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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ1_HTML.gif)
which are analytic in the open unit disk . If
in
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ5_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ6_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ8_HTML.gif)
where ,
,
,
,
and
is analytic in
with
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ9_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ11_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ13_HTML.gif)
Now we suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ14_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ15_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ16_HTML.gif)
Then, by a simple calculation, we see that the function takes the minimum value at
. Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ17_HTML.gif)
where is given by (2.6). This evidently contradicts the assumption of Theorem 2.2.
Next, we suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ18_HTML.gif)
Applying the same method as the above, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ20_HTML.gif)
where is given by (2.6) with
and
, then
.
Proof.
Taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ22_HTML.gif)
where ,
,
, and
is analytic in
with
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ23_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ24_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ25_HTML.gif)
Corollary 2.6.
Let , where
is given by (2.20) with
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ26_HTML.gif)
Proof.
Letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ28_HTML.gif)
and is given by (2.20). Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ29_HTML.gif)
Corollary 2.8.
Let ,
and
,
be real numbers with
and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ30_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ31_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ32_HTML.gif)
where is the integral operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ33_HTML.gif)
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ35_HTML.gif)
Then and
are analytic in
with
. By a simple calculation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ36_HTML.gif)
Using a similar method of the proof in Theorem 2.2, we can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ37_HTML.gif)
From (2.29) and (2.31), we easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ38_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ40_HTML.gif)
where ,
,
and
is analytic in
with
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ41_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ42_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ43_HTML.gif)
Corollary 2.11.
Let with
in
and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ44_HTML.gif)
where is given by (2.38) with
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ45_HTML.gif)
that is, is univalent (close-to-convex) in
.
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ47_HTML.gif)
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ48_HTML.gif)
then .
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ49_HTML.gif)
Then, by Theorem 2.10, condition (2.44) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ50_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ51_HTML.gif)
which completes the proof of Corollary 2.12.
Corollary 2.13.
Let with
in
and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ52_HTML.gif)
where is given by (2.38), then
.
Finally, we have the following result.
Theorem 2.14.
Let be nonzero analytic in
with
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ53_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ54_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ55_HTML.gif)
Proof.
If there exists a point satisfying the conditions of Lemma 2.1, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ56_HTML.gif)
Now we suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ57_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ58_HTML.gif)
where is given by (2.50). Also, for the case
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ59_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ60_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F601597/MediaObjects/13660_2009_Article_1979_Equ61_HTML.gif)
then .
References
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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.69.234
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/S1446788700037873
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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