Gen. Math. Notes, Vol. 16, No. 2, June, 2013, pp.93-102
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Coefficient Estimates for Certain Subclasses
of Bi-Univalent Functions
Oana Crişan
Department of Mathematics, Babeş-Bolyai University
400084 Cluj-Napoca, Romania
E-mail: oana.crisan@ubbcluj.com
(Received: 8-3-13 / Accepted: 12-4-13)
Abstract
In this paper we introduce some new subclasses of the class σ of bi-univalent
functions and obtain bounds for the initial coefficients of the Taylor series
expansion of functions from the considered classes.
Keywords: Bi-univalent functions, Starlike functions with respect to symmetric points, Convex functions with respect to symmetric points, Close-toconvex functions, Quasi-convex functions.
1
Introduction
Let A denote the class of analytic functions in the unit disk
U = {z ∈ C : |z| < 1},
that have the form
f (z) = z +
∞
X
an z n
(1)
n=2
and let S be the class of all functions from A which are univalent in U.
If the functions f and g are analytic in U, then f is said to be subordinate
to g, written as f ≺ g or f (z) ≺ g(z), if there exists a Schwarz function ω (i.e.
analytic in U, with ω(0) = 0 and |ω(z)| < 1, z ∈ U) such that f (z) = g(ω(z)).
The Koebe one-quarter theorem states that the image of U under every
function f from S contains a disk of radius 1/4. Thus every such univalent
94
Oana Crişan
function has an inverse f −1 which satisfies
f −1 (f (z)) = z,
z∈U
and
1
|w| < r0 (f ), r0 (f ) ≥ .
4
A function f ∈ A is said to be bi-univalent in U if both f and the analytic
extension of f −1 to U are univalent in U. We denote by σ the class of all
bi-univalent functions in U.
In [7], the authors introduced the class S ∗ (φ) of the so-called Ma-Minda
starlike functions and the class C(φ) of Ma-Minda convex functions, unifying
several previously studied classes related to those of starlike and convex functions. The class S ∗ (φ) consists of all the functions f ∈ A satisfying the subordination zf 0 (z)/f (z) ≺ φ(z), whereas C(φ) is formed with functions f ∈ A
for which the subordination 1 + zf 00 (z)/f 0 (z) ≺ φ(z) holds. The function φ
considered here is analytic with positive real part in U, φ(0) = 1, φ0 (0) > 0
and with the property that φ maps U onto a domain starlike with respect to
1 and symmetric with respect to the real axis.
Lewin [6] studied the class of bi-univalent functions, obtaining the bound
1.51 for the modulus of the second coefficient |a2 |. In recent papers, several
authors considered a series of subclasses of the bi-univalent function class σ,
similar to the above mentioned classes of starlike and convex functions (see
[1], [4], [11] and also [2]). In these papers, bounds of the initial coefficients |a2 |
and |a3 | of the Taylor expansion (1) were investigated.
In [10], Sakaguchi introduced and investigated the class Ss∗ of starlike functions with respect to symmetric
points in U, consisting of functions f ∈ A that
zf 0 (z)
> 0, z ∈ U. The class of functions univalent
satisfy the condition < f (z)−f
(−z)
and starlike with respect to symmetric points includes the classes of convex
functions and odd starlike functions.
Similarly, in [13], Wang et al. introduced the class Cs of convex functions
with respect to symmetric
points, formed with all functions f ∈ A for which the
(zf 0 (z))0
inequality < f 0 (z)+f 0 (−z) > 0 holds for all z ∈ U. In the style of Ma and Minda,
Ravichandran (see [12]) defined the classes Ss∗ (φ) and Cs (φ). A function f ∈ A
2zf 0 (z)
≺ φ(z) holds, whereas
is said to be in Ss∗ (φ) if the subordination f (z)−f
(−z)
f (f −1 (w)) = w,
0
0
(z))
f ∈ A belongs to Cs (φ) if the relation f 02(zf
≺ φ(z) is true.
(z)+f 0 (−z)
In view of our following investigation, we give now the definitions of closeto-convex and quasi-convex functions. A function f ∈ A is called closeto-convex if there exists a convex function h such that <(f 0 (z)/h0 (z)) > 0,
z ∈ U, or equivalently, if there exists h starlike such that the inequality
<(zf 0 (z)/h(z)) > 0 holds true for all z ∈ U. A function f ∈ A is said to
be quasi-convex if <([zf 0 (z)]0 /h(z)) > 0, z ∈ U. The classes of close-to-convex
Coefficient Estimates for Certain Subclasses...
95
and quasi-convex functions were first introduced and studied by Kaplan [5]
and Noor [8], respectively.
In the present paper, we define first two new subclasses of bi-univalent
functions by using combinations of starlike and convex functions with respect
to symmetric points. We also introduce a subclass of bi-univalent functions
defined using a combination of close-to-convex and quasi-convex functions. For
functions belonging to each of the considered classes, we investigate the bounds
of the initial coefficients of their series expansions.
In order to derive our main results, we require the following lemma.
Lemma 1.1. ([9]) If p(z) = 1 + p1 z + p2 z 2 + p3 z 3 + · · · is an analytic
function in U with positive real part, then
|pn | ≤ 2
(n ∈ N = {1, 2, . . .})
and
2
2
p
1
p2 − ≤ 2 − |p2 | .
2
2
2
(2)
Main Results
In the following, let φ be an analytic function with positive real part in U,
with φ(0) = 1 and φ0 (0) > 0. Also, let φ(U) be starlike with respect to 1
and symmetric with respect to the real axis. Thus, φ has the Taylor series
expansion
φ(z) = 1 + B1 z + B2 z 2 + · · · , B1 > 0.
(3)
∗
Definition 2.1. A function f ∈ σ is said to be in the class Ss,σ
(α, φ) if the
following subordinations hold:
2[(1 − α)zf 0 (z) + αz(zf 0 (z))0 ]
≺ φ(z)
(1 − α)(f (z) − f (−z)) + αz(f 0 (z) + f 0 (−z))
and
2[(1 − α)wg 0 (w) + αw(wg 0 (w))0 ]
≺ φ(z),
(1 − α)(g(w) − g(−w)) + αw(g 0 (w) + g 0 (−w))
where g is the extension of f −1 to U.
∗
Remark 2.2. When α = 0, the class Ss,σ
(0, φ) represents the class of all
bi-univalent Ma-Minda starlike functions with respect to symmetric points,
∗
whereas when α = 1, Ss,σ
(1, φ) is the class of all bi-univalent Ma-Minda convex
functions with respect to symmetric points, introduced in [3].
96
Oana Crişan
∗
Theorem 2.3. If f ∈ Ss,σ
(α, φ) is given by (1) then
√
B1 B1
|a2 | ≤ p
2|(1 + 2α)B12 + 2(1 + α)2 (B1 − B2 )|
and
1
|a3 | ≤ B1
2
(4)
1
1
+
B1 .
1 + 2α 2(1 + α)2
(5)
∗
Proof. Let f ∈ Ss,σ
(α, φ) and g be the analytic extension of f −1 to U. Then
there exist two functions u and v, analytic in U, with u(0) = v(0) = 0, |u(z)| <
1 and |v(w)| < 1, z, w ∈ U, such that
2(zf 0 (z) + αz 2 f 00 (z))
= φ(u(z)),
(1 − α)(f (z) − f (−z)) + α(f 0 (z) + f 0 (−z))
2(wg 0 (w) + αw2 g 00 (w))
= φ(v(w)),
(1 − α)(g(w) − g(−w)) + α(g 0 (w) + g 0 (−w))
(z ∈ U),
(6)
(w ∈ U).
(7)
Next, define the functions p1 and p2 by
1 + u(z)
= 1 + c1 z + c2 z 2 + · · ·
1 − u(z)
(8)
1 + v(w)
= 1 + b1 w + b2 w 2 + · · · .
1 − v(w)
(9)
p1 (z) =
and
p2 (w) =
Since u and v are Schwarz functions, p1 and p2 are analytic functions in U ,
with p1 (0) = p2 (0) = 1 and which have positive real part in U. The equations
(8) and (9) then give
p1 (z) − 1
1
c21 2
u(z) =
=
c1 z + c2 −
z + ···
(10)
p1 (z) + 1
2
2
and
p2 (w) − 1
1
v(w) =
=
p2 (w) + 1
2
b21
2
b1 w + b 2 −
w + ··· .
2
(11)
Using (6) and (7), we have
2(zf 0 (z) + αz 2 f 00 (z))
=φ
(1 − α)(f (z) − f (−z)) + α(f 0 (z) + f 0 (−z))
p1 (z) − 1
p1 (z) + 1
(12)
and
2(wg 0 (w) + αw2 g 00 (w))
=φ
(1 − α)(g(w) − g(−w)) + α(g 0 (w) + g 0 (−w))
p2 (w) − 1
p2 (w) + 1
.
(13)
Coefficient Estimates for Certain Subclasses...
Next, the equations (3), (10) and (11) lead to
p1 (z) − 1
1
1
c21
1
2
φ
= 1 + B1 c1 z + B1 c2 −
+ B2 c1 z 2 + · · ·
p1 (z) + 1
2
2
2
4
97
(14)
and
φ
p2 (w) − 1
p2 (w) + 1
1
1
b21
1
2
= 1 + B1 b1 w + B1 b2 −
+ B2 b1 w2 + · · ·
2
2
2
4
(15)
Because the inverse g of f is given by
g(w) = f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 + · · · ,
we find that
2(zf 0 (z) + αz 2 f 00 (z))
= 1+2(1+α)a2 z+2(1+2α)a3 z 2 +· · ·
(1 − α)(f (z) − f (−z)) + α(f 0 (z) + f 0 (−z))
(16)
and
2(wg 0 (w) + αw2 g 00 (w))
(1 − α)(g(w) − g(−w)) + α(g 0 (w) + g 0 (−w))
= 1 − 2(1 + α)a2 z + 2(1 + 2α)(2a22 − a3 )z 2 + · · · .
(17)
Therefore, from the combination of equations (12)-(17), we deduce
1
2(1 + α)a2 = B1 c1 ,
2
1
c21
1
2(1 + 2α)a3 = B1 c2 −
+ B2 c21 ,
2
2
4
1
−2(1 + α)a2 = B1 b1
2
and
2(1 +
2α)(2a22
1
b21
1
+ B2 b21 .
− a3 ) = B1 b2 −
2
2
4
(18)
(19)
(20)
(21)
Equations (18) and (20) evidently show
c1 = −b1
(22)
16(1 + α)2 a22
.
B12
(23)
and from (20) we get
b21 =
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Oana Crişan
By adding (19) with (21) and also using (22) and (23), it follows that
a22 =
B13 (b2 + c2 )
,
8[(1 + 2α)B12 + 2(1 + α)2 (B1 − B2 )]
and now, by applying Lemma 1.1 for the coefficients b2 and c2 , the last equations gives the bound of |a2 | from (4).
For the estimation of |a3 |, we subtract (21) from (19) and, in view of (18)
and (22), it follows that
a3 =
1
1
B1 (b2 − c2 ).
B12 b21 +
2
16(1 + α)
8(1 + 2α)
The bound of |a3 |, as asserted in (5), is now a consequence of Lemma 1.1, and
this completes our proof.
Definition 2.4. A function f ∈ σ is said to be in the class Ls,σ (α, φ) if the
following subordinations hold:
α 1−α
2zf 0 (z)
2(zf 0 (z))0
≺ φ(z)
f (z) − f (−z)
(f 0 (z) + f 0 (−z))
and
2wg 0 (w)
g(w) − g(−w)
α 2(wg 0 (w))0
(g 0 (w) + g 0 (−w))
1−α
≺ φ(w)
where g is the extension of f −1 to U.
Theorem 2.5. If f ∈ Ls,σ (α, φ) is given by (1) then
√
B1 B1
|a2 | ≤ p
2|(α2 − 3α + 3)B12 + 2(2 − α)2 (B1 − B2 )|
and
1
|a3 | ≤ B1
2
1
1
B1 +
2
2(2 − α)
3 − 2α
.
Proof. Let f ∈ Ls,σ (α, φ) and g = f −1 . We have
α 1−α
2zf 0 (z)
2(zf 0 (z))0
f (z) − f (−z)
(f 0 (z) + f 0 (−z))
= 1 + 2(2 − α)a2 z + [2(3 − 2α)a3 − 2α(1 − α)a22 ]z 2 + · · ·
and
1−α
α 2wg 0 (w)
2(wg 0 (w))0
g(w) − g(−w)
(g 0 (w) + g 0 (−w))
= 1 − 2(2 − α)a2 z + [2(3 − 2α)(2a22 − a3 ) − 2α(1 − α)a22 ]z 2 + · · · .
(24)
(25)
99
Coefficient Estimates for Certain Subclasses...
Since f ∈ Ls,σ (α, φ), there are two Schwarz functions u and v such that
2(zf 0 (z))0
2(wg 0 (w))0
=
φ(u(z))
and
≺ φ(u(w)).
f 0 (z) + f 0 (−z)
g 0 (w) + g 0 (−w)
Proceeding now in the same manner as in the proof of Theorem 2.3, we obtain
1
2(2 − α)a2 = B1 c1 ,
2
1
c21
1
2
2(3 − 2α)a3 − 2α(1 − α)a2 = B1 c2 −
+ B2 c21 ,
2
2
4
1
−2(2 − α)a2 = B1 b1
2
(26)
(27)
(28)
and
2(3 −
2α)(2a22
− a3 ) − 2α(1 −
α)a22
1
1
b21
+ B2 b21 .
= B1 b2 −
2
2
4
(29)
Equations (26) and (28) obviously yield c1 = −b1 , and after some further
calculations using (27)-(29) we find
a22 =
B13 (b2 + c2 )
8[(α2 − 3α + 3)B12 + 2(2 − α)2 (B1 − B2 )]
a3 =
1
1
B12 c21 +
B1 (c2 − b2 ).
2
16(20 − α)
8(3 − 2α)
and
After applying Lemma (1.1), the estimates in (24) and (25) follow.
Definition 2.6. A function f ∈ σ is said to be in the class Qs,σ (α, φ) if
following subordinations hold:
and
(1 − α)zf 0 (z) + αz(zf 0 (z))0
≺ φ(z)
(1 − α)h(z) + αzh0 (z)
(30)
(1 − α)wg 0 (w) + αw(wg 0 (w))0
≺ φ(w)
(1 − α)h(w) + αwh0 (w)
(31)
(1 − α)zh0 (z) + αz(zh0 (z))0
<
>0
(1 − α)h(z) + αzh0 (z)
(32)
where h satisfies
and g is the analytic continuation of f −1 to U.
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Oana Crişan
Remark 2.7. When α = 0, the class Qs,σ (0, φ) consists of all bi-univalent
close-to-convex functions of Ma-Minda type, whereas when α = 1, Qs,σ (1, φ)
represents the class of all bi-univalent quasi-convex functions of Ma-Minda
type.
Theorem 2.8. If f ∈ Qs,σ (α, φ) is given by (1), then
s
B12 + B13 + 4|B1 − B2 |
|a2 | ≤
|3(1 + 2α)B12 + 4(1 + α)2 (B1 − B2 )|
and
|a3 | ≤ B1
1
B1
+
1 + 2α 4(1 + α)2
+
4|B1 − B2 |
3(1 + 2α)B1
(33)
(34)
Proof. If f ∈ Qs,σ (α, φ), then there exist two Schwarz functions u and v such
that
(1 − α)zf 0 (z) + αz(zf 0 (z))0
= φ(u(z))
(1 − α)h(z) + αzh0 (z)
and
(1 − α)wg 0 (w) + αw(wg 0 (w))0
= φ(v(w)).
(1 − α)h(w) + αwh0 (w)
A computation gives
(1 − α)f 0 (z) + α(zf 0 (z))0
= 1 + (1 + α)(2a2 − h2 )z
h0 (z)
+ [3(1 + 2α)a3 − 2(1 + α)2 a2 h2 − (1 + 2α)h3 + (1 + α)2 h22 ]z 2 + · · ·(35)
and
(1 − α)g 0 (w) + α(wg 0 (w))0
= 1 − (1 + α)(2a2 + h2 )
h0 (w)
+ [3(1 + 2α)(2a22 − a3 ) + 2(1 + α)2 h2 a2 − (1 + 2α)h3 + (1 + α)2 h22 ]z 2 + ·(36)
·· .
From (35) and (36) in combination with (10), (11), (14) and (15) it now follows
that
1
(1 + α)(2a2 − h2 ) = B1 c1 ,
(37)
2
1
c21
1
2
2 2
3(1 + 2α)a3 − 2(1 + α) a2 h2 − (1 + 2α)h3 + (1 + α) h2 = B1 (c2 − ) + B2 c21 ,
2
2
4
(38)
1
−(1 + α)(a2 + h2 ) = B1 b1 ,
(39)
2
and
1
b2 1
3(1+2α)(2a22 −a3 )+2(1+α)2 a2 h2 −(1+2α)h3 +(1+α)2 h22 = B1 (b2 − 1 )+ B2 b21 .
2
2
4
(40)
101
Coefficient Estimates for Certain Subclasses...
By squaring and adding equations (37) and (39) we obtain
c21 + b21 =
8(1 + α)2 (4a21 + h22 )
.
B12
(41)
Because h satisfies (32), there exists an analytic function p(z) = 1 + p1 z +
p2 z 2 + · · · with <p(z) > 0 such that
(1 − α)zh0 (z) + αz(zh0 (z))0
= p(z).
(1 − α)h(z) + αzh0 (z)
The above relation yields
h2 =
p1
1+α
and h3 =
p2 + p21
.
2(1 + 2α)
(42)
From relations (38), (40), (41) and (42) it now follows that
a22 =
2(p2 − p21 )B12 + (c2 + b2 )B13 − 4p21 (B1 − B2 )
.
4[3(1 + 2α)B13 + 4(1 + α)2 (B1 − B2 )]
(43)
By applying Lemma 1.1, we get the desired estimate of |a2 | from (33).
For the estimation of |a3 |, observe first that from (37) and (39) we obtain
a2 =
(c1 − b1 )B1
8(1 + α)
and c21 − b21 = −
4(1 + α)(c1 − b1 )h2
.
B1
(44)
Using (38), (40), (42) and (44) lead to
a3 =
(c1 − b1 )2 B12 [(c1 − b1 )p1 + c2 − b2 ]B1 (c1 − b1 )(B1 − B2 )p1
+
+
,
64(1 + α)2
12(1 + 2α)
6(1 + 2α)B1
which, in view of Lemma 1.1, yields the bound of |a3 | as asserted in (34)
3
Acknowledgements
This work was possible with the financial support of the Sectoral Operation
Programme for Human Resources Development 2007-2013, co-financed by the
European Social Fund, under the project number POSDRU/107/1.5/S/76841
with the title ”Modern Doctoral Studies: Internationalization and Interdisciplinarity”.
102
Oana Crişan
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