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# Conditions for Carathéodory Functions

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 601597 (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.

## 1. Introduction

Let be the class of functions of the form

which are analytic in the open unit disk . If in satisfies

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 .

For , let and denote the classes of functions which are strongly convex and starlike of order ; that is, which satisfy

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.

Let be analytic in and in . Suppose that there exists a point such that

Then we have

where

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

Theorem 2.2.

Let be nonzero analytic in with and let satisfy the differential equation

where , , , , and is analytic in with . If

where

then

Proof.

If there exists a point such that the conditions (2.1) are satisfied, then (by Lemma 2.1) we obtain (2.2) under the restrictions (2.3). Then we obtain

Now we suppose that

Then we have

where

Then, by a simple calculation, we see that the function takes the minimum value at . Hence, we have

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

Next, we suppose that

Applying the same method as the above, we have

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.

Let and , . If

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

Proof.

Taking

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.

Let be nonzero analytic in with and let satisfy the differential equation

where , , , and is analytic in with . If

where

then

Corollary 2.6.

Let , where is given by (2.20) with and . Then

Proof.

Letting

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.

Let , where

and is given by (2.20). Then

Corollary 2.8.

Let , and , be real numbers with and . If

where

then

where is the integral operator defined by

Proof.

Let

Then and are analytic in with . By a simple calculation, we have

Using a similar method of the proof in Theorem 2.2, we can obtain that

From (2.29) and (2.31), we easily see that

Since

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.

Let be nonzero analytic in with and let satisfy the differential equation

where , , and is analytic in with . If

where

then

Corollary 2.11.

Let with in and . If

where is given by (2.38) with and , then

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

Proof.

Let

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.

Let with and be the solution of the equation given by

If

then .

Proof.

Let

Then, by Theorem 2.10, condition (2.44) implies that

Therefore, we have

which completes the proof of Corollary 2.12.

Corollary 2.13.

Let with in and . If

where is given by (2.38), then .

Finally, we have the following result.

Theorem 2.14.

Let be nonzero analytic in with . If

then

Proof.

If there exists a point satisfying the conditions of Lemma 2.1, then we have

Now we suppose that

Then we have

where is given by (2.50). Also, for the case

we obtain

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.

Let with in and . If

then .

## References

Brannan DA, Kirwan WE:

**On some classes of bounded univalent functions.***Journal of the London Mathematical Society*1969,**1:**431–443.Mocanu PT:

**On strongly-starlike and strongly-convex functions.***Studia Universitatis Babes-Bolyai—Series Mathematica*1986,**31**(4):16–21.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.69.234Nunokawa 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/S1446788700037873Mocanu PT:

**Alpha-convex integral operator and strongly-starlike functions.***Studia Universitatis Babes-Bolyai—Series Mathematica*1989,**34**(2):19–24.Mocanu PT:

**Some starlikeness conditions for analytic functions.***Revue Roumaine de Mathématiques Pures et Appliquées*1988,**33**(1–2):117–124.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)90082-8

## 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).

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Cho, N.E., Kim, I.H. Conditions for Carathéodory Functions.
*J Inequal Appl* **2009**, 601597 (2009). https://doi.org/10.1155/2009/601597

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

### Keywords

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