Advances in Pure Mathematics
Vol.4 No.5(2014), Article ID:45780,9 pages DOI:10.4236/apm.2014.45025
Some Properties of Analytic Functions with the Fixed Second Coefficients
Oh Sang Kwon
Department of Mathematics, Kyungsung University, Busan, Korea
Email: oskwon@ks.ac.kr
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 20 March 2014; revised 20 April 2014; accepted 26 April 2014
ABSTRACT
In this paper, we investigate some argument properties for analytic functions with fixed second coefficient and positive real part. And we apply the argument properties to the functions that are analytic and normalized. In particular, the order of strongly starlikeness of strongly convex functions with fixed second coefficients is given.
Keywords:Analytic Functions, Argument Estimate, Fixed Second Coefficients, Strongly Starlike Functions, Strongly Convex Functions
1. Introduction
Let be the set of functions that are analytic in and normalized by
And let be the class of all functions that are analytic and have positive real part in with.
Nunokawa investigated some properties of analytic functions which are not Caratheodory, that is, which are not in. Furthermore, he has found the order of strongly starlikeness of strongly convex functions in (see: [1] [2] ).
Now, for a fixed, let consist of functions of the form
And let consist of analytic functions of the form
where the second coefficient is fixed constant.
In [3] [4] , Ali et al. have extended the theory of differential subordination developed by Miller and Mocanu [5] , to the functions with fixed second coefficients. And Lee et al. [6] and Nagpal et al. [7] have applied the results, to obtain several extensions of properties for univalent functions with fixed second coefficients.
In this paper, we investigate some argument properties for analytic functions with fixed second coefficients and positive real part. And we apply our results to the normalized univalent functions with fixed second coefficients.
We need the following Lemma for functions with fixed initial coefficient.
Lemma 1 [3] Let and be continuous in, analytic in with and. If
then
and
where
(1)
Here, we note that the inequality (1) implies that
since.
2. Lemmas
Lemma 2 Let be analytic in and in. Suppose that there exists a point such that
and
Then
where and.
Proof. Let us put
Then, for and. And we note that
By Lemma 1, we have
Hence
And this inequality implies is a negative real number which satisfies
Now, we put. For the case,
(2)
For the case,
(3)
Hence, by (2) and (3),
where and
Hence the proof of Lemma 2 is completed.
Lemma 3 Let be analytic in and in. Suppose that there exists a such that
and
Then
where
and
with
Proof. Let us put
Then
and
Let us put
Applying Lemma 2, we get
where with
and
3. Argument Estimates for Functions with Fixed Second Coefficient
Theorem 4 Let and satisfy
Then
where
(4)
Proof. Suppose that there exists a point such that
and
By Lemma 3, we can obtain that
where
and
with. For the case,
which is a contradiction to the assumption. For the case, using the same method, we can obtain a contradiction to the assumption.
Remark 5 If, then Theorem 4 reduces the result in [[8] , Theorem 3].
Theorem 6 Let and satisfy
(5)
Then
where
(6)
Proof. If there exists a point such that
and
then Lemma 3 gives us that
If, then we have. Therefore, we see that
with
Hence
Now, we define a function by
Then
Hence takes the minimum value at. Therefore,
Thus we have
which contradicts the condition (5). And if, applying the same method we have
which contradicts the condition (5). And this completes the proof of the Theorem 6.
Remark 7 If, then Theorem 6 reduces the result in [[8] , Theorem 1].
Theorem 8 Let and
(7)
for some, , where is given by
Then
Proof. Suppose that there exists a such that
and
By Lemma 3, we can obtain that
where with
and
For the case,
(8)
since
Now, we define
Then
Define
Then and. Furthermore,
for all. Hence for all. And
By (8), we have
which is a contradiction to the hypothesis. For the case , using the same method, we can obtain a contradiction to the assumption.
Remark 9 If, then Theorem 8 reduces the result in [[9] , Theorem 2.1].
4. Corollaries
For a function, is called strongly starlike of order, , if
And is called strongly convex of order, , if
Using these definitions and Theorem in Section 3, we can obtain the following corollaries.
Corollary 10 Let and
Then is strongly starlike of order, where is given by (4).
Putting in Corollary 1, we can obtain the following Corollary.
Corollary 11 Let be a strongly convex function of order. Then is a strongly starlike function of order, where is given by
Corollary 12 Let and satisfy
Then is strongly starlike of order, where is given by (6).
Corollary 13 Let and
where is given by (7). Then is strongly starlike of order.
References
- Nunokawa, M. (1992) On Properties of Non-Caratheodory Functions. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 68, 152-153.
- Nunokawa, M. (1993) On the Order of Strongly Starlikeness of Strongly Convex Functions. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 69, 234-237.
- Ali, R.M., Nagpal, S. and Ravichandran, V. (2011) Second-Order Differential Subordination for Analytic Functions with Fixed Initial Coefficient. Bulletin of the Malaysian Mathematical Sciences Society, 34, 611-629.
- Ali, R.M., Cho, N.E., Jain, N. and Ravichandran, V. (2012) Radii of Starlikeness and Convexity of Functions Defined by Subordination with Fixed Second Coefficients, Filomat, 26, 553-561. http://dx.doi.org/10.2298/FIL1203553A
- Miller, S.S. and Mocanu, P.T. (2000) Differential Subordinations. Dekker, New York.
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- Nunokawa, M., Owa, S. and Saitoh, H. (2003) Argument Estimates for Certain Analytic Functions. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 79, 163-166.