Some Properties of Analytic Functions with the Fixed Second Coefficients
Oh Sang Kwon

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.

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Kwon, O. (2014) Some Properties of Analytic Functions with the Fixed Second Coefficients. Advances in Pure Mathematics, 4, 194-202. doi: 10.4236/apm.2014.45025.

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.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Nunokawa, M. (1992) On Properties of Non-Caratheodory Functions. Proceedings of the Japan Academy, Series A, Mathematical Sciences, 68, 152-153. [2] 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. [3] 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. [4] 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 [5] Miller, S.S. and Mocanu, P.T. (2000) Differential Subordinations. Dekker, New York. [6] Lee, S.K., Ravichandran, V. and Supramaniam, S. (2012) Applications of Differential Subordination for Functions with Fixed Coefficient to Geometric Function Theory. arXiv:1209.0896. [7] Nagpal, S. and Ravichandran, V. (2012) Applications of Theory of Differential Subordinatioin for Functions with Fixed Initial Coefficient to Univalent Functions. Annales Polonici Mathematici, 105, 225-238. http://dx.doi.org/10.4064/ap105-3-2 [8] Nunokawa, M., Owa, S., Hayami, T., Kuroki, K. and Uyanik, N. (2011) Some Conditions for Strongly Starlikeness of Certain Analytic Functions. International Mathematical Forum, 6, 1199-1208. [9] 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.