- Research Article
- Open Access

# Conditions for Carathéodory Functions

- Nak Eun Cho
^{1}Email author and - In Hwa Kim
^{1}

**2009**:601597

https://doi.org/10.1155/2009/601597

© N. E. Cho and I. H. Kim 2009

**Received:**12 April 2009**Accepted:**13 October 2009**Published:**15 October 2009

## Abstract

The purpose of the present paper is to derive some sufficient conditions for Carathéodory functions in the open unit disk. Our results include several interesting corollaries as special cases.

## Keywords

- Differential Equation
- Real Number
- Integral Operator
- Unit Disk
- Simple Calculation

## 1. Introduction

then we say that is the Catathéodory function.

Let denote the class of all functions analytic in the open unit disk with the usual normalization . If and are analytic in , we say that is subordinate to , written or , if is univalent, and .

respectively. We note that (1.3) and (1.4) can be expressed, equivalently, by the argument functions. The classes and were introduced by Brannan and Kirwan [1] and studied by Mocanu [2] and Nunokawa [3, 4]. Also, we note that if , then coincides with , the well-known class of starlike(univalent) functions with respect to origin, and if , then consists only of bounded starlike functions [1], and hence the inclusion relation is proper. Furthermore, Nunokawa and Thomas [4] (see also [5]) found the value such that .

In the present paper, we consider general forms which cover the results by Mocanu [6] and Nunokawa and Thomas [4]. An application of a certain integral operator is also considered. Moreover, we give some sufficient conditions for univalent (close-to-convex) and (strongly) starlike functions (of order ) as special cases of main results.

## 2. Main Results

To prove our results, we need the following lemma due to Nunokawa [3].

Lemma 2.1.

With the help of Lemma 2.1, we now derive the following theorem.

Theorem 2.2.

Proof.

where is given by (2.6). This evidently contradicts the assumption of Theorem 2.2.

where is given by (2.6), which is a contradiction to the assumption of Theorem 2.2. Therefore, we complete the proof of Theorem 2.2.

Corollary 2.3.

where is given by (2.6) with and , then .

Proof.

in Theorem 2.2, we can see that (2.4) is satisfied. Therefore, the result follows from Theorem 2.2.

Corollary 2.4.

Let and . Then , where is given by (2.6) with and .

By a similar method of the proof in Theorem 2.2, we have the following theorem.

Theorem 2.5.

Corollary 2.6.

Proof.

in Theorem 2.5, we have Corollary 2.6 immediately.

If we combine Corollaries 2.4 and 2.6, then we obtain the following result obtained by Nunokawa and Thomas [4].

Corollary 2.7.

Corollary 2.8.

Proof.

the conclusion of Corollary 2.8 immediately follows.

Remark 2.9.

Letting in Corollary 2.8, we have the result obtained by Miller and Mocanu [7].

The proof of the following theorem below is much akin to that of Theorem 2.2 and so we omit for details involved.

Theorem 2.10.

Corollary 2.11.

that is, is univalent (close-to-convex) in .

Proof.

in Theorem 2.10. Then (2.36) is satisfied and so the result follows.

By applying Theorem 2.10, we have the following result obtained by Mocanu [6].

Corollary 2.12.

then .

Proof.

which completes the proof of Corollary 2.12.

Corollary 2.13.

where is given by (2.38), then .

Finally, we have the following result.

Theorem 2.14.

Proof.

where is given by (2.50). These contradict the assumption of Theorem 2.14 and so we complete the proof of Theorem 2.14.

Corollary 2.15.

then .

## Declarations

### Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2009-0066192).

## Authors’ Affiliations

## References

- Brannan DA, Kirwan WE:
**On some classes of bounded univalent functions.***Journal of the London Mathematical Society*1969,**1:**431–443.MathSciNetView ArticleMATHGoogle Scholar - Mocanu PT:
**On strongly-starlike and strongly-convex functions.***Studia Universitatis Babes-Bolyai—Series Mathematica*1986,**31**(4):16–21.MathSciNetMATHGoogle Scholar - Nunokawa M:
**On the order of strongly starlikeness of strongly convex functions.***Proceedings of the Japan Academy, Series A*1993,**69**(7):234–237. 10.3792/pjaa.69.234MathSciNetView ArticleMATHGoogle Scholar - Nunokawa M, Thomas DK:
**On convex and starlike functions in a sector.***Journal of the Australian Mathematical Society (Series A)*1996,**60**(3):363–368. 10.1017/S1446788700037873MathSciNetView ArticleMATHGoogle Scholar - Mocanu PT:
**Alpha-convex integral operator and strongly-starlike functions.***Studia Universitatis Babes-Bolyai—Series Mathematica*1989,**34**(2):19–24.MathSciNetMATHGoogle Scholar - Mocanu PT:
**Some starlikeness conditions for analytic functions.***Revue Roumaine de Mathématiques Pures et Appliquées*1988,**33**(1–2):117–124.MathSciNetMATHGoogle Scholar - Miller SS, Mocanu PT:
**Univalent solutions of Briot-Bouquet differential equations.***Journal of Differential Equations*1985,**56**(3):297–309. 10.1016/0022-0396(85)90082-8MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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