Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

Let C be a set of complex numbers and let N={1,2,3,}= N 0 {0} be a set of positive integers. Let A be a class of functions of the form

f(z)=z+ n = 2 a n z n ,
(1.1)

which are analytic in the open unit disk D={zC:|z|<1}. Also, let S denote a subclass of all functions in A which are univalent in D (for details, see [1, 2]).

Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk D. However, the famous Koebe one-quarter theorem [1] ensures that the image of the unit disk D under every function fS contains a disk of radius 1/4. Thus, every univalent function fS has an inverse f 1 satisfying

f 1 ( f ( z ) ) =z(zD)

and

f ( f 1 ( w ) ) =w ( | w | < r 0 ( f ) ; r 0 ( f ) 1 4 ) ,

where

f 1 (w)=w a 2 w 2 + ( 2 a 2 2 a 3 ) w 3 ( 5 a 2 3 5 a 2 a 3 + a 4 ) w 4 +.
(1.2)

A function fA is said to be bi-univalent in D if both f and f 1 are univalent in D. Let σ denote the class of bi-univalent functions defined in the unit disk D. In 1967, Lewin [3] first introduced the class σ of bi-univalent functions and showed that | a 2 |1.51 for every fσ. Subsequently, Branan and Clunie [4] conjectured that | a 2 | 2 for fσ. Later, Netanyahu [5] proved that max f σ | a 2 |=3/4. The coefficient estimate problem for each of | a n | (nN{1,2}) is still an open problem.

Brannan and Taha [6] (see also [7]) introduced certain subclasses of a bi-univalent function class σ similar to the familiar subclasses S (α) and K(α) of starlike and convex functions of order α (0<α1), respectively (see [8]). Thus, following Brannan and Taha [6] (see also [7]), a function fA is in the class S σ [α] of strongly bi-starlike functions of order α (0<α1) if both functions f and f 1 are strongly starlike functions of order α. The classes S σ (α) and K σ (α) of bi-starlike functions of order α and bi-convex functions of order α, corresponding (respectively) to the function classes S (α) and K(α), were also introduced analogously. For each of the function classes S σ (α) and K σ (α), they found non-sharp estimates on the first two Taylor-Maclaurin coefficients | a 2 | and | a 3 | (for details, see [6, 7]).

An analytic function f is subordinate to an analytic function g, written fg, if there is an analytic function w with |w(z)||z| such that f=(g(w)). If g is univalent, then fg if and only if f(0)=g(0) and f(D)g(D). Ma and Minda [9] unified various subclasses of starlike and convex functions for which either of the quantities z f (z)/f(z) or 1+z f (z)/ f (z) is subordinate to a more general superordinate function. For this purpose, they considered an analytic function φ with positive real part in the unit disk D, φ(0)=1, φ (0)>0, and φ maps D onto a region starlike with respect to 1 and symmetric with respect to the real axis. The classes S (φ) and K(φ) of Ma-Minda starlike and Ma-Minda convex functions are respectively characterized by z f (z)/f(z)φ(z) or 1+z f (z)/ f (z)φ(z). A function f is bi-starlike of Ma-Minda type or bi-convex of Ma-Minda type if both f and f 1 are respectively Ma-Minda starlike or convex. These classes are denoted respectively by S σ (φ) and K σ (φ). Recently, Srivastava et al. [10], Frasin and Aouf [11] and Caglar et al. [12] introduced and investigated various subclasses of bi-univalent functions and found estimates on the coefficients | a 2 | and | a 3 | for functions in these classes. Very recently, Ali et al. [13], Kumar et al. [14], Srivastava et al. [15] and Xu et al.[16] unified and extended some related results in [7, 1012, 17] by generalizing their classes using subordination.

Motivated by Ali et al. [13] and Kumar et al. [14], we investigate the estimates for the initial coefficients | a 2 | and | a 3 | of bi-univalent functions of Ma-Minda type belonging to the classes H σ μ (λ,φ) and M σ γ (λ,μ,φ) defined in Section 2. Our results generalize several well-known results in [1014] and these are also pointed out.

2 Coefficient estimates

Throughout this paper, we assume that φ is an analytic univalent function with positive real part in D, φ(D) is symmetric with respect to the real axis and starlike with respect to φ(0)=1, and φ (0)>0. Such a function has series expansion of the form

φ(z)=1+ B 1 z+ B 2 z 2 + B 3 z 3 +( B 1 >0).
(2.1)

With this assumption on φ, we now introduce the following subclasses of Ma-Minda bi-univalent functions.

Definition 2.1 A function fσ given by (1.1) is said to be in the class H σ μ (λ,φ) if it satisfies

(1λ) ( f ( z ) z ) μ +λ f (z) ( f ( z ) z ) μ 1 φ(z)(λ1,μ1,zD)
(2.2)

and

(1λ) ( g ( w ) w ) μ +λ g (w) ( g ( w ) z ) μ 1 φ(w)(λ1,μ1,wD),
(2.3)

where the function g is given by

g(w)= f 1 (w)=w a 2 w 2 + ( 2 a 2 2 a 3 ) w 3 ( 5 a 2 3 5 a 2 a 3 + a 4 ) w 4 +.
(2.4)

We note that, for suitable choices λ, μ and φ, the class H σ μ (λ,φ) reduces to the following known classes.

  1. (1)

    H σ μ (λ, ( 1 + z 1 z ) α )= H σ μ (λ,α) (λ1, 0<α1, μ0) (see Caglar et al. [[12], Definition 2.1]);

  2. (2)

    H σ μ (λ, 1 + ( 1 2 β ) z 1 z )= H σ μ (λ,β) (λ1, 0β<1, μ0) (see Caglar et al. [[12], Definition 3.1]);

  3. (3)

    H σ 1 (λ,φ)= H σ (λ,φ) (λ1) (see Kumar et al. [[14], Definition 1.1]);

  4. (4)

    H σ μ (1,φ)= H σ μ (φ) (μ0) (see Kumar et al. [[14], Definition 2.1]);

  5. (5)

    H σ 1 (1,φ)= H σ (φ) (see Ali et al. [[13], p.345]);

  6. (6)

    H σ 1 (λ, ( 1 + z 1 z ) α )= H σ (λ,α) (λ1, 0<α1) (see Frasin and Aouf [[11], Definition 2.1]);

  7. (7)

    H σ 1 (λ, 1 + ( 1 2 β ) z 1 z )= H σ (λ,β) (λ1, 0β<1) (see Frasin and Aouf [[11], Definition 3.1]);

  8. (8)

    H σ 1 (1, ( 1 + z 1 z ) α )= H σ (α) (0<α1) (see Srivastava et al. [[10], Definition 1]);

  9. (9)

    H σ 1 (1, 1 + ( 1 2 β ) z 1 z )= H σ (β) (0β<1) (see Srivastava et al. [[10], Definition 2]).

For functions in the class H σ μ (λ,φ), the following estimates are obtained.

Theorem 2.1 Let the function f given by (1.1) be in the class H σ μ (λ,φ), λ1 and μ0. Then

| a 2 |min { B 1 λ + μ , 2 ( B 1 + | B 2 B 1 | ) ( 1 + μ ) ( 2 λ + μ ) }
(2.5)

and

| a 3 |{ min { B 1 2 λ + μ + B 1 2 ( λ + μ ) 2 , 2 ( B 1 + | B 2 B 1 | ) ( 1 + μ ) ( 2 λ + μ ) } , 0 μ < 1 , B 1 2 λ + μ + 2 | B 2 B 1 | ( 1 + μ ) ( 2 λ + μ ) , μ 1 .
(2.6)

Proof Since f H σ μ (λ,φ), there exist two analytic functions u,v:DD, with u(0)=v(0)=0, such that

(1λ) ( f ( z ) z ) μ +λ f (z) ( f ( z ) z ) μ 1 =φ ( u ( z ) )
(2.7)

and

(1λ) ( g ( w ) w ) μ +λ g (w) ( g ( w ) z ) μ 1 =φ ( v ( w ) ) .
(2.8)

Define the functions p and q by

p ( z ) = 1 + u ( z ) 1 u ( z ) = 1 + p 1 z + p 2 z 2 + and q ( z ) = 1 + v ( z ) 1 v ( z ) = 1 + q 1 z + q 2 z 2 + ,
(2.9)

or, equivalently,

u(z)= p ( z ) 1 p ( z ) + 1 = 1 2 ( p 1 z + ( p 2 p 1 2 2 ) z 2 + )
(2.10)

and

v(z)= q ( z ) 1 q ( z ) + 1 = 1 2 ( q 1 z + ( q 2 q 1 2 2 ) z 2 + ) .
(2.11)

It is clear that p and q are analytic in D and p(0)=q(0)=1. Since u,v:DD, the functions p and q have positive real part in D, and hence | p i |2 and | q i |2 (i=1,2,). By virtue of (2.7), (2.8), (2.10) and (2.11), we have

(1λ) ( f ( z ) z ) μ +λ f (z) ( f ( z ) z ) μ 1 =φ ( p ( z ) 1 p ( z ) + 1 )
(2.12)

and

(1λ) ( g ( w ) w ) μ +λ g (w) ( g ( w ) z ) μ 1 =φ ( q ( w ) 1 q ( w ) + 1 ) .
(2.13)

Using (2.10), (2.11), together with (2.1), we easily obtain

φ ( p ( z ) 1 p ( z ) + 1 ) =1+ 1 2 B 1 p 1 z+ ( 1 2 B 1 ( p 2 1 2 p 1 2 ) + 1 4 B 2 p 1 2 ) z 2 +
(2.14)

and

φ ( q ( w ) 1 q ( w ) + 1 ) =1+ 1 2 B 1 q 1 w+ ( 1 2 B 1 ( q 2 1 2 q 1 2 ) + 1 4 B 2 q 1 2 ) w 2 +.
(2.15)

Since fσ has the Maclaurin series given by (1.1), a computation shows that its inverse g= f 1 has the expansion given by (1.2). Also, since

f ( z ) = 1 + 2 a 2 z + 3 a 3 z 2 + and g ( w ) = 1 2 a 2 w + 3 ( 2 a 2 a 3 ) w 2 ,

it follows from (2.12)-(2.15) that

(λ+μ) a 2 = 1 2 B 1 p 1 ,
(2.16)
(2λ+μ) a 3 + ( μ 1 ) ( 2 λ + μ ) 2 a 2 2 = 1 2 B 1 ( p 2 1 2 p 1 2 ) + 1 4 B 2 p 1 2 ,
(2.17)
(λ+μ) a 2 = 1 2 B 1 q 1
(2.18)

and

(2λ+μ) a 3 + ( 3 + μ ) ( 2 λ + μ ) 2 a 2 2 = 1 2 B 1 ( q 2 1 2 q 1 2 ) + 1 4 B 2 q 1 2 .
(2.19)

From (2.16) and (2.18), we get

p 1 = q 1
(2.20)

and

8 ( λ + μ ) 2 a 2 2 = B 1 2 ( p 1 2 + q 1 2 ) .
(2.21)

Also, from (2.17) and (2.19), we obtain

(1+μ)(2λ+μ) a 2 2 = 1 2 B 1 ( p 2 + q 2 )+ 1 4 ( B 2 B 1 ) ( p 1 2 + q 1 2 ) ,

or

a 2 2 = 2 B 1 ( p 2 + q 2 ) + ( B 2 B 1 ) ( p 1 2 + q 1 2 ) 4 ( 1 + μ ) ( 2 λ + μ ) .
(2.22)

Since | p i |2 and | q i |2 (i=1,2), it follows from (2.21) and (2.22) that

| a 2 | B 1 λ + μ
(2.23)

and

| a 2 | 2 ( B 1 + | B 2 B 1 | ) ( 1 + μ ) ( 2 λ + μ ) ,
(2.24)

which yields the desired estimate on | a 2 | as asserted in (2.5).

Next, in order to find the bound on | a 3 |, by subtracting (2.19) from (2.17), we get

2(2λ+μ) ( a 3 a 2 2 ) = 1 2 B 1 ( p 2 q 2 )+ 1 4 ( B 2 B 1 ) ( p 1 2 q 1 2 ) .
(2.25)

Using (2.20) and (2.21) in (2.25), we have

a 3 = 1 4 ( 2 λ + μ ) B 1 ( p 2 q 2 )+ 1 4 ( λ + μ ) 2 B 1 2 p 1 2 ,

which evidently yields

| a 3 | B 1 2 λ + μ + B 1 2 ( λ + μ ) 2 .
(2.26)

On the other hand, by using (2.20) and (2.22) in (2.25), we obtain

a 3 = B 1 [ ( μ + 3 ) p 2 + ( 1 μ ) q 2 ] + ( B 2 B 1 ) ( p 1 2 + q 1 2 ) 4 ( 1 + μ ) ( 2 λ + μ ) ,
(2.27)

and applying | p i |2 and | q i |2 (i=1,2) for (2.27), we get

| a 3 | B 1 2 ( 2 λ + μ ) [ μ + 3 1 + μ + | 1 μ | 1 + μ ] + 2 | B 2 B 1 | ( 1 + μ ) ( 2 λ + μ ) .
(2.28)

Now, we consider the bounds on | a 3 | according to μ.

Case 1. If 0μ<1, then from (2.28)

| a 3 | 2 ( B 1 + | B 2 B 1 | ) ( 1 + μ ) ( 2 λ + μ ) .
(2.29)

Case 2. If μ1, then from (2.28)

| a 3 | B 1 2 λ + μ + 2 | B 2 B 1 | ( 1 + μ ) ( 2 λ + μ ) .
(2.30)

Thus, from (2.26), (2.29) and (2.30), we obtain the desired estimate on | a 3 | given in (2.6). This completes the proof of Theorem 2.1. □

Putting μ=1 and λ=μ=1 in Theorem 2.1, we respectively get the following Corollaries 2.1 and 2.2.

Corollary 2.1 If f H σ (λ,φ) (λ1), then

| a 2 |min { B 1 λ + 1 , B 1 + | B 2 B 1 | 2 λ + 1 }

and

| a 3 |{ min { B 1 2 λ + 1 + B 1 2 ( λ + 1 ) 2 , B 1 + | B 2 B 1 | 2 λ + 1 } , 0 μ < 1 , B 1 + | B 2 B 1 | 2 λ + 1 , μ 1 .

Corollary 2.2 If f H σ (φ), then

| a 2 |min { B 1 2 , B 1 + | B 2 B 1 | 3 }

and

| a 3 |{ min { B 1 3 + B 1 2 4 , B 1 + | B 2 B 1 | 3 } , 0 μ < 1 , B 1 + | B 2 B 1 | 3 , μ 1 .

Remark 2.1 The estimates of the coefficients | a 2 | and | a 3 | of Corollaries 2.1 and 2.2 are the improvement of the estimates obtained in [[14], Theorem 2.1] and [[13], Theorem 2.1], respectively.

Remark 2.2 If we set

φ(z)= 1 + ( 1 2 β ) z 1 z =1+2(1β)z+2(1β) z 2 +(0β<1)

in Corollaries 2.1 and 2.2, the results obtained improve the results in [[11], Theorem 3.2, inequalities (3.3) and (3.4)] and [[10], Theorem 2, inequality (3.3)], respectively.

Definition 2.2 Let γ C =C{0}, λ0 and μ0. A function fσ given by (1.1) is said to be in the class M σ γ (λ,μ,φ), if the following subordinations hold:

1+ 1 γ ( z f ( z ) + ( 2 λ μ + λ μ ) z 2 f ( z ) + λ μ z 3 f ( z ) ( 1 λ + μ ) f ( z ) + ( λ μ ) z f ( z ) + λ μ z 2 f ( z ) 1 ) φ(z)

and

1+ 1 γ ( w g ( w ) + ( 2 λ μ + λ μ ) w 2 g ( w ) + λ μ w 3 g ( w ) ( 1 λ + μ ) g ( w ) + ( λ μ ) w g ( w ) + λ μ w 2 g ( w ) 1 ) φ(w),

where the function g is defined by (2.4).

We note that, by choosing appropriate values for λ, μ, γ and φ, the class M σ γ (λ,μ,φ) reduces to several earlier known classes.

  1. (1)

    M σ γ (λ,0,φ)= N σ , γ λ (φ) (λ0, γ C ) (see Kumar et al. [[14], Definition 2.2]);

  2. (2)

    M σ 1 (0,0, 1 + ( 1 2 β ) z 1 z )= S σ (β) (0β<1) (see Brannan and Taha [[6], Definition 3.1]);

  3. (3)

    M σ 1 (1,0, 1 + ( 1 2 β ) z 1 z )= K σ (β) (0β<1) (see Brannan and Taha [[6], Definition 4.1]);

  4. (4)

    M σ 1 (0,0, ( 1 + z 1 z ) α )= S σ (α) (0<α1) (see Taha [7]).

For functions in the class M σ γ (λ,μ,φ), the following estimates are derived.

Theorem 2.2 Let γ C , λ0 and μ0. If f M σ γ (λ,μ,φ), then

| a 2 | | γ | B 1 B 1 | [ 2 ( 6 λ μ + 2 λ 2 μ + 1 ) ( 2 λ μ + λ μ + 1 ) 2 ] B 1 2 γ + 2 ( 2 λ μ + λ μ + 1 ) 2 ( B 1 B 2 ) |
(2.31)

and

| a 3 | | γ | ( B 1 + | B 2 B 1 | ) | 2 ( 6 λ μ + 2 λ 2 μ + 1 ) ( 2 λ μ + λ μ + 1 ) 2 | .
(2.32)

Proof If f M σ γ (λ,μ,φ), then there are analytic functions u,v:DD, with u(0)=v(0)=0, satisfying

1+ 1 γ ( z f ( z ) + ( 2 λ μ + λ μ ) z 2 f ( z ) + λ μ z 3 f ( z ) ( 1 λ + μ ) f ( z ) + ( λ μ ) z f ( z ) + λ μ z 2 f ( z ) 1 ) =φ ( u ( z ) )
(2.33)

and

1+ 1 γ ( w g ( w ) + ( 2 λ μ + λ μ ) w 2 g ( w ) + λ μ w 3 g ( w ) ( 1 λ + μ ) g ( w ) + ( λ μ ) w g ( w ) + λ μ w 2 g ( w ) 1 ) =φ ( v ( w ) ) .
(2.34)

Let p and q be defined as in (2.8), then it is clear from (2.33), (2.34), (2.9) and (2.10) that

1 + 1 γ ( z f ( z ) + ( 2 λ μ + λ μ ) z 2 f ( z ) + λ μ z 3 f ( z ) ( 1 λ + μ ) f ( z ) + ( λ μ ) z f ( z ) + λ μ z 2 f ( z ) 1 ) = φ ( p ( z ) 1 p ( z ) + 1 )
(2.35)

and

1 + 1 γ ( w g ( w ) + ( 2 λ μ + λ μ ) w 2 g ( w ) + λ μ w 3 g ( w ) ( 1 λ + μ ) g ( w ) + ( λ μ ) w g ( w ) + λ μ w 2 g ( w ) 1 ) = φ ( q ( w ) 1 q ( w ) + 1 ) .
(2.36)

It follows from (2.35), (2.36), (2.14) and (2.15) that

( 2 λ μ + λ μ + 1 ) a 2 = 1 2 B 1 p 1 γ ,
(2.37)
( 2 λ μ + λ μ + 1 ) 2 a 2 2 + 2 ( 6 λ μ + 2 λ 2 μ + 1 ) a 3 = γ [ 1 2 B 1 ( p 2 1 2 p 1 2 ) + 1 4 B 2 p 1 2 ] ,
(2.38)
( 2 λ μ + λ μ + 1 ) a 2 = 1 2 B 1 q 1 γ
(2.39)

and

[ 4 ( 6 λ μ + 2 λ 2 μ + 1 ) ( 2 λ μ + λ μ + 1 ) 2 ] a 2 2 2 ( 6 λ μ + 2 λ 2 μ + 1 ) a 3 = γ [ 1 2 B 1 ( q 2 1 2 q 1 2 ) + 1 4 B 2 q 1 2 ] .
(2.40)

Equations (2.37) and (2.39) yield

p 1 = q 1
(2.41)

and

8 ( 2 λ μ + λ μ + 1 ) 2 a 2 2 = B 1 2 γ 2 ( p 1 2 + q 1 2 ) .
(2.42)

From (2.38), (2.40), (2.41) and (2.42), it follows that

a 2 2 = γ 2 B 1 3 ( p 2 + q 2 ) 4 [ ( 2 ( 6 λ μ + 2 λ 2 μ + 1 ) ( 2 λ μ + λ μ + 1 ) 2 ) B 1 2 γ + ( 2 λ μ + λ μ + 1 ) 2 ( B 1 B 2 ) ]

which yields the desired estimate on | a 2 | as described in (2.31).

Similarly, it can be obtained from (2.38), (2.40) and (2.41) that

a 3 = γ B 1 [ p 2 ( 4 ( 6 λ μ + 2 λ 2 μ + 1 ) ( 2 λ μ + λ μ + 1 ) 2 ) + q 2 ( 2 λ μ + λ μ + 1 ) 2 ] 8 [ 2 ( 6 λ μ + 2 λ 2 μ + 1 ) ( 2 λ μ + λ μ + 1 ) 2 ] ( 6 λ μ + 2 λ 2 μ + 1 ) + 2 γ ( B 2 B 1 ) ( 6 λ μ + 2 λ 2 μ + 1 ) p 1 2 8 [ 2 ( 6 λ μ + 2 λ 2 μ + 1 ) ( 2 λ μ + λ μ + 1 ) 2 ] ( 6 λ μ + 2 λ 2 μ + 1 )

which easily leads to the desired estimate (2.32) on | a 3 |. □

Taking μ=0 in Theorem 2.2, we obtain the following corollary.

Corollary 2.3 [[14], Theorem 2.3]

If f N σ , γ λ (φ), then

| a 2 | | γ | B 1 B 1 | ( 1 + 2 λ λ 2 ) B 1 2 γ + ( 1 + λ ) 2 ( B 1 B 2 ) | and| a 3 | | γ | ( B 1 + | B 2 B 1 | ) | 1 + 2 λ λ 2 | .

Further, for γ=1, putting λ=0 and λ=1 in Corollary 2.3, respectively, we have the following Corollaries 2.4 and 2.5.

Corollary 2.4 [[13], Corollary 2.1]

If f M σ 1 (0,0,φ)=S T σ (φ), then

| a 2 | B 1 B 1 | B 1 2 + B 1 B 2 | and| a 3 | B 1 +| B 2 B 1 |.

Corollary 2.5 [[13], Corollary 2.2]

If f M σ 1 (1,0,φ)=C V σ (φ), then

| a 2 | B 1 B 1 2 | B 1 2 + 2 B 1 2 B 2 | and| a 3 | 1 2 ( B 1 + | B 2 B 1 | ) .

Remark 2.3 If we set

φ(z)= ( 1 + z 1 z ) α =1+2αz+2 α 2 z 2 +(0<α1)

and

φ(z)= 1 + ( 1 2 β ) z 1 z =1+2(1β)z+2(1β) z 2 +(0β<1)

in Corollaries 2.4 and 2.5, we obtain the results of Brannan and Taha [[6], Theorems 2.1, 3.1 and 4.1, respectively].