New Applications of Nunokawa’s Lemmas

Abstract

We derive several univalent and -valent conditions on certain meromorphic functions by applying well-known Nunokawa’s lemmas.

1. A Sufficient Condition on -Valent Analytic Functions

Let a function be univalent in the unit disk normalized by the conditions and . Then, maps onto a starlike domain with respect to the origin if and only ifIf a function is -valent in and satisfies (1), then is called -valent starlike in .

In [1], Ozaki proved that if is analytic in and for some real , then is at most -valent in . Ozaki’s -valent condition is a generalization of the well-known Noshiro-Warschawski univalence condition. Very recently, Nunokawa, Cho, Kwon, and Sokól (an extension of Ozaki’s -valent conditions, preprint) considered similar problems in the unit disc . In this section, we will extend these -valent conditions.

To prove our results, we need the following well-known Nunokawa lemmas.

Lemma 1 (see [2, 3]). Let be analytic function in of the form with in . If there exists a point , , such that for and for some , then where

Lemma 2 (see [4]). Let and . Also let be analytic in . If there exists -valent starlike function that satisfiesthen is -valent in .

Theorem 3. Let and also let be analytic in . Suppose that, for any positive integer with ,where Then, is -valent in .

Proof. First we prove thatFor , we defineIf there exists a point such that then, from Lemma 1, we haveFor the case , we have This contradicts (10). For the case , applying the same method as the above, we have This also contradicts (10). Therefore,Because , we cannot apply Lemma 2 at this moment. We need the following step.
PutThen, it follows thatWe want to show . If there exists a point such that then, from Lemma 1, we haveFor the case , by (20), we have It contradicts (18). For the case arg , applying the same method as the above, we have It also contradicts (18). This shows that or Therefore, from Lemma 2, we obtain that is -valent in . Now the proof is complete.

Taking in Theorem 3, we have the following corollary.

Corollary 4. Let be analytic in . Suppose that, for positive integer , where Then, is -valent in .

Example 5. The function satisfies condition (10) in Theorem 3 and is -valent in .

2. A Sufficient Condition on Univalency of Meromorphic Functions

Let be the class of functions of the formwhich are analytic in . For two functions and analytic in , the function is said to be subordinate to , written as , if there exists a function analytic in with and such that . In particular, if the function is univalent in , then is equivalent to and .

Also let denote the class of functions of the formwhich are analytic in the punctured unit disc .

In [5], Nunokawa et al. proved the following theorem.

Theorem A. Let with for and letThen, is univalent in .

In [6], Yang and Liu generalized the result of Theorem A and obtained sharp bound of (31). Many interesting properties of analytic functions were considered by several authors in some recent works (e.g., [7–9]). In this section, we will derive a sufficient condition on univalency of meromorphic functions in .

For our purpose, we need the following lemmas.

Lemma 6 (see [10]). Let and be analytic in with . If is starlike in and , then

Lemma 7 (see [11]). Let be analytic and convex univalent in and let be analytic in . If , then where and .

Theorem 8. Let with for . Ifwhere and , then is meromorphic univalent in .

Proof. LetThen, the function is analytic in . By integration from to , we getThus, we havewhere .
One can see easily from (34) and (35) that and , which shows that for . Therefore,for and .
If , it follows from (37) and (38) thatHence, is meromorphic univalent in . The proof is complete.

Taking in Theorem 8, we obtain the following.

Corollary 9. Let with for . Ifthen is meromorphic univalent in .

Theorem 10. Let with for . Also let satisfy condition (34). Then, for ,Equalities in (41) are attained if we take

Proof. In view of (34), we haveApplying Lemma 6 to (43), we find thatBy using Lemma 7, (44) gives thatthat is,where is analytic in and by Schwarz lemma.
Now, from (46), we can easily derive inequalities (41). The proof is complete.

Example 11. The function satisfies the conditions in Theorems 8 and 10.

An Open Problem. The maximum value of in Theorem 8 for which (34) implies meromorphic univalence is unknown.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11571299) and the Natural Science Foundation of Jiangsu Province (Grant no. BK20151304).