On a Class of Function of Bounded Turning with Negative Coefficients
Associated with a Generalized Multiplier Transformation
M. O. OLUWAYEMI1,2, B. O. MOSES3, J. O. HAMZA4
1Landmark University SDG 4 (Quality Education Research Group), Omu-Aran, NIGERIA
2Department of Mathematics, Landmark University, P.M.B. 1001, Omu-Aran, NIGERIA
3Department of Mathematics, American University of Nigeria, Yola, NIGERIA 4Department of
Mathematics, University of Lagos, Akoka, NIGERIA
Abstract:- The study of univalent functions and its applications is an hallmark of geometric function theory.
Since univalent functions are analytic and has one-to-one mapping, it has a wide range of applications in the
fields of studies where transformations (enlargements and reductions) are done. The functions also have angle
and orientation preserving properties among other uses. Many authors have defined and studied various classes
of univalent functions using different approaches and tools. In this study however, the authors used a
generalized multiplier transformation to define a and investigate a new class of functions . Various
properties of the class of functions were investigated. The results extend some known results in literature.
Key-Words: - Univalent, unit disc, multiplier transformation, function of bounded turning, starlike functions,
convex functions, and coefficients
Received: April 24, 2021. Revised: January 10, 2022. Accepted: January 27, 2022. Published: February 25, 2022.
1 Introduction and Preliminaries
Let
(1)
which are analytic in the open unit disk
and normalised by and
.
We denote by the subclass of S which are
normalized univalent function of the form
(2)
that are analytic and univalent in the open unit disk
. The function of the form (2) was first introduced
in [13]. Functions of negative coefficients were also
investigated in [2], [5], [6], [8], [9] and [10].
Univalent functions have applications in many areas
of science and engineering, The simplest example of
a univalent function is the identity function
. Other examples are the circular function
which is univalent in the disc
and the
Koebe function
1.1 Multiplier Transformations (Differential
Operator)
Since the introduction of differentiatial operator as
multiplier transformation in the geometric function
theories, various authors such as [1], [4], [11], [12],
[14] and [15] have continued to extend the
transformation and used as a tool to define, study
and establish properties of the class of functions.
Let
(3)
For convenience, we let
(4)
So that
(5)
The function defined in (4) is associated with
the multiplier transformation
defined in [8] where with
such that and . The
operator generalises the operators defined in [4],
[12], [14], and [15]. The operator also reduces to (2)
when .
1.2 Class
Definition 1.1 A function defined by (2) is
said to belong to class if
is a function of bounded
turning. That is, if
(6)
Investigation of classes of univalent functions is a
great deal in geometric functions theory (GFT).
Various authors have investigated different forms of
functions such as starlike functions, convex
functions , close to convex functions and
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functions of bounded turning among others.
Ibrahim and Kılıçman in 16] considered a
relationship between and as classes of
univalent functions. Deng in [3] studied some
geometric properties of a univalent functions with
negative coefficients which was first defined in [13]
while Fadipe-Joseph and Oluwayemi in [7] defined
and investigated class as a class of convex
univalent functions. Motivated by the work of [3]
and [8], the following results were presented using a
differential operator. Other approaches such as
subordination principle and convolution among
others could also be used in investigating a class of
function.
2 Main Results
Theorem 2.1 A function defined by
belongs to class if and only if
(7)
Proof:
It’s necessary to show that
if .
Suppose , then is univalent in U.
Consequently,
is non negative which implies that . Thus,
Assume
then there exists such that
Hence, we have such that
Thus,
But is continuous and . Hence, we
can find such that by
intermediate value theorem which contradicts the
assumption. Hence,
as required.
Conversely, assume
Then
Thus,
The result is sharp for
(8)
Corollary 2.1 A function defined by
belongs to the class if and only if
(9)
Corollary 2.2 A function defined by
belongs to the class if and only if
(10)
Corollary 2.3 A function defined by
belongs to the class if and only if
(11)
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Remark 2.1 The class investigated in
[3].
Theorem 2.2 The class is convex.
Proof:
Let be defined as (4). Then, there exists a
function
(12)
in such that
(13)
where and are non-negative and .
Then,
and
(14)
Since . Then,
(15)
Using in ,
Since , it follows that
Consequently
which implies that the class is convex.
Theorem 2.3 Let . Then,
with equality for
Proof:
Let , then we have that (8) holds. Also,
also from Theorem 2.1. Hence,
and
The results follows from the above expressions.
Corollary 2.4 Let . Then,
with equality for
Theorem 2.4 Let . The disk is
mapped onto a domaiin that contains the disk
with equality for
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Proof:
Suppose , we have from Theorem 2.3
that
The result follows as .
Theorem 2.5 Let . Then,
with equality for
Proof:
Suppose . Then,
and
as required.
Corollary 2.5 Let . Then,
with equality for
Let such that . We define the
functions
(16)
Theorem 2.6 Let the functions defined by
be in the class . Then, the function
defined by
(17)
Then, where
Proof:
Let . Then using (16) and by Theorem
2.1,
.
Hence,
Theorem 2.7 Suppose a function defined by
belongs to class and there exists
such that . Then, the function defined
by
(18)
also belongs to the class .
Proof:
Let . Then, we define
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where
So that
The result follows from Theorem 2.1.
Theorem 2.8 Let such that . If
, then the function defined by
(18) is univalent in
where
(19)
Proof:
Let
Then from (18),
We now need to show that in
. But
We have that since . Then,
(20)
implies that . Using
(20) will hold true if
which implies that
which completes the proof.
The result is sharp for
Corollary 2.6 Let such that . If
, then the function defined by
(18) is univalent in
where
(21)
3 Conclusion
The result extend some known results in literature.
In particular, the class of function defined in [3] is
extended in the present study. The results in the
study are generalised in this paper. The class of
functions studied in the work find applications in
the areas where mappings are done.
Acknowledgement:
The authors acknowledge their respective
institution.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Oluwayemi M.O. did conceptualization, the initial
investigation of the work and also prepared the
original draft, Hamzat J. O. reviewed the work after
the initial draft to ascertain the correctness of the
work and while Moses B. O. worked on the
methodology and validation of the study. All
authors agreed upon the final work and approved the
final draft accordingly.
Sources of Funding for Research Presented
in a Scientific Article or Scientific Article
Itself
The authors report that they do not have any form of
financial support for the work.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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