ESTIMATES FOR THE INITIAL COEFFICIENTS OF BI-UNIVALENT FUNCTIONS
arXiv:1203.5480v2 [math.CV] 28 Feb 2013
S. SIVAPRASAD KUMAR, VIRENDRA KUMAR, AND V. RAVICHANDRAN
A BSTRACT. A bi-univalent function is a univalent function defined on the unit disk with its inverse also univalent on the unit disk. In the present investigation, estimates for the initial coefficients are obtained for bi-univalent functions belonging to certain classes defined by subordination and relevant connections with earlier results are pointed out.
1. I NTRODUCTION Let A be the class of analytic functions defined on the open unit disk D := {z ∈ C : |z| < 1} and normalized by the conditions f (0) = 0 and f ′ (0) = 1. A function f ∈ A has Taylor’s series expansion of the form ∞
(1.1)
f (z) = z + ∑ an zn . n=2
The class of all univalent functions in the open unit disk D of the form (1.1) is denoted by S . Determination of the bounds for the coefficients an is an important problem in geometric function theory as they give information about the geometric properties of these functions. For example, the bound for the second coefficient a2 of functions in S gives the growth and distortion bounds as well as covering theorems. Some coefficient related problems were investigated recently in [1, 3, 8, 9, 17, 27]. Since univalent functions are one-to-one, they are invertible but their inverse functions need not be defined on the entire unit disk D. In fact, the famous Koebe one-quarter theorem ensures that the image of the unit disk D under every function f ∈ S contains a disk of radius 1/4. Thus, inverse of every function f ∈ S is defined on a disk, which contains the disk |z| < 1/4. It can also be easily verified that (1.2)
F(w) := f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 − (5a22 − 5a2 a3 + a4 )w4 + · · ·
in some disk of radius at least 1/4. A function f ∈ A is called bi-univalent in D if both f and f −1 are univalent in D. In 1967, Lewin [16] introduced the class σ of bi-univalent analytic functions and showed that the second coefficient of every f ∈ σ satisfy the inequality |a2 | ≤ 1.51. Let σ1 be the class of all functions f = φ ◦ ψ −1 where φ , ψ map D onto a domain containing D and φ ′ (0) = ψ ′ (0). In 1969, Suffridge [26] gave a function in σ1 ⊂ σ , satisfying a2 = 4/3 and conjectured that |a2 | ≤ 4/3 for all functions in σ . In 1969, Netanyahu [19] proved this conjecture for the subclass σ1 . Later in 1981, Styer and Wright [25] disproved the 2010 Mathematics Subject Classification. 30C45, 30C80. Key words and phrases. Univalent functions, bi-univalent functions, coefficient estimate, subordination. 1
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S. SIVAPRASAD KUMAR, V. KUMAR, AND V. RAVICHANDRAN
conjecture of Suffridge [26] by showing a2 > 4/3 for some function in σ . Also see [7] for an example to show σ 6= σ1 . For results √ on bi-univalent polynomial, see [14, 22]. In 1967, Brannan [4] conjectured that |a2 | ≤ 2 for f ∈ σ . In 1985, Kedzierawski [13, Theorem 2] proved this conjecture for a special case when the function f and f −1 are starlike functions. In 1985, Tan [28] obtained the bound for a2 namely |a2| ≤ 1.485 which is the best known estimate for functions in the class σ . For some open problems and survey, see [11, 23]. In 1985, Kedzierawski [13] proved the following: 1.5894, f ∈ S , f −1 ∈ S ; √ 2, f ∈ S ∗ , f −1 ∈ S ∗ ; |a2 | ≤ 1.507, f ∈ S ∗ , f −1 ∈ S ; 1.224, f ∈ K , f −1 ∈ S , where S ∗ and K denote the well-known classes of starlike and convex functions in S .
Let us recall now various definitions required in sequel. An analytic function f is subordinate to another analytic function g, written f ≺ g, if there is an analytic function w with |w(z)| ≤ |z| such that f = g ◦ w. If g is univalent, then f ≺ g if and only if f (0) = g(0) and f (D) ⊆ g(D). Let ϕ be an analytic univalent function in D with positive real part and ϕ (D) be symmetric with respect to the real axis, starlike with respect to ϕ (0) = 1 and ϕ ′ (0) > 0. Ma and Minda [18] gave a unified presentation of various subclasses of starlike and convex functions by introducing the classes S ∗ (ϕ ) and K (ϕ ) of functions f ∈ S satisfying z f ′ (z)/ f (z) ≺ ϕ (z) and 1 + z f ′′ (z)/ f ′ (z) ≺ ϕ (z) respectively, which includes several wellknown classes as special case. For example, when ϕ (z) = (1 + Az)/(1 + Bz) (−1 ≤ B < A ≤ 1), the class S ∗ (ϕ ) reduces to the class S ∗ [A, B] introduced by Janowski [12]. For 0 ≤ β < 1, the classes S ∗ (β ) := S ∗ ((1 + (1 − 2β )z)/(1 − z)) and K (β ) := K ((1 + (1 − 2β )z)/(1 − z)) are starlike and convex functions of order β . Further let S ∗ := S ∗ (0) and K := K (0) are the classes of starlike and convex functions respectively. The class of strongly starlike functions Sα∗ := S ∗ (((1 + z)/(1 − z))α ) of order α , 0 < α ≤ 1. Denote by R(ϕ ) the class of all functions satisfying f ′ (z) ≺ ϕ (z) and let R(β ) := R((1 + (1 − 2β )z)/(1 − z)) and R := R(0). For 0 ≤ β < 1, a function f ∈ σ is in the class Sσ∗ (β ) of bi-starlike function of order β , or Kσ (β ) of bi-convex function of order β if both f and f −1 are respectively starlike or convex functions of order β . For 0 < α ≤ 1, the function f ∈ σ is strongly bi-starlike function of order α if both the functions f and f −1 are strongly starlike functions of order α . The class of all such functions is denoted by Sσ∗,α . These classes were introduced by Brannan and Taha [6] in 1985 (see also [5]). They obtained estimates on the initial coefficients a2 and a3 for functions in these classes. Recently, Ali et al. [2] extended the results of Brannan and Taha [6] by generalizing their classes using subordination. For some related results, see [10, 24, 29]. For the various applications of subordination one can refer to [1, 3, 9, 17, 27] and the references cited therein.
Motivated by Ali et al. [2] in this paper estimates for the initial coefficient a2 of bi-univalent functions belonging to the class Rσ (λ , ϕ ) as well as estimates on a2 and a3 for functions in classes S ∗ σ (ϕ ) and Kσ (ϕ ), defined later, are obtained. Further work of Kedzierawski [13] actuates us to derive the estimates on initial coefficients a2 and a3 when f is in the some subclass of univalent functions and f −1 belongs to some other subclass of univalent functions. Our results generalize several well-known results in [2, 10, 13, 24], which are pointed out here.
ESTIMATES FOR THE INITIAL COEFFICIENTS OF BI-UNIVALENT FUNCTIONS
2. C OEFFICIENT
3
ESTIMATES
Throughout this paper, we assume that ϕ is an analytic function in D of the form (2.1)
ϕ (z) = 1 + B1 z + B2 z2 + B3 z3 + · · · with B1 > 0, and B2 is any real number.
Definition 2.1. Let λ ≥ 0. A function f ∈ σ given by (1.1) is in the class Rσ (λ , ϕ ), if it satisfies (1 − λ )
f (z) + λ f ′ (z) ≺ ϕ (z) z
and
(1 − λ )
F(w) + λ F ′ (w) ≺ ϕ (w). w
The class Rσ (λ , ϕ ) includes many earlier classes, which are mentioned below: (1) (2) (3) (4) (5)
Rσ (λ , (1 + (1 − 2β )z)/(1 − z)) = Rσ (λ , β ) (λ ≥ 1; 0 ≤ β < 1) [10, Definition 3.1] Rσ (λ , ((1 + z)/(1 − z))α ) = Rσ ,α (λ ) (λ ≥ 1; 0 < α ≤ 1) [10, Definition 2.1] Rσ (1, ϕ ) = Rσ (ϕ ) [2, p. 345]. Rσ (1, (1 + (1 − 2β )z)/(1 − z)) = Rσ (β ) (0 ≤ β < 1) [24, Definition 2] Rσ (1, ((1 + z)/(1 − z))α ) = Rσ ,α (0 < α ≤ 1) [24, Definition 1]
Our first result provides estimate for the coefficient a2 of functions f ∈ Rσ (λ , ϕ ).
Theorem 2.2. If f ∈ Rσ (λ , ϕ ), then
|a2 | ≤
(2.2)
r
B1 + |B1 − B2 | 1 + 2λ
Proof. Since f ∈ Rσ (λ , ϕ ), there exist two analytic functions r, s : D → D, with r(0) = 0 = s(0), such that f (z) F(w) (2.3) (1 − λ ) + λ f ′ (z) = ϕ (r(z)) and (1 − λ ) + λ F ′ (w) = ϕ (s(z)). z w Define the functions p and q by (2.4) p(z) =
1 + r(z) 1 + s(z) = 1 + p1 z + p2 z2 + p3 z3 + · · · and q(z) = = 1 + q1 z + q2 z2 + q3 z3 + · · · , 1 − r(z) 1 − s(z)
or equivalently, (2.5) and (2.6)
p(z) − 1 1 = r(z) = p(z) + 1 2
p1 p2 3 p21 2 p1 p21 p1 z + p2 − z + p3 + z +··· − p2 − 2 2 2 2
q1 q2 3 q21 2 q1 q21 q(z) − 1 1 q1 z + q2 − z + q3 + z +··· . = − q2 − s(z) = q(z) + 1 2 2 2 2 2
It is clear that p and q are analytic in D and p(0) = 1 = q(0). Also p and q have positive real part in D, and hence |pi | ≤ 2 and |qi | ≤ 2. In the view of (2.3), (2.5) and (2.6), clearly p(z) − 1 q(w) − 1 f (z) F(w) ′ ′ (2.7) (1 − λ ) and (1 − λ ) . + λ f (z) = ϕ + λ F (w) = ϕ z p(z) + 1 w q(w) + 1
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S. SIVAPRASAD KUMAR, V. KUMAR, AND V. RAVICHANDRAN
On expanding (2.1) using (2.5) and (2.6), it is evident that p(z) − 1 1 1 1 2 1 2 (2.8) ϕ = 1 + B1 p 1 z + B1 p 2 − p 1 + B2 p 1 z 2 + · · · . p(z) + 1 2 2 2 4 and
(2.9) ϕ
q(w) − 1 q(w) + 1
1 1 + B1 q 1 w + 2
=
1 1 2 1 2 B1 q2 − q1 + B2 q1 w2 + · · · . 2 2 4
Since f ∈ σ has the Maclaurin series given by (1.1), a computation shows that its inverse F = f −1 has the expansion given by (1.2). It follows from (2.7), (2.8) and (2.9) that 1 (1 + λ )a2 = B1 p1 , 2 (2.10)
(2.11)
1 1 1 (1 + 2λ )a3 = B1 p2 − p21 + B2 p21 , 2 2 4
1 −(1 + λ )a2 = B1 q1 , 2 1 1 2 1 2 (1 + 2λ )(2a2 − a3 ) = B1 q2 − q1 + B2 q21 . 2 2 4
Now (2.10) and (2.11) yield (2.12)
8(1 + 2λ )a22 = 2(p2 + q2 )B1 + (B2 − B1 )(p21 + q21 ).
Finally an application of the known results, |pi | ≤ 2 and |qi | ≤ 2 in (2.12) yields the desired estimate of a2 given by (2.2). Remark 2.3. Let ϕ (z) = (1 + (1 − 2β )z)/(1 −p z), 0 ≤ β < 1. So B1 = B2 = 2(1 − β ). When λ = 1, Theorem 2.2 gives the estimate |a2 | ≤ 2(1 − β )/3 for functions in the class Rσ (β ) which coincides with the result p [29, Corollary 2] of Xu et al. In particular if β = 0, then above estimate becomes |a2 | ≤ 2/3 ≈ 0.816 for functions f ∈ Rσ (0).√Since the estimate on |a2 | for f ∈ Rσ (0) is improved over the conjectured estimate |a2| ≤ 2 ≈ 1.414 for f ∈ σ , the functions in Rσ (0) are not the candidate for the sharpness of the estimate in the class σ . Definition 2.4. A function f ∈ σ is in the class S ∗ σ (ϕ ), if it satisfies z f ′ (z) ≺ ϕ (z) and f (z)
wF ′ (w) ≺ ϕ (w). F(w)
Note that for a suitable choice of ϕ , the class S ∗ σ (ϕ ), reduces to the following well-known classes: (1) S ∗ σ ((1 + (1 − 2β )z)/(1 − z)) = S ∗ σ (β ) (0 ≤ β < 1). (2) S ∗ σ (((1 + z)/(1 − z))α ) = Sσ∗,α (0 < α ≤ 1).
ESTIMATES FOR THE INITIAL COEFFICIENTS OF BI-UNIVALENT FUNCTIONS
Theorem 2.5. If f ∈ S ∗ σ (ϕ ), then s √ p B21 + B1 + |B2 − B1 | B1 B1 ,q |a2 | ≤ min B1 + |B2 − B1 |, 2 B2 + |B − B 1
and
where
B21 + B1 + |B2 − B1 | |a3| ≤ min B1 + |B2 − B1 |, ,R , 2 B1 − 4B2 1 R := . B1 + 3B1 max 1; 4 3B1
1
2|
5
Proof. Since f ∈ Sσ∗ (ϕ ), there are analytic functions r, s : D → D, with r(0) = 0 = s(0), such that z f ′ (z) wF ′ (w) (2.13) = ϕ (r(z)) and = ϕ (s(z)). f (z) F(w) Let p and q be defined as in (2.4), then it is clear from (2.13), (2.5) and (2.6) that z f ′ (z) wF ′ (w) p(z) − 1 q(z) − 1 (2.14) and . =ϕ =ϕ f (z) p(z) + 1 F(w) q(z) + 1 It follows from (2.14), (2.8) and (2.9) that 1 (2.15) a 2 = B1 p 1 , 2 1 1 B1 p 1 1 2 a2 + B1 p2 − p1 + B2 p21 , (2.16) 2a3 = 2 2 2 4 1 − a 2 = B1 q 1 2
(2.17) and
B1 q 1 1 1 1 2 (2.18) =− a2 + B1 q2 − q1 + B2 q21 . 2 2 2 4 The equations (2.15) and (2.17) yield 4a22 − 2a3
(2.19)
p1 = −q1 ,
(2.20)
8a22 = (p21 + q21 )B21
and B1 (p1 − q1 ) . 2 From (2.16), (2.18) and (2.21), it follows that (2.21)
(2.22)
2a2 =
8a22 = 2B1 (p2 + q2 ) + (B2 − B1 )(p21 + q21 ).
Further a computation using (2.16), (2.18), (2.15) and (2.19) gives (2.23)
16a22 = 2B21 q21 + 2B1 (p2 + q2 ) + (B2 − B1 )(p21 + q21 ).
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S. SIVAPRASAD KUMAR, V. KUMAR, AND V. RAVICHANDRAN
Similarly a computation using (2.16), (2.18), (2.21) and (2.20) yields 4(B21 − B2 + B1 )a22 = B31 (p2 + q2 ).
(2.24)
Now (2.22), (2.23) and (2.24) yield the desired estimate on a2 as asserted in the theorem. To find estimate for a3 subtract (2.16) from (2.18), to get B1 (q2 − p2 ) (2.25) − 4a3 = −4a22 + . 2 Now a computation using (2.23) and (2.25) leads to 16a3 = 2B21 q21 + 4B2 p2 + (B1 − B2 )(p21 + q21 ).
(2.26)
From (2.15), (2.16), (2.17) and (2.18), it follows that
B1 (3p2 + q2 ) + (B2 − B1 )p21 2 2(B1 − B2 ) 2 B1 q2 3B1 p2 − (2.28) p1 . = + 2 2 3B1 On applying the result of Keogh and Merkes [15](see also [20]), that is for any complex number v, |p2 − vp21 | ≤ 2 max{1; |2v − 1|}, along with |q2 | ≤ 2 in (2.28), we obtain B1 − 4B2 . (2.29) 4|a3 | ≤ B1 + 3B1 max 1; 3B1 4a3 =
(2.27)
Now the desired estimate on a3 follows from (2.26), (2.27) and (2.29) at once.
Remark 2.6. If f ∈ S ∗ σ (β ) (0 ≤ β < 1), then from Theorem 2.5 it is evident that
(2.30)
|a2 | ≤ min
np
2(1 − β ),
p
p 2(1 − β ), 0 ≤ β ≤ 1/2; (1 − β )(3 − 2β ) = p (1 − β )(3 − 2β ), 1/2 ≤ β < 1. o
p Recall Brannan and Taha’s [5, Theorem 3.1] coefficient estimate, |a2 | ≤ 2(1 − β ) for functions f ∈ S ∗ σ (β ), who claimed that their estimate is better than the estimate |a2 | ≤ 2(1 − β ), given by Robertson [21]. But their claim is true only when 0 ≤ β ≤ 1/2. Also it may noted that our estimate for a2 given in (2.30) improves the estimate given by Brannan and Taha [5, Theorem 3.1]. Further if we take ϕ (z) = ((1 + z)/(1 − z))α , 0 < α ≤ 1 in Theorem 2.5, we have B1 = 2α and B2 = 2α 2 . Then we obtain the estimate on a2 for functions f ∈ S ∗ σ ,α as: p p 2α 2α 2 2 4α − 2α , α + 2α , √ . |a2 | ≤ min =√ 1+α 1+α √ Note that Brannan and Taha [5, Theorem 2.1] gave the same estimate |a2 | ≤ 2α / 1 + α for functions f ∈ S ∗ σ ,α .
Definition 2.7. A function f given by (1.1) is said to be in the class Kσ (ϕ ), if f and F satisfy the subordinations wF ′′ (w) z f ′′ (z) ≺ ϕ (z) and 1 + ′ ≺ ϕ (w). 1+ ′ f (z) F (w)
ESTIMATES FOR THE INITIAL COEFFICIENTS OF BI-UNIVALENT FUNCTIONS
7
Note that Kσ ((1 + (1 − 2β )z)/(1 − z))) =: Kσ (β ) (0 ≤ β < 1).
Theorem 2.8. If f ∈ Kσ (ϕ ), then
and
s B2 + B + |B − B | B 1 2 1 1 1 , |a2 | ≤ min 6 2
B21 + B1 + |B2 − B1 | B1 (3B1 + 2) |a3 | ≤ min . , 6 12
Proof. Since f ∈ Kσ (ϕ ), there are analytic functions r, s : D → D, with r(0) = 0 = s(0), satisfying wF ′′ (w) z f ′′ (z) = = ϕ (s(z)). ϕ (r(z)) and 1 + f ′ (z) F ′ (w) Let p and q be defined as in (2.4), then it is clear from (2.31), (2.5) and (2.6) that z f ′′ (z) wF ′′ (w) p(z) − 1 q(z) − 1 (2.32) 1+ ′ and 1 + ′ . =ϕ =ϕ f (z) p(z) + 1 F (w) q(z) + 1 It follows from (2.32), (2.8) and (2.9) that 1 (2.33) 2a2 = B1 p1 , 2 1 1 1 2 (2.34) 6a3 = B1 p1 a2 + B1 p2 − p1 + B2 p21 , 2 2 4
(2.31)
1+
1 − 2a2 = B1 q1 2
(2.35) and (2.36)
6(2a22 − a3 ) =
Now (2.33) and (2.35) yield (2.37) and
1 1 1 2 −B1 q1 a2 + B1 q2 − q1 + B2 q21 . 2 2 4 p1 = −q1
B1 (p1 − q1 ) . 2 From (2.34), (2.36), (2.37) and (2.33), it follows that
(2.38)
(2.39)
4a2 =
48a22 = 2B21 p21 + 2B1 (p2 + q2 ) + (B2 − B1 )(p21 + q21 ).
In view of |pi | ≤ 2 and |qi | ≤ 2 together with (2.38) and (2.39) yield the desired estimate on a2 as asserted in the theorem. In order to find a3 , we subtract (2.34) from (2.36) and use (2.37) to obtain B1 (q2 − p2 ) . (2.40) − 12a3 = −12a22 + 2
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S. SIVAPRASAD KUMAR, V. KUMAR, AND V. RAVICHANDRAN
Now a computation using (2.39) and (2.40) leads to (2.41)
− 48a3 = 2B21 p21 − 4B2 p2 + (B1 − B2 )(p21 + q21 ).
From (2.38) and (2.40), it follows that B1 (q2 − p2 ) 3(p1 − q1 )2 B21 − . 2 16 Now (2.41) and (2.42) yield the desired estimate on a3 as asserted in the theorem. − 12a3 =
(2.42)
Remark 2.9. If f ∈ Kσ (β ) (0 ≤ β < 1), then theorem 2.8 gives (r ) (1 − β )(3 − 2β ) |a2 | ≤ min ,1−β = 1−β 3 and
(1 − β )(3 − 2β ) (1 − β )(4 − 3β ) |a3 | ≤ min , 3 3
(1 − β )(3 − 2β ) , 3 p which improves the Brannan and Taha’s [5, Theorem 4.1] estimates |a2 | ≤ 1 − β and |a3 | ≤ 1 − β for functions f ∈ Kσ (β ).
=
Theorem 2.10. Let f ∈ σ be given by (1.1). If f ∈ K (ϕ ) and F ∈ R(ϕ ), then r 3[B1 + |B2 − B1 |] |a2 | ≤ 8 and 5[B1 + |B2 − B1 |] . |a3 | ≤ 12
Proof. Since f ∈ K (ϕ ) and F ∈ R(ϕ ), there exist two analytic functions r, s : D → D, with r(0) = 0 = s(0), such that (2.43)
1+
z f ′′ (z) = ϕ (r(z)) and F ′ (w) = ϕ (s(z)). f ′ (z)
Let the functions p and q are defined by (2.4). It is clear that p and q are analytic in D and p(0) = 1 = q(0). Also p and q have positive real part in D, and hence |pi | ≤ 2 and |qi | ≤ 2. Proceeding as in the proof of Theorem 2.2 it follow from (2.43), (2.8) and (2.9) that
(2.44)
1 2a2 = B1 p1 , 2 1 1 2 1 2 6a3 − 4a2 = B1 p2 − p1 + B2 p21 , 2 2 4 1 −2a2 = B1 q1 2
and (2.45)
1 1 1 3(2a22 − a3 ) = B1 q2 − q21 + B2 q21 . 2 2 4
ESTIMATES FOR THE INITIAL COEFFICIENTS OF BI-UNIVALENT FUNCTIONS
9
A computation using (2.44) and (2.45), leads to (2.46)
a22
2(p2 + 2q2 )B1 + (p21 + 2q21 )(B2 − B1 ) = . 32
and 2(3p2 + 2q2 )B1 + (3p21 + 2q21 )(B2 − B1 ) . 48 Now the desired estimates on a2 and a3 , follow from (2.46) and (2.47) respectively.
(2.47)
a3 =
Remark 2.11. If f ∈ K (β ) and F ∈ R(β ), then from Theorem 2.10 we see that p |a2 | ≤ 3(1 − β )/2 and |a3 | ≤ 5(1 − β )/6. √ In particular if f ∈ K and F ∈ R, then |a2 | ≤ 3/2 ≈ 0.867 and |a3 | ≤ 5/6 ≈ 0.833.
Theorem 2.12. Let f ∈ σ be given by (1.1). If f ∈ S ∗ (ϕ ) and F ∈ R(ϕ ), then p 5[B1 + |B2 − B1 |] 7[B1 + |B2 − B1 |] , and |a3 | ≤ . |a2 | ≤ 3 9
Proof. Since f ∈ S ∗ (ϕ ) and F ∈ R(ϕ ), there exist two analytic functions r, s : D → D, with r(0) = 0 = s(0), such that z f ′ (z) = ϕ (r(z)) and F ′ (w) = ϕ (s(z)). f (z) Let the functions p and q be defined as in (2.4). Then (2.48)
(2.49)
z f ′ (z) =ϕ f (z)
p(z) − 1 p(z) + 1
It follow from (2.49), (2.8) and (2.9) that
and F ′ (w) = ϕ
q(w) − 1 . q(w) + 1
1 a 2 = B1 p 1 , 2 1 1 1 (2.50) 2a3 − a22 = B1 p2 − p21 + B2 p21 , 2 2 4 1 −2a2 = B1 q1 , 2 1 1 1 (2.51) 3(2a22 − a3 ) = B1 q2 − q21 + B2 q21 . 2 2 4 A computation using (2.50) and (2.51) leads to (2.52)
a22 =
2(3p2 + 2q2 )B1 + (3p21 + 2q21 )(B2 − B1 ) 36
and 2(6p2 + q2 )B1 + (6p21 + q21 )(B2 − B1 ) (2.53) a3 = . 36 Now the bounds for a2 and a3 are obtained from (2.52) and (2.53) respectively using the fact that |pi | ≤ 2 and |qi | ≤ 2.
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S. SIVAPRASAD KUMAR, V. KUMAR, AND V. RAVICHANDRAN
Remark 2.13. If f ∈ S ∗ (β ) and F ∈ R(β ), then from Theorem 2.12 it is easy to see that p |a2| ≤ 10(1 − β )/3 and |a3 | ≤ 14(1 − β )/9. √ In particular if f ∈ S ∗ and F ∈ R, then |a2 | ≤ 10/3 ≈ 1.054 and |a3 | ≤ 14/9 ≈ 1.56.
Theorem 2.14. Let f ∈ σ given by (1.1). If f ∈ S ∗ (ϕ ) and F ∈ K (ϕ ), then r B1 + |B2 − B1 | |a2 | ≤ 2 and |a3 | ≤
B1 + |B2 − B1 | . 2
Proof. Assuming f ∈ S ∗ (ϕ ) and F ∈ K (ϕ ) and proceeding in the similar way as in the proof of Theorem 2.10, it is easy to see that 1 a 2 = B1 p 1 , 2 (2.54)
3a3 − a22
1 1 2 1 = B1 p2 − p1 + B2 p21 , 2 2 4 1 −2a2 = B1 q1 , 2
(2.55)
8a22 − 6a3
1 1 2 1 = B1 q2 − q1 + B2 q21 . 2 2 4
A computation using (2.54) and (2.55) leads to (2.56)
a22 =
2(2p2 + q2 )B1 + (2p21 + q21 )(B2 − B1 ) 24
a3 =
2(8p2 + q2 )B1 + (8p21 + q21 )(B2 − B1 ) . 72
and (2.57)
Now using the result |pi | ≤ 2 and |qi | ≤ 2, the estimates on a2 and a3 follow from (2.56) and (2.57) respectively. Remark 2.15. Let f ∈ S ∗ (β ) and F ∈ K (β ), 0 ≤ β < 1. Then from Theorem 2.14, it is easy to see that p |a2 | ≤ 1 − β and |a3 | ≤ 1 − β . In particular if f ∈ S ∗ and F ∈ K , then |a2 | ≤ 1 and |a3 | ≤ 1.
Acknowledgements. The research is supported by a grant from University of Delhi.
ESTIMATES FOR THE INITIAL COEFFICIENTS OF BI-UNIVALENT FUNCTIONS
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S. SIVAPRASAD KUMAR, V. KUMAR, AND V. RAVICHANDRAN
[27] S. Supramaniam, R.M. Ali, S. K. Lee and V. Ravichandran, Convolution and differential subordination for multivalent functions, Bull. Malays. Math. Sci. Soc. (2) 32 (2009), no. 3, 351–360. [28] D. L. Tan, Coefficient estimates for bi-univalent functions, Chinese Ann. Math. Ser. A 5 (1984), no. 5, 559– 568. [29] Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and biunivalent functions, Appl. Math. Lett. 25 (2012), 990–994. D EPARTMENT OF A PPLIED M ATHEMATICS , D ELHI T ECHNOLOGICAL U NIVERSITY, D ELHI —110042, I N DIA
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