Some Properties of Analytic Functions with the Fixed Second Coefficients ()
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
.