1. Introduction and Definitions
Let
be the class of holomorphic or analytic functions
f defined in the open unit disk
and have series representation of the form
From the above series representation, it is obvious that
Let
denote the class of Carathéodory functions
p such that
and
The Möbius function
is a special case of the known bilinear fractional transformations. This function, or its rotation, acts as an extremal function for the class
and it maps the open unit disk to the right half-plane. For functions
we have
Furthermore, recall the class
,
, consisting of functions
such that
For holomorphic or analytic mappings
f and
h in
,
f is subordinate to
h, and we write
, and if there is a Schwarz mapping or function
:
, where
. For a univalent function
h, we see that
Additionally,
represents the class of univalent functions, whereas
and
denote the subclasses of univalent starlike and univalent convex functions, respectively. For more detail and further explanation, see [
1]. These families have various applications as seen in [
2], and are related with the change in argument of the radius vector and tangent vector of the image of
as non-decreasing functions of the angle
, respectively. We recall that a function
if and only if
where
is such that
A function
is strongly starlike of order
if and only if
where
and
is defined above by (
3). For reference, see [
3]. A function
is strongly starlike with symmetric points if and only if
Let
,
Then,
if and only if
where
is defined above by (
3) and
Remark 1. so by using binomial expansion, we write We note thatwhere denotes the Guass hypergeometric function. For detail, we refer to [
4]. The family
is introduced in [
5]. Obviously,
and also
. A function
if
f satisfies the inequalities
We note that
l as defined above is convex and it maps
onto
where
Let
and
Then,
if and only if
where
For detail, we refer to [
6]. We note that
The function
defined above is convex univalent in
with
and it maps onto
or onto the convex hull of three points (one of which may be that point at infinity) on the boundary of
. Thus,
is a subfamily of starlike functions of order
where
, see [
7].
In dynamics, we examine how a system behaves under a certain iterative scheme. Asymptotic behaviour under these iteration is of great interest. We concentrate on models that are continuous mappings of actual data. The Schwarzian and pre-Schwarzian derivatives were introduced to one-dimensional dynamics for any sufficiently smooth map. We may categorize maps based on the sign of their Schwarzian derivative. Now, let
denote the subfamily of
such that
For a function
, the pre-Schwarzian along with Schwarzian derivatives of
f are defined as
respectively. The Schwarzian derivative of any Möbius transformation
is zero. Conversely, the Schwarzian derivative is the only derivative which measures the degree to which a function fails to be a Möbius transformation. The second-order ordinary differential equation and Schwarzian derivative can be used to determine the Riemann mapping between any bounded polygon and the upper half-plane or unit circle. This reduces to the Schwarz–Christoffel mapping for polygons with straight edges, which can be derived directly without using the Schwarzian derivative. The pre-Schwarzian as well as Schwarzian derivatives are popular tools for studying the geometric properties of these mappings. They can, for example, be used to obtain either necessary or sufficient conditions for overall univalence, or to obtain specific geometric conditions on the image domain of such mappings. Many researchers have investigated estimates of the pre-Schwarzian as well as Schwarzian norms for a family of injective or univalent functions. Because of the computational difficulty, the pre-Schwarzian norm has received more attention than the Schwarzian norm.
Obradović [
8] found bounds on the first- and second-order derivatives of starlike and convex functions respectively. Same results were then investigated by Tuneski [
9]. He, along with Irmak [
10], studied conditions for some other subfamilies of analytic functions. Singh and Tuneski [
11] further extended these results for Janowski-type functions.
Historically, certain differential operators were first known to Riemann, but the first person who actually studied these operators extensively was Schwarz. He investigated and found that such differential operators were invariant with respect to Mobius transformations. These were then known as the Schwarzians. Much later, Lavie [
12] showed that under the assumption that
all differential operators of order
n on
f are invariant with respect to Möbius transformation, written as
and its derivative of orders up to
. In complex function theory, there are several branches depending on the Schwarzian or pre-Schwarzian derivatives. For a univalent or injective function
f, it is obvious that
,
and these results are the best possible. In [
13], Fait et al. proved that every function
is generalized to a
-quasiconformal automorphism of
. Thus, we have
and
. Moreover, Chiang [
14] determined that
However, such structures are slightly different. We see that
and
are analytic when
f is analytic and meromorphic and
in
. The family
defined above by (
7) is a vector space as seen in [
15], so we define the norm on
by
The
has a significance for Teichmüller spaces and it is assumed as an element of a Banach space, see [
16]. It is obvious that
if and only if
in
. Thus, there is a
such that
f for each set
as seen in [
17]. The norm
also leads to the univalence of a meromorphic function
f in
. In fact, if
, then
f in
and for
f in
,
and
for
,
and
k is the known Koebe function, see [
18]. Additionally, if
f of order
, then we have the sharp estimate
as found in [
17]. For more details on the norm
, see [
19]. Moreover, others have norm estimates as seen in [
3,
20,
21,
22].
Bazilevič, as studied in [
23], introduced a family of functions
defined by the following integral representation:
where
:
,
in
,
is real and
. If we put
in (
8), then
From this expression, we note that
A function
f observing the condition given by (
9) is called a Bazilevic function of type
For more detail, see [
24] and others. Another important and well-known subclass of
which has been highly investigated in recent years, as seen in [
25,
26,
27,
28,
29,
30,
31] and references therein, is the class
defined subsequently.
Definition 1. For f given by (1), f if and only ifwhere and is defined above in (3). Although the above class is not strictly related to either or , its definition resembles the class of non-Bazilevič functions. Like other fundamental subclasses of this class is rotationally invariant.
We now see this class in a more general setting.
Definition 2. For f given by (1), f if and only ifwhere and is defined above in (3). For various properties of the class of
, we refer to [
28,
31,
32,
33] and others. It is noted that the class
is not invariant under the
root transformation but preserved under rotation, dilation, omitted-value transformations and conjugation. In the following, we move to another known family as seen in [
34].
Definition 3. For f given by (1), f if and only ifwhere and is defined above in (3). In view of the above structure, we have the following:
Definition 4. For f given by (1), we say if and only ifwhere g is starlike, and is defined above as seen in (3). Definition 5. For f given by (1), we say if and only ifwhere g is starlike, and is defined as in (3). For details of the related work, we refer to [
1,
23,
24,
31].
2. Main Results
Based on the observation and motivation of our main discussion, we find the best norm estimates of pre-Schwarzian derivative for functions f in the subfamilies defined above.
Theorem 1. Let . If a function such that where is given above in (6), then This result is sharp.
Proof. Suppose that
such that it satisfies the condition
where
is defined by (
3). Then in view of (
10), we consider a function
with
such that
Logarithmically differentiating (
15), we obtain
By using the condition
where
is given above in (
6), we find that
Using the Schwarz–Pick lemma as seen in [
1], we have
and we further see that
Now according to (
19), we discuss the two cases subsequently:
By using (
17), (
18) and (
20), we note that
Thus, the pre-Schwarzian norm
of function
f takes the form
We also consider that
Using (
19), we observe that
After simplification, we note that
Therefore, we have the same estimates. So, in both cases, we have the desired sharp result. □
Corollary 1. As an application of the Theorem 1, we define We note that Additionally,such that for So, is increasing and thus we find that This estimate shows that if , then it is uniformly locally univalent.
Theorem 2. Let If such that is given above in (5), thenwhere is defined above by (10). These bounds are sharp. Proof. Let
, where
is defined above by (
10) as
where
is given by (
3). Then, in view of (
10), we consider a function
with
such that
which on simplification leads to
By using the condition
as given above in the statement of the theorem, we find that
where
By using (
22) in (
21), we have
Using the inequality (
17) and the identity
, we see that
Since s is a Schwarz function and
, thus the last inequality yields
By the definition of the pre-Schwarzian norm, we note that
These bounds are sharp □
In light of the above theorem and condition , we find the following estimates:
Theorem 3. Let . If a function such that where is given above in (6), thenwhere and is defined above by (11). This result is sharp. Proof. Let
. Then, by rewriting (
11), we have
where
is represented in (
3). In view of (
11), there exists a function
such that
and note that
On logarithmic differentiation, we see that
By using the condition
as described above, we have
Substituting (
24) in (
23), we note that
On simplification, we write
In view of (
19), we discuss two cases subsequently:
Therefore, we obtain
or it takes the form
By using (
25), we note that
or we write
Thus, the pre-Schwarzian norm
takes the form
Moreover, we consider that
Using (
19) and on simplification, we note that
Therefore, in these cases we have the same estimates. So, in view of these cases, we obtain the required sharp result. □
By using the condition
and in view of (
11), we have the following estimates:
Theorem 4. Let If such that is given above in (5), thenwhere is defined above by (11). These bounds are sharp. Proof. For
suppose
such that
where
is defined above as in (
3). Then there exists a function
such that
such that
On differentiating (
26), we note that
By using the condition
as given above in the statement of the theorem, we find that
By applying both the inequality (
17) and the identity
the above equation leads to
or we can write
We use
in the last inequality to obtain
This leads to the desired proof. □
In view of the condition , we find the following estimates for functions in the class .
Theorem 5. Let . If a function such that where is given above in (6), thenwhere and is defined above by (13). These bounds are sharp. Proof. Let
be defined above by (
13) such that
where
is given above in (
3). Then in view of Definition 4, there exists a function
with
such that
Logarithmically differentiating the above expression, we have
By using the condition
as described above in (
16), we write
or we see that
In view of (
19), we discuss the following cases subsequently:
Therefore, we note that
which leads to
In view of (
29) and (
28), we note that
By using the upper bounds for
as given by (
2), the pre-Schwarzian norm
of a function
f takes the form
Using (
19) and then simplifying, we write
Continuing as above, again we have
Therefore, in these cases, we have equal sharp estimates. □
In view of the condition
, where
l is given above in (
5), we find the following estimates for functions in the class
.
Theorem 6. If such that is given above in (5), thenwhere is defined above by (13). These bounds are sharp. Proof. We assume that
In view of Definition 4, f satisfies the condition
where
is defined above as in (
3). Then, in view of Definition 4, there exists a function
with
such that
Logarithmically differentiating the above expression, we have
By using (
22) in (
30), we obtain
By using the inequality (
17) and the identity
in the above equation, we note that
In view of the inequality
, the last expression leads to
As
by using the bounds for
given above in (
2), we have
□
Remark 2. In view of the subfamilies and defined by (12) and (14), we can add some more problems by using similar procedures as seen in the above theorems.