Conditions for Carathéodory Functions

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

(1.1)

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

(1.2)

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

(1.3)

(1.4)

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

(2.1)

Then we have

(2.2)

where

(2.3)

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

(2.4)

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

(2.5)

where

(2.6)

(2.7)

then

(2.8)

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

(2.9)

Now we suppose that

(2.10)

Then we have

(2.11)

where

(2.12)

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

(2.13)

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

Next, we suppose that

(2.14)

Applying the same method as the above, we have

(2.15)

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

(2.16)

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

Proof.

Taking

(2.17)

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

(2.18)

where , , , and is analytic in with . If

(2.19)

where

(2.20)

then

(2.21)

Corollary 2.6.

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

(2.22)

Proof.

Letting

(2.23)

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

(2.24)

and is given by (2.20). Then

(2.25)

Corollary 2.8.

Let , and , be real numbers with and . If

(2.26)

where

(2.27)

then

(2.28)

where is the integral operator defined by

(2.29)

Proof.

Let

(2.30)

(2.31)

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

(2.32)

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

(2.33)

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

(2.34)

Since

(2.35)

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

(2.36)

where , , and is analytic in with . If

(2.37)

where

(2.38)

then

(2.39)

Corollary 2.11.

Let with in and . If

(2.40)

where is given by (2.38) with and , then

(2.41)

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

Proof.

Let

(2.42)

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

(2.43)

If

(2.44)

then .

Proof.

Let

(2.45)

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

(2.46)

Therefore, we have

(2.47)

which completes the proof of Corollary 2.12.

Corollary 2.13.

Let with in and . If

(2.48)

where is given by (2.38), then .

Finally, we have the following result.

Theorem 2.14.

Let be nonzero analytic in with . If

(2.49)

(2.50)

then

(2.51)

Proof.

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

(2.52)

Now we suppose that

(2.53)

Then we have

(2.54)

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

(2.55)

we obtain

(2.56)

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

(2.57)

then .