1. Introduction
Consider the class of functions A defined as
where f(z) is analytic in the open unit disk
Let s be the subset of functions f∈A that are univalent in .
The Koebe one-quarter theorem [3] states that for every f∈s, image of under f contains a disk of radius
. Thus, every f∈s has an inverse f-1, defined as
and
with
A function f∈A is bi-univalent in
if both f and f-1 are univalent in
. Let ∑ be the set of bi-univalent functions in given by [1,6].
Several authors have investigated bounds for various subclasses of biunivalent functions [2-5,7-14]. However, the estimation of the Taylor-Maclaurin coefficients |an| for
remains an open problem.
In 1983, Salagean [13] introduced the differential operator
defined by
That is, the Salagean differential operator Dk applied to a function f(z) is defined recursively as the derivative of Dk-1f(z) multiplied by z. This operator is employed in the study of analytic functions, particularly for estimating coefficients of certain classes of functions. It should be noted that
This paper aims to introduce a new subclass of the function class ∑ associated with the Salagean differential operator and derive estimates for the coefficients |a2| and |a3| functions within these new subclasses of the function class ∑.
2. The subclass
In this section, we introduce and investigate the general subclass
.
Definition 2.1: Let the functions
be so constrained that
Also let the function f, defined by (1.1), be in the analytic function class A. we say that
if the following conditions are satisfied:
and
where the function g(w) is given by (1.2).
Remark 2.2: There are many choices of the functions h(z) and p(z) which would provide interesting subclasses of the analytic function class A.
- For
and k=l=0, we have
and k=1,l=0,
where the classes
and
of bi-starlike functions of order a and bi-convex functions of order a corresponding, was introduced and studied by Brannan and Taha [1].
- For
and k=l=0, we have
and k=1,l=0,
where the classes
and
of bi-starlike functions of order b and bi-convex functions of order b corresponding, was introduced and studied by Brannan and Taha [1].
- For
we have
and taking
,
where the classes
and
was introduced and studied by J.Jothibaso [6].
3. Coefficient estimates
For proof of the theorem, we need the following lemma.
Lemma 3.1: (see [3]). If p∈P, then |ck|≤2 for each k, where p is the family of all functions p(z) analytic in
for which
for
.
Theorem 3.2: Let f(z) given by the Taylor Maclaurin series expansion (1.1) be in the class
. Then,
and
Proof. First of all, it follows from the conditions (2.1) and (2.2) that
and
where the function g(w) is given by (1.2). respectively, where h(z) and p(w) satisfy the conditions of Definition (2.1). Furthermore, the functions h(z) and p(w) have the following Taylor-Maclaurin series expansions:
and
Now, equating the coefficients in (3.2) and (3.3), we get
From (3.4) and (3.6), we obtain
and
Also, From (3.5) and (3.7), we find that
Therefore, we find from the equations (3.8) and (3.9) that
and
respectively. So we get the desired estimate on the coefficient |a2| as asserted in (3.1). Next, to find the bound on the coefficient |a3|, we subtract (3.7) from (3.5). We thus get
Upon substituting the value of
from (3.8) into (3.10), it follows that
We thus find that
On the other hand, upon substituting the value of
from (3.9) into (3.10), it follows that
Consequently, we have
This evidently completes the proof of Theorem 3.2.
4. Corollaries and consequences
By setting
, (0<α≤1) in Theorem 3.2. we get the following consequence.
Corollary 4.1: Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class
, (0<1). Then
and
Remark 4.2: Corollary 4.1 is an improvement of the following estimates obtained by J.Jothibaso [6].
Corollary 4.3: (see[6]) Let the function f(z) given by the Taylor-Maclaurin series expansion (1) be in the bi-univalent function class
. Then
and
Remark 4.4: It is easy to see that [(i)]
1. For the coefficient |a2|, If k = 0 and 0<α≤1, we have
In another case, if k = 1,2,3… and 0<α≤1, we have
2. For the coefficient |a3|, we make the following observations: If k=0 and 0<α≤1, we have
and
Then
In another case, if k = 1,2,3… and 0<1, we have
Then
Thus Theorem 3.2 clearly improves the estimate of coefficients |a2| and |a3| obtained by J.Jothibaso [6].
By setting
, k=λ=0 in Theorem 3.2. we get the following consequence.
Corollary 4.5: Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class
. Then
Remark 4.6: Corollary 4.5 is an improvement of the following estimates obtained by the coefficient estimates for a well-known class
of strongly bi-starlike functions of order α as in [1].
By setting
in Theorem 3.2. we get the following consequence.
Corollary 4.7: Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class
, (0≤<1). Then
and
Remark 4.8: Corollary 4.7 is an improvement of the following estimates obtained by J.Jothibaso [6].
Corollary 4.9: (see[6]) Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class
. Then
and
Corollary 4.10: By setting k=λ=0 in Corollary 4.7, we have the coefficients estimates for the well-known class
of bi-starlike functions of order as in [1]. Further, taking k=1,λ=0 in Corollary 4.7, we obtain the estimates for the well-known class k∑(β) of bi-convex functions of order and our results reduce to. [1].
Conclusion
This paper introduces a new subclass of the function class ∑ involving analytic and bi-univalent functions associated with the Salagean differential operator. Our study provides estimates for the Taylor-Maclaurin coefficients |a2| and |a3| for functions within this subclass, contributing to the advancement of knowledge in this area. The findings enhance recent research in the field and open up new avenues for further exploration and development in the theory of analytic and bi-univalent functions.