma 1 yields , and so by Theorem 7 it follows that is an angular limit of a function at a point which is its Fatou point.
Remark 9. Theorem 8 shows that the condition on the existence of asymptotic values of normal meromorphic function from Theorem 2 in [4] by Lehto and Virtanen can be replaced by a weaker condition on the existence of its boundary value along arbitrarily “small” arcs of a Jordan curve, where the hyperbolic distance between the consecutive arcs is uniformly bounded from above.
Theorem 10. Let be a normal meromorphic function in the disk , and let be a sequence in satisfying and (16). If for any with there exist countably many sequences , with , , , , for which for each , then is a Fatou point of a function , and is its angular limit value.
Proof. Put , with . Since , we have with and any fixed . Each of sequences , , has a subsequence which tends to a limit value which lies on the circle . This shows that and , with and .
Since a function sequence forms a normal family of meromorphic functions on the disk , it follows that has a subsequence that converges uniformly on compact subsets of the disk to a meromorphic function . The remainder of the proof is the same as that of Theorem 8, where terms of a sequence and a function have the same notations and roles as the corresponding notions related to the proof of Theorem 8.
Using the results obtained, we can prove Theorem 1 in [6, p. 4] by Bagemihl and Seidel.
Theorem 11 (Bagemihl and Seidel [6, Theorem 1]). Let be a normal meromorphic function in the disk , and let be a sequence in satisfying , (16) and with . Then is a Fatou point of a function , and is its angular limit value.
Proof. We use the same notations as those in the proof of Theorem 8. By the condition it follows that on for some with (see Figure 1). This shows that, for each term of a subsequence with of the sequence such that which converges uniformly on compacts of the disk to a meromorphic function , there holds for all . Since , it follows from Lemma 5 that on each compact subset of . Further, it follows that on each compact subset of (see proof of Theorem 8). This together with Lemma 1 yields , and so by Theorem 6, we conclude that is an angular limit value of a function at a point , and is its Fatou point. This completes the proof.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.