Coefficient estimates for two subclasses of analytic and bi-univalent functions
We introduce two new subclasses of the class $\sigma$ of analytic and bi-univalent functions in the open unit disk $U$. Furthermore, we obtain the estimates for the first two Taylor – Maclaurin coefficients $|a_2|$ and $|a_3|$ for the functions from these new subclasses.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507456034045952 |
|---|---|
| author | Lashin, A. Y. Лашин, А. Ю. |
| author_facet | Lashin, A. Y. Лашин, А. Ю. |
| author_sort | Lashin, A. Y. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:21:25Z |
| description | We introduce two new subclasses of the class $\sigma$ of analytic and bi-univalent functions in the open unit disk $U$. Furthermore,
we obtain the estimates for the first two Taylor – Maclaurin coefficients $|a_2|$ and $|a_3|$ for the functions from these new
subclasses. |
| first_indexed | 2026-03-24T02:09:36Z |
| format | Article |
| fulltext |
UDC 517.5
A. Y. Lashin (Mansoura Univ., Egypt)
COEFFICIENT ESTIMATES FOR TWO SUBCLASSES
OF ANALYTIC AND BI-UNIVALENT FUNCTIONS
КОЕФIЦIЄНТНI ОЦIНКИ ДЛЯ ДВОХ ПIДКЛАСIВ
АНАЛIТИЧНИХ ТА БIУНIВАЛЕНТНИХ ФУНКЦIЙ
We introduce two new subclasses of the class \sigma of analytic and bi-univalent functions in the open unit disk U. Furthermore,
we obtain the estimates for the first two Taylor – Maclaurin coefficients | a2| and | a3| for the functions from these new
subclasses.
Введено два нових пiдкласи класу \sigma аналiтичних та бiунiвалентних функцiй у вiдкритому одиничному крузi U.
Крiм того, отримано оцiнки для перших двох коефiцiєнтiв Тейлора – Маклорена | a2| та | a3| для функцiй iз цих
нових пiдкласiв.
1. Introduction. Let A denote the class of all functions of the form
f(z) = z +
\infty \sum
n=2
anz
n (1)
which are analytic in the open unit disk U = \{ z : z \in \BbbC and | z| < 1\} , \BbbC being, as usual, the set
of complex numbers. We also denote by S the subclass of all functions in A which are univalent in
U. Some of the important and well-investigated subclasses of the univalent function class S include
(for example) the class S\ast (\alpha ) of starlike functions of order \alpha in U and the class K(\alpha ) of convex
functions of order \alpha in U. By definition, we have
S\ast (\alpha ) :=
\biggl\{
f \in S : \Re
\biggl(
zf \prime (z)
f(z)
\biggr)
> \alpha , 0 \leq \alpha < 1, z \in U
\biggr\}
and
K(\alpha ) :=
\biggl\{
f \in S : f \prime (0) \not = 0, \Re
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
> \alpha , 0 \leq \alpha < 1, z \in U
\biggr\}
.
If f and g are analytic functions in U, we say that f is subordinate to g, written f(z) \prec g(z)
if there exists a Schwarz function \varphi , which (by definition) is analytic in U with \varphi (0) = 0 and
| \varphi (z)| < 1 for all z \in U, such that f(z) = g(\varphi (z)), z \in U. Furthermore, if the function g is
univalent in U, then we have the following equivalence:
f(z) \prec g(z)(z \in U) \leftrightarrow f(0) = g(0) and f(U) \subset g(U).
By using the method of differential subordination Obradovic [21] gave some criteria for univalence
expressing by \Re \{ f \prime (z)\} > 0, for the linear combinations
\alpha
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
+ (1 - \alpha )
1
f \prime (z)
.
In [24] Silverman investigated an expression involving the quotient of the analytic representations
of convex and starlike functions. Precisely, for 0 < b \leq 1 he considered the class
c\bigcirc A. Y. LASHIN, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9 1289
1290 A. Y. LASHIN
Gb :=
\Biggl\{
f \in A :
\bigm| \bigm| \bigm| \bigm| \bigm| 1 + zf
\prime \prime
(z)/f \prime (z)
zf \prime (z)/f(z)
- 1
\bigm| \bigm| \bigm| \bigm| \bigm| < b, z \in U
\Biggr\}
and proved that Gb \subset S\ast (2/(1 +
\surd
1 + 8b).
For each f \in S, the Koebe one-quarter theorem [11] ensures the image of U under f contains a
disk of radius 1/4. Thus every univalent function f \in S has an inverse f - 1, which is defined by
f - 1(f(z)) = z, z \in U,
and
f(f - 1(w)) = w, | w| < r0(f), r0(f) \leq
1
4
.
In fact, the inverse function g = f - 1 is given by
g(w) = f - 1(w) = w - a2w
2 + (2a22 - a3)w
3 - (5a22 - 5a2a3 + a4)w
4 + . . . .
A function f \in A is said to be bi-univalent in U if f and f - 1 are univalent in U. Let \sigma denote
the class of bi-univalent functions in U given by (1). The familiar Koebe function is not a member
of \sigma because it maps the unit disk U univalently onto the entire complex plane minus a slit along
the line - 1
4
to - \infty . Hence the image domain does not contain the unit disk U.
In 1985 Louis de Branges [3] proved the celebrated Bieberbach Conjecture which states that,
for each f(z) \in S given by the Taylor – Maclaurin series expansion (1), the following coefficient
inequality holds true:
| an| \leq n (n \in N - \{ 1\} ),
N being the set of positive integers. The class of analytic bi-univalent functions was first introduced
and studied by Lewin [15], where it was proved that | a2| < 1.51. Subsequently, Brannan and
Clunie [4] improved Lewin’s result to | a2| \leq
\surd
2. Brannan and Taha [6] and Taha [29] considered
certain subclasses of bi-univalent functions, similar to the familiar subclasses of univalent functions
consisting of strongly starlike, starlike and convex functions. They introduced bi-starlike functions
and bi-convex functions and found non-sharp estimates on the first two Taylor – Maclaurin coefficients
| a2| and | a3| . For further historical account of functions in the class \sigma , see the work by Srivastava et
al. [26] (see also [5, 6]). In fact, the above-cited recent pioneering work of Srivastava et al. [26] has
apparently revived the study of analytic and bi-univalent functions in recent years; it was followed
by such works as those by Frasin and Aouf [12], Xu et al. [31, 32], Hayami [14], and others (see, for
example, [1, 2, 7 – 10, 13, 16 – 19, 22, 23, 25, 27, 28, 30]).
In the present investigation, we derive estimates on the initial coefficients | a2| and | a3| of two
new subclass of the bi-univalent function class \sigma .
2. Coefficient estimates. In the section, it is assumed that \phi is an analytic function with positive
real part in the unit disk U, satisfying \phi (0) = 1, \phi \prime (0) > 0, and \phi (U) is symmetric with respect to
the real axis. Such a function has a Taylor series of the form
\phi (z) = 1 +B1z +B2z
2 +B3z
3 + . . . , B1 > 0.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
COEFFICIENT ESTIMATES FOR TWO SUBCLASSES OF ANALYTIC AND BI-UNIVALENT FUNCTIONS 1291
Suppose that u(z) and v(z) are analytic in the unit disk U with u(0) = v(0) = 0, | u(z)| < 1,
| v(z)| < 1, and suppose that
u(z) = b1z +
\infty \sum
n=2
bnz
n, v(z) = c1z +
\infty \sum
n=2
cnz
n, z \in U. (2)
It is well known that (see [20, p. 172])
| b1| \leq 1, | b2| \leq 1 - | b1| 2 , | c1| \leq 1, | c2| \leq 1 - | c1| 2 . (3)
By a simple calculation, we have
\phi (u(z)) = 1 +B1b1z + (B1b2 +B2b
2
1)z
2 + . . . , z \in U, (4)
and
\phi (v(w)) = 1 +B1c1w + (B1c2 +B2c
2
1)w
2 + . . . , w \in U. (5)
Definition 1. A function f \in \sigma is said to be in the class H\lambda
\sigma (\phi ), \lambda \geq 1 if the following
subordinations holds:
\lambda
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
+ (1 - \lambda )
1
f \prime (z)
\prec \phi (z), \lambda \geq 1, z \in U,
and
\lambda
\biggl(
1 +
wg\prime \prime (w)
g\prime (w)
\biggr)
+ (1 - \lambda )
1
g\prime (w)
\prec \phi (w), \lambda \geq 1, w \in U,
where g(w) := f - 1(w).
Theorem 1. If f given by (1) is in the class H\lambda
\sigma (\phi ), \lambda \geq 1, then
| a2| \leq
B1
\surd
B1\sqrt{}
4(2\lambda - 1)2B1 +
\bigm| \bigm| (\lambda + 1)B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm|
and
| a3| \leq
\left\{
B1
\lambda + 1
, if | B2| \leq B1,
4(2\lambda - 1)2B1 | B2| +B1
\bigm| \bigm| (\lambda + 1)B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm|
(\lambda + 1)
\bigl[
4(2\lambda - 1)2B1 +
\bigm| \bigm| (\lambda + 1)B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm| \bigr] , if | B2| > B1.
Proof. Let f \in H\lambda
\sigma (\phi ), \lambda \geq 1. Then there are analytic functions u, v : U \rightarrow U given by (2)
such that
\lambda
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
+ (1 - \lambda )
1
f \prime (z)
= \phi (u(z)), \lambda \geq 1, (6)
and
\lambda
\biggl(
1 +
wg\prime \prime (w)
g\prime (w)
\biggr)
+ (1 - \lambda )
1
g\prime (w)
= \phi (v(w)), \lambda \geq 1, (7)
where g(w) = f - 1(w). Since
\lambda
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
+ (1 - \lambda )
1
f \prime (z)
=
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1292 A. Y. LASHIN
= 1 + 2(2\lambda - 1)a2z +
\bigl[
(9\lambda - 3)a3 + 4(1 - 2\lambda )a22
\bigr]
z2 + . . .
and
\lambda
\biggl(
1 +
wg\prime \prime (w)
g\prime (w)
\biggr)
+ (1 - \lambda )
1
g\prime (w)
= 1 - 2(2\lambda - 1)a2w+
+
\bigl[
(10\lambda - 2)a22 - (9\lambda - 3)a3
\bigr]
w2 + . . . ,
it follows from (4), (5), (6) and (7) that
2(2\lambda - 1)a2 = B1b1, (8)
(9\lambda - 3)a3 + 4(1 - 2\lambda )a22 = B1b2 +B2b
2
1, (9)
- 2(2\lambda - 1)a2 = B1c1, (10)
(10\lambda - 2)a22 - (9\lambda - 3)a3 = B1c2 +B2c
2
1. (11)
From (8) and (10), we get
b1 = - c1, (12)
8(2\lambda - 1)2a22 = B2
1
\bigl(
b21 + c21
\bigr)
. (13)
By adding (11) to (9), further computations using (13) lead to\bigl[
2(1 + \lambda )B2
1 - 8(2\lambda - 1)2B2
\bigr]
a22 = B3
1(b2 + c2), (14)
(12), (14), together with (3), give that\bigm| \bigm| (1 + \lambda )B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm| \bigm| \bigm| a22\bigm| \bigm| \leq B3
1(1 -
\bigm| \bigm| b21\bigm| \bigm| ). (15)
From (8) and (15) we obtain
| a2| \leq
B1
\surd
B1\sqrt{}
4(2\lambda - 1)2B1 +
\bigm| \bigm| (1 + \lambda )B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm| .
Next, from (11) and (9), we have
(1 + \lambda )(9\lambda - 3)a3 = (5\lambda - 1)B1b2 + 2(2\lambda - 1)B1c2 + (9\lambda - 3)B2b
2
1.
Then, in view of (3), we get
(1 + \lambda ) | a3| \leq B1 + [| B2| - B1]
\bigm| \bigm| b21\bigm| \bigm| .
Notice that \bigm| \bigm| b21\bigm| \bigm| = 4(2\lambda - 1)2
B2
1
\bigm| \bigm| a22\bigm| \bigm| \leq 4(2\lambda - 1)2B1
4(2\lambda - 1)2B1 +
\bigm| \bigm| (1 + \lambda )B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm| ,
we obtain
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
COEFFICIENT ESTIMATES FOR TWO SUBCLASSES OF ANALYTIC AND BI-UNIVALENT FUNCTIONS 1293
| a3| \leq
\left\{
B1
\lambda + 1
, if | B2| \leq B1,
4(2\lambda - 1)2B1 | B2| +B1
\bigm| \bigm| (\lambda + 1)B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm|
(\lambda + 1)
\bigl[
4(2\lambda - 1)2B1 +
\bigm| \bigm| (\lambda + 1)B2
1 - 4(2\lambda - 1)2B2
\bigm| \bigm| \bigr] , if | B2| > B1.
Theorem 1 is proved.
If we set
\lambda = 1, \phi (z) =
1 + z
1 - z
= 1 + 2z + 2z2 + . . . , z \in U,
in Definition 1 of the bi-univalent function class H\lambda
\sigma (\phi ), we obtain the class of bi-convex functions
H\sigma (\phi ) given by Definition 2 below.
Definition 2. A function f \in \sigma is said to be in the class H\sigma (\phi ), if the following conditions
hold true:
1 +
zf \prime \prime (z)
f \prime (z)
\prec \phi (z), z \in U,
and
1 +
wg\prime \prime (w)
g\prime (w)
\prec \phi (w), w \in U,
where g(w) := f - 1(w).
Using the parameter setting of Definition 2 in the Theorem 1, we get the following corollary.
Corollary 1. Let the function f \in H\sigma (\phi ), be given by (1). Then
| a2| \leq 1 and | a3| \leq 1.
This is a special case of Theorem 4.1 (with \beta = 0) in Brannan and Taha [6].
Definition 3. A function f \in \sigma is said to be in the class K\sigma (\phi ) if and only if
1 +
zf \prime \prime (z)
f \prime (z)
zf \prime (z)
f(z)
\prec \phi (z),
1 +
wg\prime \prime (w)
g\prime (w)
wg\prime (w)
g(w)
\prec \phi (w),
where g(w) := f - 1(w).
Theorem 2. If f given by (1) is in the class K\sigma (\phi ), then
| a2| \leq
B1
\surd
B1\sqrt{}
B1 + | B2|
,
and
| a3| \leq
\left\{
B1
4
, if B1 \leq
1
4
,
\biggl[
1 - 1
4B1
\biggr]
B3
1
B1 + | B2|
+
B1
4
, if B1 >
1
4
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1294 A. Y. LASHIN
Proof. Let f \in K\sigma (\phi ). Then there are analytic functions u, v : U \rightarrow U given by (2) such that
1 +
zf \prime \prime (z)
f \prime (z)
zf \prime (z)
f(z)
= \phi (u(z)) (16)
and
1 +
wg\prime \prime (w)
g\prime (w)
wg\prime (w)
g(w)
= \phi (v(w)), (17)
where g(w) = f - 1(w). Since
1 +
zf \prime \prime (z)
f \prime (z)
zf \prime (z)
f(z)
= 1 + a2z + 4
\bigl(
a3 - a22
\bigr)
z2 + . . .
and
1 +
wg\prime \prime (w)
g\prime (w)
wg\prime (w)
g(w)
= 1 - a2w - 4
\bigl(
a3 - a22
\bigr)
w2 + . . . ,
it follows from (4), (5), (16) and (17) that
a2 = B1b1, (18)
4
\bigl(
a3 - a22
\bigr)
= B1b2 +B2b
2
1, (19)
- a2 = B1c1, (20)
- 4
\bigl(
a3 - a22
\bigr)
= B1c2 +B2c
2
1. (21)
From (18) and (20), we get
b1 = - c1 (22)
and
2a22 = B2
1
\bigl(
b21 + c21
\bigr)
. (23)
By adding (19) to (21), further computations using (23) lead to
2B2a
2
2 = - B3
1(b2 + c2), (24)
(22), (24), together with (3), give that
| B2|
\bigm| \bigm| a22\bigm| \bigm| \leq B3
1(1 -
\bigm| \bigm| b21\bigm| \bigm| ). (25)
From (18) and (25) we get
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
COEFFICIENT ESTIMATES FOR TWO SUBCLASSES OF ANALYTIC AND BI-UNIVALENT FUNCTIONS 1295
| a2| \leq
B1
\surd
B1\sqrt{}
B1 + | B2|
.
Next, from (19) and (21), we have
8a3 = 8a22 +B1(b2 - c2). (26)
From (3), (18), (22) and (26), it follows that
| a3| \leq a22 +
B1
4
\bigl(
1 -
\bigm| \bigm| b21\bigm| \bigm| \bigr) = \biggl(
1 - 1
4B1
\biggr)
a22 +
B1
4
\leq
\leq
\left\{
B1
4
, if B1 \leq
1
4
,\biggl[
1 - 1
4B1
\biggr]
B3
1
B1 + | B2|
+
B1
4
, if B1 >
1
4
.
Theorem 2 is proved.
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Received 12.09.15
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
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| id | umjimathkievua-article-1635 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:36Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/36/1e3f4ab7cb18bdef87fc85b274a91c36.pdf |
| spelling | umjimathkievua-article-16352019-12-05T09:21:25Z Coefficient estimates for two subclasses of analytic and bi-univalent functions Коефiцiєнтнi оцiнки для двох пiдкласiв аналiтичних та бiунiвалентних функцiй Lashin, A. Y. Лашин, А. Ю. We introduce two new subclasses of the class $\sigma$ of analytic and bi-univalent functions in the open unit disk $U$. Furthermore, we obtain the estimates for the first two Taylor – Maclaurin coefficients $|a_2|$ and $|a_3|$ for the functions from these new subclasses. Введено два нових пiдкласи класу $\sigma$ аналiтичних та бiунiвалентних функцiй у вiдкритому одиничному крузi $U$. Крiм того, отримано оцiнки для перших двох коефiцiєнтiв Тейлора – Маклорена $|a_2|$ та $|a_3|$ для функцiй iз цих нових пiдкласiв. Institute of Mathematics, NAS of Ukraine 2018-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1635 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 9 (2018); 1289-1296 Український математичний журнал; Том 70 № 9 (2018); 1289-1296 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1635/617 Copyright (c) 2018 Lashin A. Y. |
| spellingShingle | Lashin, A. Y. Лашин, А. Ю. Coefficient estimates for two subclasses of analytic and bi-univalent functions |
| title | Coefficient estimates for two subclasses of analytic and bi-univalent functions |
| title_alt | Коефiцiєнтнi оцiнки для двох пiдкласiв аналiтичних та бiунiвалентних функцiй |
| title_full | Coefficient estimates for two subclasses of analytic and bi-univalent functions |
| title_fullStr | Coefficient estimates for two subclasses of analytic and bi-univalent functions |
| title_full_unstemmed | Coefficient estimates for two subclasses of analytic and bi-univalent functions |
| title_short | Coefficient estimates for two subclasses of analytic and bi-univalent functions |
| title_sort | coefficient estimates for two subclasses of analytic and bi-univalent functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1635 |
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