1. Introduction
Let
denote the family of complex valued functions
f which are holomorphic (analytic) in
and are normalized through the conditions
and
. That is, for
one may have its series form
The class
is comprised those univalent functions
by which every circular arc
, with center at
, is mapped onto the convex arc and such functions are known as uniformly convex functions. This class was first introduced by Goodman [
1]. The interesting analytic condition of class
was given in [
2] and is stated as follows:
Kanas et al. [
3] further generalized the class
by introducing the class of
k-uniformly convex functions, named as
k-
and the class
k-
of corresponding
k-starlike functions. The class
k-
is defined as follows:
They, in addition, discussed these classes geometrically and established connections with the conic domains
It is important to mention that the class k- was studied much earlier with some extra conditions but without geometrical interpretation. The class k- is defined geometrically in a way that the common region of and the disk is mapped onto a convex domain by these univalent functions. Thus, the notion of convexity got the generalized version of k-uniform convexity. If , Then, the center shifts to origin and thus k- takes the form of the family of convex univalent functions.
The domain
represents conic regions for certain values of parameter
that is, it gives an elliptic region for
, the hyperbolic region (right branch) for
and the parabolic region when
For more details, see [
3,
4,
5,
6]. The domain
which is generalization of
is given as:
where
For details, see [
7]. The function which gives the boundary curves of these conical regions is denoted by
which is holomorphic in
and maps
onto
such that
and
and is defined as:
For the detailed study of this function, we refer the readers to see [
3,
6].
Let
k-
denote the family of holomorphic functions
with
and
for
where the notion “≺” denotes the familiar subordinations. It is pertinent to have
where
is the family of functions with a positive real part. In addition, for
-
we have
where
Definition 1. Let the function be holomorphic in with . Then, - if for and β is given by Equation (3), we havewhere - [8]. Taking
and
the class
introduced by Pinchuk [
9] is obtained. In addition,
k-
-
0-
and 0-
where
and
were introduced in [
9].
It is noted that
k-
is a convex set. Noor [
8] introduced the classes
k-
and
k-
of
k-uniformly bounded boundary and radius rotation of order
corresponding to the class
k-
.
Now, we consider the following new subclasses of holomorphic functions.
Definition 2. A function is known to be in k-, and β is given by Equation (3), if Definition 3. A function is known to be in the class k-, and β is given by Equation (3), if there exists - such thator equivalently It is pertinent to note that, by assigning specific values to parameters and k in k- and k- several well-known subclasses of holomorphic and univalent functions are obtained, from which some are listed below:
0-
introduced by Bhargava et al. [
10].
For
and
we obtain the class
k-
, and
k-
for details, we refer to [
8].
0-
for details, see [
11].
Throughout the article, we shall consider, unless otherwise stated, that
,
and
is given by Equation (
3).
3. Main Results
Theorem 1. Let -. Then, the odd functionbelongs to k-. Proof. Let
-
and consider
Logarithmic differentiation of the above relation yields
or, equivalently,
where
Because
-
, then, there exist
-
such that
Since
k-
is a convex set, we have
Thus, we have that
and hence
-
□
When we take
the following result, proved by Noor [
8], is obtained.
Corollary 1. Let -. Then,belongs to k-. Corollary 2. Let . Then,belongs to . Theorem 2. If -, thenfor some - Proof. Let
-
. Then, by definition, one may have
Simple computation leads us to
Using (9) in (10), we can easily obtain (8). □
When we take
the above result takes the following form, proved by Noor [
8].
Corollary 3. If -, thenfor - When
and
. Then, we have the following result, proved in [
11].
Theorem 3. Let - be of the form (1). Then,where is given by (6). Proof. Let
-
and let it be of the form (9). Then,
From (12), we have
. It is well known that
in the class
k-
is
where
is given by (
6). Thus, we get (11). □
Corollary 4. The following disk is contained in the range of every function from k-.where is given by (6). Proof. According to the Koebe theorem, each omitted value
w satisfies
Using (13) and Theorem 3, we get the required result. □
By using the similar technique as used in [
11], we have the following result.
Theorem 4. Let -. Then, for and ,for Theorem 5. Let -. Then, for ,where is defined by (14) and Proof. Let
where
where
is an odd function of the form
Since
-
and by Theorem 1
-
, therefore, by using Theorem 4, we have
In addition, we observe that, for
Therefore,
which takes the form
Thus, the values
are contained in the circle of Apollonius with diameter end points
and
and radius
. Thus, the maximum of
is attained at points where tangent ray from origin to the circle can be drawn, that is, when
This completes the proof. □
For talking we obtain the integral representation for the class
Corollary 5. Let . Then, for ,where Theorem 6. Let - Then, for where , and is a constant depending upon and only. Proof. Since
-
thus
By Theorem 1, we have for
-
the function
which yields
Since
-
, we have
Since
k-
so we can write
Thus, for odd functions
we have
Now, by making use of Holder’s inequality, with
and
such that
, we have
By using Lemma 2 and distortion results, we obtain
This implies that
where
is a constant depending upon
and
only. Similarly, for
we have
□
Theorem 7. Let - Then, for and where is given by (14) and are the same as in Theorem 6 and is a constant. Proof. Since
Cauchy theorem gives
which reduces to
Now, using Theorem 6 for
we have
Putting
we have
Similarly, we obtain the required result for □
Theorem 8. Let - Then, for where and is a constant depending upon and only. Proof. We know that, for
and
,
As
-
thus
Thus, for
and
, we have
Since
-
, therefore, for
we have
Using Lemma 3
we have
Now, using Lemma 3
, we have
Now, using Cauchy–Schwarz inequality, we have
By using Lemma 2 and distortion results, we obtain
where
is a constant. Now, putting
we obtain
Now, taking
we have
Similarly for
we have
Thus, the result follows. □
Theorem 9. Let - for Then, is convex for Proof. Let
where
-
and
Differentiating logarithmically, we obtain
Now, using the distortion results for the classes
and
we have
taking
This completes the proof. □