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ABSTRACT
In recent years, using the idea of analytic and bi-univalent functions, many ideas have been developed by different well-known authors, but the using Gegenbauer polynomials along with certain bi-univalent functions is very rare in the literature. We are essentially motivated by this recent research going on, here in our present investigation, we make use of certain q-derivative operator and Gegenbauer polynomials and define a new subclass of analytic and bi-univalent functions. We then obtain certain coefficient bounds, the Fekete–Szegö inequalities and upper bounds for the second-order Hankel determinant for the defined functions class.
2020 MATHEMATICS SUBJECT CLASSIFICATIONS:
1. Introduction and preliminaries
Geometric Functions Theory is a fascinating area of research in Complex Analysis, with applications in a variety of mathematical areas, including Mathematical Physics. Researchers in the field of Complex Analysis have been looking into holomorphic functions because of their various applications in analytical solutions to problems like electrostatics and fluid mechanics.
Analytic functions such as can be stated in Taylor's series expansion about the origin
as
which can be normalized in the following way:
(1)
(1)
where
,
,
and
is convergent for
. Let
indicate a class of functions
that are holomorphic in
, having form (Equation1
(1)
(1) ), and normalized by the constraints
.
It is well known that every function has an inverse
defined by
and
where
(2)
(2)
A function is said to be bi-univalent in
if both f and
are univalent in
.
Let Σ denote the class of bi-univalent function in given by (Equation2
(2)
(2) ). Examples of functions in the class Σ are
However, the familiar Koebe function is a member of class Σ. Other common examples of functions in
such as
are also not members of Σ.
Lewin [Citation1] investigated a bi-univalent functions class Σ and showed that . Subsequently, Brannan and Clunie [Citation2] conjectured that
. Netanyahu [Citation3], on the other hand, showed that
. The coefficient for each of the Taylor–Maclaurin coefficients
is presumably still an open problem.
Similar to the familiar subclass and
of starlike and convex functions of order
, respectively. Brannan and Taha [?] introduced certain subclasses of the bi-univalent function class
and
of bi-starlike functions and bi-convex functions of order
, respectively. For each of the function classes
and
they found non-sharp bounds on the first two Taylor–Maclaurin coefficients
and
.
Let and
are analytic functions in open unit disc
, then the function
is subordinated to
symbolically denoted as
,
, if there occur an analytic function
with properties that
and
, such that suppose ω holomorphic in
, such that
. If the function
is univalent in
then above condition is equivalent to
and
.
Jackson [Citation5, Citation6] introduced and studied the q-derivative operator of a function as follows:
(3)
(3)
and
. In case
for j is a positive integer, the q-derivative of
is given by
(4)
(4)
(5)
(5)
where
, for more details on the concepts of q-derivative (see [Citation7]).
The quantum (or q-) calculus is an essential tool for studying diverse families of analytic functions, and its applications in mathematics and related fields have inspired researchers. Srivastava [Citation8] was the first person to apply it in the context of univalent functions. Numerous scholars conducted substantial work on q-calculus and examined its various applications due to the applicability of q-analysis in mathematics and other domains. More importantly, the convolution theory enable us to investigate various properties of analytic functions. Due to the large range of applications of q-calculus and the importance of q-operators instead of regular operators, many researchers have explored q-calculus in depth. In addition, Srivastava [Citation9] recently published survey-cum-expository review paper which might be useful for researchers and scholars working on these subjects. Also, Srivastava's recent survey-cum-expository review article [Citation9] further motivates the use of the q-analysis in Geometric Function Theory, as well as commenting on the triviality of the so-called ()-analysis involving an insignificant and redundant parameter
(see especially [Citation9, p.340]). For some recent investigation about q-calculus, we may refer the readers to [Citation10–15]
The class of functions φ that is holomorphic in and has the form
with
is denoted by
.
The nth coefficient of a class function is well known to be restricted by n, and the coefficient limits give information about the functions' geometric characteristics. The famous problem solved by Fekete–Szegö [Citation16] is to determine the greatest value of the coefficient functional
over the class
for each
, which was demonstrated using the Loewner technique.
Noonan and Thomas [Citation17] introduced and investigated the mth Hankel determinant of f for and
as
(6)
(6)
Several writers, notably Noor [Citation18], have investigated this determinant, with topics ranging from the rate of development of
(as
) to the determinant of exact limits for particular subclasses of analytic functions on the unit disk
with specified values of j and m. When
j = 1, and
, the Hankel determinant is
. The Hankel determinant simplifies to
when j = m = 2. Fekete and Szegö [Citation19] consider the Hankel determinant
and refer to
as the second Hankel determinant. If f is univalent in
, then the sharp upper inequality
is known (see [Citation16]). Janteng et al. [Citation20] obtained sharp bounds for the functional
for the function f in the subclass
of
, which was introduced by Mac Gregor [Citation21] and consists of functions whose derivative has a positive real part. They demonstrated that
for each
. They also discovered the sharp second Hankel determinant for the classical subclass of
, namely the
and
which are the class of starlike and convex functions (see [Citation20]). These two classes have bounds of
and
. The Hankel determinants for starlike and convex functions with respect to symmetric points were recently discovered by Ready and Krishna [Citation22]. For functions belonging to subclasses of Ma–Minda starlike and convex functions, Lee et al. [Citation23] found the second Hankel determinant. Mishra and Gochhayat [Citation24] found the sharp bound to the nonlinear functional
for the subclass of analytic functions.
Deniz et al. [Citation25] discussed the upper bounds of for the classes
and
lately. Later, Altinkaya and Yalcin [Citation26], Caglar et al. [Citation27], Kanas et al. [Citation28], and Orhan et al. [Citation29] determined the upper bounds of
for several subclasses of Σ.
Gegenbauer polynomials, also known as ultraspherical polynomials , are orthogonal polynomials with regard to the weight function
on the interval
. They are particular instances of Jacobi polynomials and generalize Legendre and Chebyshev polynomials. They were given the name Leopold Gegenbauer. The following generating function of polynomials can be used to define them.
(7)
(7)
The recurrence relation is satisfied by the polynomials.
Gegenbauer polynomials are specific solutions to
differential equation. The equation becomes the Legendre equation when
, and the Gegenbauer polynomials become Legendre polynomials. When
, the equation becomes a Chebyshev differential equation and the Gegenbauer polynomials become second-order Chebyshev polynomials.
The Gegenbauer polynomials naturally emerge as extensions of Legendre polynomials in the context of potential theory and harmonic analysis. The Gegenbauer polynomial looks to be fascinating and significant in the subject of mathematical physics. Gegenbauer polynomials have lately been studied in the setting of mathematical physics by a number of authors (see [Citation30–35]).
Many scholars have recently started investigating bi-univalent functions related to orthogonal polynomials, with a few to name [Citation36–38]. As far as we know, there is minimal work-related to bi-univalent functions in the literatures for the Gegenbauer polynomial. The major objective of this work is to begin an investigation into the characteristics of bi-univalent functions linked with Gegenbauer polynomials.
Definition 1.1
Let be defined as follows:
A function
given by (Equation1
(1)
(1) ) is said to be in the class
if the following subordination conditions are fulfilled:
(8)
(8)
and
(9)
(9)
where
,
, 0<q<1 and the function g is given by (Equation2
(2)
(2) ).
We use the Gegenbauer polynomials expansions to determine the initial coefficient estimates, Fekete Szegö problem and estimate of Hankel determinant for a subclass of analytic and bi-univalent functions in this work.
Lemma 1.1
[Citation19]
Let , then
Lemma 1.2
[Citation39]
Let , then
for some complex number satisfying x, z,
and
.
2. Coefficient estimates for the class ![](//:0)
![](//:0)
Theorem 2.1
Let ,
,
, 0<q<1,
. Then
(10)
(10)
(11)
(11)
(12)
(12)
and for some
,
where
(13)
(13)
Proof.
Let given by (Equation1
(1)
(1) ) be in the class
. Then
(14)
(14)
and
(15)
(15)
where
and let
be define as
(16)
(16)
and
(17)
(17)
It follows from (Equation16
(16)
(16) ) and (Equation17
(17)
(17) ) that
(18)
(18)
and
(19)
(19)
From (Equation18
(18)
(18) ) and (Equation19
(19)
(19) ), applying
as given in (Equation7
(7)
(7) ), we see that
and
(20)
(20)
It the following follows from (Equation14
(14)
(14) ), (Equation20
(20)
(20) ) and (Equation15
(15)
(15) ) that
(21)
(21)
(22)
(22)
(23)
(23)
(24)
(24)
(25)
(25)
(26)
(26)
Adding (Equation21
(21)
(21) ) and (Equation24
(24)
(24) ), we have
(27)
(27)
and
(28)
(28)
Also, adding (Equation22
(22)
(22) ), (Equation25
(25)
(25) ) and applying (Equation27
(27)
(27) ) yields
(29)
(29)
Applying (Equation27
(27)
(27) ) in (Equation28
(28)
(28) ) gives
(30)
(30)
Putting (Equation30
(30)
(30) ) into (Equation29
(29)
(29) ) and with some calculations, we have
Applying triangular inequality and Lemma 1.1, we have
(31)
(31)
Subtracting (Equation25
(25)
(25) ) from (Equation22
(22)
(22) ) and with some calculations, we have
(32)
(32)
and
(33)
(33)
Applying triangular inequality, and Lemma 1.1, we have
(34)
(34)
Subtracting (Equation26
(26)
(26) ) from (Equation23
(23)
(23) ), we have
(35)
(35)
Applying triangular inequality and Lemma 1.1, we have
From (Equation32
(32)
(32) ), we have
By triangular inequality, we have
(36)
(36)
Suppose
then, we have
(37)
(37)
where
and suppose
then, we have
where
and
is given in (Equation13
(13)
(13) ).
Remark 2.1
If we let in the above result, we can get the same bounds for the function class
of analytic and bi-univalent functions, involving the Gegenbauer polynomials.
3. Second Hankel determinant for the class ![](//:0)
![](//:0)
Theorem 3.1
Let the function given by (Equation1
(1)
(1) ) be in the class
,
,
, 0<q<1,
. Then
Where
and
Proof.
From (Equation21(21)
(21) ) and (Equation35
(35)
(35) ), we have
With some calculations, we have
(38)
(38)
By using Lemma 1.2,
(39)
(39)
(40)
(40)
and
(41)
(41)
for some x, h, z, w with
and substituting
and
, and after some straightforward simplifications, we have
Let
, assume without any restriction that
,
and applying triangular inequality, we have
and equivalently, we have
(42)
(42)
where
where
. Now, we maximize the function
in the closed square
For a fixed value of t, the coefficients of the function
in (Equation42
(42)
(42) ) are dependent on m, thus the maximum of
with regard to m must be investigated, taking into account the cases when m = 0, r = 2 and
.
First Case: When m = 0,
It is obvious that the function
reaches its maximum at
and
(43)
(43)
Second Case: When m = 2, is expressed as a constant function with respect to m, we have
Third Case: When , let
and
in this case, then (Equation42
(42)
(42) ) can be of the form
(44)
(44)
where
and
. Now, we need to investigate the maximum of
(45)
(45)
By differentiating
partially, we have
These results reveal that
does not have a critical point in Λ, and so
does not have a critical point in the square Δ.
As a result, the function cannot have its maximum value in the interior of Δ. The maximum of
on the boundary of the square Δ will be investigated next.
For (also, for
) and
(46)
(46)
Now, since
, then we have
which implies that
is an increasing function. Therefore, for a fixed
and
, the maximum occurs at
. Thus, from (Equation46
(46)
(46) ),
(47)
(47)
For
(also, for
,
) and
(48)
(48)
(49)
(49)
We know that
, then
Therefore, the function
is an increasing function and the maximum occurs at
. From (Equation48
(48)
(48) ), we have
(50)
(50)
Hence, for every
, taking it from (Equation47
(47)
(47) ) and (Equation50
(50)
(50) ), we have
Therefore,
Since,
then
occurs on the boundary of square Δ.
Let defined by
(51)
(51)
Now, inserting the values of
and
into (Equation51
(51)
(51) ) and with some calculations, we have
By simplifying, we have
(52)
(52)
(53)
(53)
where
If
has a maximum value in the interior of
and by applying some elementary calculus, we have
Now, we need to examine the sign of the function
depending on the signs of
and
as follows:
First Result: Suppose and
then,
. This shows that
is an increasing function on the boundary of
that is m = 2. Therefore,
Second Result: If
and
then
(54)
(54)
at critical point
(55)
(55)
is a critical point of the function
. Now,
Therefore,
is the minimum point of the function
. Hence,
cannot have a maximum.
Third Result: If and
then
Therefore,
is a decreasing function on the interval
. Hence,
(56)
(56)
Fourth Result: If
and
Therefore,
. Hence,
is the maximum point of the function
and the maximum value occurs at
. Thus,
4. Conclusion
Many researchers have recently started investigating bi-univalent functions related to orthogonal polynomials as described in the introduction section. But as far as we know, there is minimal work-related with bi-univalent functions in the literatures for the Gegenbauer polynomial. In our present study make used of certain q-derivative operator and Gegenbauer polynomials, we have first defined a new subclass of analytic and bi-univalent functions. We have then obtained certain coefficients bounds, the Fekete–Szegö inequalities and upper bounds for the second-order Hankel determinant for our defined functions class.
Author's contributions
All authors jointly worked on the results, and they read and approved the final manuscript
Disclosure statement
The authors declare that they have no competing interest(s).
Data availability statement
No data were used to support this study.
Additional information
Funding
References
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