NEW SUBCLASSES OF ANALYTIC AND BI-UNIVALENT FUNCTIONS INVOLVING BECKER

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202

Vol. 07, No. 01 (2015), pp. 39–49 http://jurnal.bull-math.org

NEW SUBCLASSES OF ANALYTIC AND

BI-UNIVALENT FUNCTIONS INVOLVING

  

BECKER

_Abstract.

Saibah Siregar and Christina Pushpa Joan

In the present paper, we introduce two new subclasses of the function class

  Σ of bi-univalent functions involving Becker defined in the open unit disc U := _{2 3}

{z : z ∈ C, |z| < 1}. The estimates on the coefficients |a | and |a | for functions

in these new subclasses of the function class Σ are obtained in our investigation.

  1. INTRODUCTION Let A be the class of the function of the form _X ∞ k f (z) = z + a z (1)

  k k =2

  which are analytic in the open unit disc U = {z : |z| < 1}. Further, by S we shall denote the class of all functions in A which are univalent in U.

  α

  Using the fractional derivative of order α, D [1], Owa and Srivastava

  z α

  [2] introduced the operator Ω : A → A which is known as an extension of fractional derivative and fractional integral, as follows

  Received 05-08-2011, Accepted 27-12-2014. 2010 Mathematics Subject Classification : 30C45 Key words and Phrases

  : Analytic functions, Univalent functions, Bi-univalent functions, Starlike and convex function, Coefficients bounds Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker α α α

  f D f α Ω (z) = Γ(2 − α)z (z), 6= 2, 3, 4, ...

  

z

_X ∞

  Γ(k + 1)Γ(2 − α)

  k

  a z = z + k

  Γ(k + 1 − α)

  k =2

  = ϕ(2, 2 − α; z) ∗ f (z). (2) f Note that Ω (z) = f (z).

  n,ν

  For function f in A, Darus and Faizal [9] defined D (α, β, µ)f (z) :

  λ

  A → A the linear fractional differential operator as follows

  0,ν

  I (α, γ, µ)f (z) = f (z)

  λ

  ν µ − µ + β − λ + λ

  1,ν α α ′

  I (α, γ, µ)f (z) = ( )(Ω f (z)) + ( )z(Ω f (z))

  λ

  ν ν

• γ + γ

  2,ν α 1,ν

  I (α, γ, µ)f (z) = I (I (α, γ, µ)f (z))

  λ λ λ ..

  .

  n,ν n −1,ν α

  I (α, γ, µ)f (z) = I (I (α, γ, µ)f (z)) (3)

  λ λ λ

  If f given by (1) then by (2) and (3), Darus and Faizal [9] introduced the following operator, that is _{! n X} ∞ Γ(k + 1)Γ(2 − α) ν + (µ + λ(k − 1) + γ)

  n,ν k

  I a z .

  (α, γ, µ)f (z) = z+ ( ( ) k

  λ

  Γ(k + 1 − α)) ν + β

  k =2

  (4)

  −1

  It is well known that every function f ∈ S has inverse f , defined by

  −1

  f (f (z)) = z (z ∈ U) and

  1

  −1

  f (f (w)) = w (|w| < r (f ) ≥ )

  4 where

  −1

  2

  2

  3

  3

  4

  f w a = w − a + (a − a )w − (5a − 5a + a )w + . . . .

  2

  3

  2

  3

  4

  2

  2 −1

  A function f ∈ A is said to bi-univalent in U if both f (z) and f are uni- valent in U. Let Σ denote the class of bi-univalent in U given by (1). For a

  Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker brief history and interesting examples in the class Σ, see [5].

  Brannan and Taha [2] (see also [7]) introduced certain subclasses of bi-

  ∗

  univalent function class Σ similar to the familiar subclasses S (α) and K(α) of starlike and convex functions of order α (0 ≤ α < 1), respectively (see [3]). Thus, following Brannan and Taha [2] (see also [7]), a function f ∈ A

  ∗

  is in the class S (α) of strongly bi-starlike functions of order α (0 ≤ α < 1)

  Σ

  if each of the following conditions is satisfied:

  ′

  zf απ (z) f ∈ Σ and arg < , (0 < α ≤ 1, z ∈ U) f

  (z)

  2 and

  ′

  zg απ (w)

  < , arg (0 < α ≤ 1, w ∈ U), g

  (w)

  2

  −1 ∗

  where g is the extension of f to U. The classes S (α) and K (α) of

  Σ Σ

  bi-starlike functions of order α and bi-convex functions of order α, corre-

  ∗

  sponding (respectively) to the function classes S (α) and K(α), were also

  ∗

  introduced analogously. For each of the function classes S (α) and K (α),

  Σ Σ

  they found non-sharp estimates on the first two Taylor-Maclaurin coeffi- cients |a

  2 | and |a 3 | (for detail, see [2, 7]).

  The object of the present paper is to introduce two new subclasses of the functions class Σ involving generalized Salagean operator and find esti- mates on the coefficients |a | and |a | for functions in these new subclasses

  2

  3

  of the function class Σ. The techniques of proofing used earlier by Srivastava et. al [5] and by Frasin and Aouf [1].

  To prove our main results, we will need the following lemma [8]. Lemma 1.1 If h ∈ P then |c | ≤ 2 for each k, where P is the family of all

  k

  functions h analytic in U for which Re h(z) > 0

  2

  3

  h z z z (z) = 1 + c + c + c + . . .

  1

  2

  3 for z ∈ U.

  2. COEFFICIENT BOUNDS FOR THE FUNCTION CLASS

  n

  B (λ, α)

  Σ Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker

  To prove our main results, we need to introduce the following defini- tion. Definition 2.1 A function f (z) given by (1) is said to be in the class in

  n

  the class B (λ, α) if the following conditions are satisfied

  Σ

′

  απ

  n,ν

  f I < , ∈ Σ and arg (α, γ, µ)f (z) (0 < α ≤ 1, λ ≥ 0, z ∈ U)

  λ

  2 (5) and

  ′

  απ

  n,ν

  I < , arg (α, γ, µ)g(w) (0 < α ≤ 1, λ ≥ 0, w ∈ U) (6)

  λ

  2 where the function g is given by

  2

  2

  

3

  3

  4

  g w a (w) = w − a + (a − a )w − (5a − 5a + a )w + . . . . (7)

  2

  3

  2

  3

  4

  2

  2 α

  We note that for n = 0, the class B (λ, α) reduces to the class H

  Σ Σ introduced and studied by Srivastava et al. [5].

  We begin by finding the estimates on the coefficients |a | and |a | for

  2

  3 n

  functions in the class B (λ, α).

  Σ n

  Theorem 2.1 Let f (z) given by (1) be in the class B (λ, α), 0 < α ≤ 1,

  Σ

  and λ ≥ 1. Then _s

  2 |a | ≤ α

  (8)

  2

n

2n

  3α(φ) − 2(α − 1)(P ) and

  

2

  α 2α . |a +

  3 | ≤

  (9)

  2n n n

  (P ) (φ) 3(φ) Where _{" # n}

  2Γ(2 − α) ν + (µ + λ + γ) P =

  (10) ν

  Γ(3 − α) + β and _{" # n} 3Γ(2 − α) ν + (µ + 2λ + γ)

  φ = . (11) ν

  Γ(4 − α) + β

• c
• c
• . . . (14) and
• l
• l
• . . . (15) respectively. Now, equating the coefficients of
• c

  − αc

  ) (20) Also, from (19) and (17), we find that

  (φ)

  n

  6a

  2

  2

  2

  2

  2

  1

  α (α − 1)

  2 = αl

  2

  2

  1

  1

  α (α − 1)

  2 ,

  2

  − a

  3 ) = αl 2 + l

  2

  1

  α (1 − α)

  (19) where P and φ given by (10) and (11), respectively. From (18) and (16), we get c

  2

  1 = −l 1 and 8(P )

2n

  a

  2

  2

  = α

  2

  (c

  1

  2 A rearrangement together with the second identity in (20) yields (φ)

  (2a

  = α(c

  2n

  a

  2

  2

  α

  2

  2

  (α − 1)

  2

  ) + α(α − 1) 4(P )

  2n

  a

  2

  2

  2 8(P )

  ) + α

  n

  ) + α

  6a

  2

  2

  = α (c

  2

  

2

  (α − 1)

  2

  2 (c

  2

  1

  2

  1

  ) = α(c

  2

  2

  n

  2 Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker

  1

  z

  2

  3

  z

  3

  L (w) = 1 + l

  z

  z

  

2

  z

  2

  3

  z

  3

  

2

  1

  n,ν λ

  = ^h Q

  Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker

  Proof 2.1 It follows from (5) and (6) that

  I

  n,ν λ

  (α, γ, µ)f (z)

  ′

  (z) ^{i α} and

  (z) = 1 + c

  I

  n,ν λ

  (α, γ, µ)g(w)

  ′

  = ^h L

  (w) ^{i α} (12) where Q(z) and L(w) satisfy the following inequalities: Re (Q(z)) > 0 (z ∈ U) and Re (L(w)) > 0 (w ∈ U). (13)

  Furthermore, the functions Q(z) and L(w) have the form Q

  I

  (α, γ, µ)f (z)

  (18) and 3(φ)

  α (α − 1)

  a

  3

  = αc

  

2

  2

  1

  2 (17)

  (16) 3(φ)

  −2(P )

  n

  a

  2

  = αl

  

1

  n

  

1

  ′

  (α, γ, µ)g(w)

  with ^h Q

  (z) ⁱ

  α

  and the coefficients of

  I

  n,ν λ

  ′

  = αc

  with ^h L

  (w) ⁱ

  α

  , we get 2(P )

  n

  a

  2

• l
• c
• l
• l
• l
• l
• l

  Therefore, we have

  2

  α (c + l )

  2

  2

  2 a .

  =

  2 n 2n

  6α(φ) − 4(α − 1)(P ) Applying Lemma 1.1 for the coefficients c _s 2 and l 2 , we will obtain

  2 . |a | ≤ α

  2

n

2n

  3α(φ) − 2(α − 1)(P ) This gives the bound on |a | as asserted in (8).

2 Next, in order to find the bound on |a |, by subtracting (19) from (17),

  3

  we get α α

  (α − 1) (α − 1)

  n n

  2

  2

  2 (φ) 6a − (φ) 6a = αc + c − αl + l .

  3

  2

  2

  2

  1

  1

  2

  2

2 Upon substituting the value of a from (20) and observing that

  2

  2

  2

  c , = l

  1

  1

  it follows that

  2

  2

  α c

  1

  1 + a = α (c − l ).

  3

  2

  2 n n 2n

  φ 4(P ) 6(φ)

  Applying Lemma 1.1 for the coefficients c , c , l and l , we readily get

  1

  2

  1

  2

  

2

  α 2α |a | ≤ + .

  3 n n 2n

  (P ) (φ) 3(φ) This completes the proof of Theorem 2.1. Putting n = 0 in Theorem 2.1, we arrived to the result by Srivastava et al. [5], as follows:

  α

  Corollary 2.1 , Let f (z) given by (1) be in the class H 0 < α ≤ 1. Then _r Σ

  2 |a | ≤ α (21)

  2

  α + 2 and α

  (3α + 2) |a | ≤ . (22)

  3

  3 Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker Taking n = 1, in Theorem 2.1, we obtain the following corollary.

  1 Corollary 2.2 Let f (z) given by (1) be in the class B (1, α) 0 < α ≤ 1. Σ

  Then _s

  2 |a | ≤ α (23)

  2

  2

  3α(φ) − 2(α − 1)(P ) and

  2

  α 2α

  |a | ≤ (24) +

  3

  

2

P φ

  3φ where P and φ given by (10) and (11), respectively.

  3. COEFFICIENT BOUNDS FOR THE FUNCTION CLASS

  n

  B (λ, β)

  Σ

  To prove our main results, we need to introduce the following defini- tion. Definition 3.2 A function f (z) given by (1) is said to be in the class in

  n

  the class B (λ, β) if the following conditions are satisfied

  Σ

′

n,ν

  f ∈ Σ and Re I (α, γ, µ)f (z) > β, (0 ≤ β < 1, λ ≥ 0, z ∈ U)

  λ

  (25) and

  ′ n,ν

  I > β, Re (α, γ, µ) g(w) (0 ≤ β < 1, λ ≥ 0, w ∈ U) (26)

  λ

  where the function g is given by

  2

  2

  3

  3

  4

  g w a (w) = w − a + (a − a )w − (5a − 5a + a )w + . . . . (27)

  2

  3

  2

  3

  4

  2

  2 n

  We note that for n = 0, the class B (λ, β) reduces to the class H (β)

  Σ Σ

  introduced and studied by Srivastava et al. [5].

  n

  Theorem 3.2 Let f (z) given by (1) be in the class B (λ, β), 0 ≤ β < 1

  Σ

  and λ ≥ 1. Then _s 2(1 − β)

  |a | ≤ (28)

  2 n

  3(φ) Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker

  and

  2

  (1 − β) 2(1 − β) ,

  |a | ≤ + (29)

  3 n 2n

  (P ) 3(φ) where P and φ given by (10) and (11), respectively.

  Proof 3.2 It follows from (25) and (26) that

  ′ ′ n,ν n,ν

  I I (α, γ, µ)f (z) = β+(1−β)Q(z) and (α, γ, µ)g(w) = β+(1−β)L(w)

  λ λ

  (30) respectively, where Q(z) and L(w) satisfy the following inequalities: Re (Q(z)) > 0 (z ∈ U) and Re (L(w)) > 0 (w ∈ U). Furthermore, the functions Q(z) and L(w) have the form

  2

  3 Q z z z

  (z) = 1 + c

  1 + c 2 + c 3 + . . .

  and

  2

  3 L z z z (w) = 1 + l + l + l + . . . .

  1

  

2

  3 As in the proof of Theorem 2.1, by suitably comparing coefficients, we find n

  a 2(P ) = (1 − β)c (31)

  2

  1 n

  3(φ) a = (1 − β)c (32)

  3

  2 n

  a −2(P ) = (1 − β)l (33)

  2

  1

  and

  n

  2

  3(φ) (2a − a ) = (1 − β)l (34)

  3

  2

  2 From (31) and (33), we obtain 2n

  

2

  2

  2

  2

  c = −l and 8(P ) a = (1 − β) (c + l ) (35)

  1

  1

  

2

  1

  1 Also, from (32) and (34), we find that n

  2

  a 6(φ) = (1 − β)(c + l ) (36)

  2

  2

2 Thus, we have

  (1 − β)(|c | + |l |)

  2

  2

  2

  |a | ≤

  2 n

  6(φ) 2(1 − β)

  = ,

  

n

  3(φ) Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker

  and so _s 2(1 − β) . |a | ≤

  2 n

  3(φ) This gives the bound on |a | as asserted in (28).

2 Next, in order to find the bound on |a |, by subtracting (34) from (32),

  3

  we get

  n

  2 6(φ) (a − a ) = (1 − β)(c − l ).

  3

  2

  2

  2

2 Upon substituting the value of a from (35), yields

  2

  (1 − β)(c − l )

  

2

  2

  2

  = a

• a

  3

  2 n

  6(φ)

  2

  2

  2

  (1 − β) (c + l ) (1 − β)(c − l )

  2

  2

  1

  1

  = +

  n 2n

  8(P ) 6(φ) Applying Lemma 1.1 for the coefficients c , c , l and l , we readily obtain

  1

  2

  1

  2

  

2

  (1 − β) 2(1 − β)

  |a | ≤

• .

  3 n 2n

  (P ) 3(φ) This completes the proof of Theorem 3.2. Putting n = 0 in Theorem 3.2, we arrived to the result by Srivastava et al. [5], as follows: Corollary 3.3 Let f (z) given by (1) be in the class H (β), 0 ≤ β < 1.

  Σ

  Then _r 2(1 − β)

  |a | ≤ (37)

  2

  3 and (1 − β)(5 − 3β)

  |a | ≤ . (38)

  3

  3 Taking n = 1, in Theorem 3.2, we obtain the following corollary.

  1 Corollary 3.4 Let f (z) given by (1) be in the class B (1, β) 0 ≤ β < 1. Σ

  Then _s 2(1 − β)

  |a | ≤ (39)

  2

  3φ and

  

2

  (1 − β) 2(1 − β) |a | ≤ (40) +

  3

2 P

  3φ where P and φ given by (10) and (11), respectively. Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker

REFERENCES

  References

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  Sˇ lˇ gean operator, Int.J. Math. Math. Sci., 27(2004), 1429–1436 [5] H. M. Srivastava, A. K. Misra and P. Gochhayat, Certain sublasses of analytic and bi-univalent functions Apllied Math. Letters, 23, (2010),

  1188-1192 a a [6] G. S. Sˇ lˇ gean, Subclasses of univalent functions. Complex Analysis-

  Fifth Romania-Finnish Seminar, part 1(Bucharest,1981), Lecturer Notes in Math.(Springer-Verlag), 1013(1983),362-372. [7] T. S. Taha, Topics in Univalent Function Theory, Ph.D Thesis, Uni- versity of London, 1981 [8] Ch. Pommerenke, Univalent Functions, Vanderhoeck and Ruprecht,

  G¨ottingen, 1975 [9] M. Darus and I. Faisal, A Study on Becker Univalence Criteria, Uni- versity Kebangsaan Malaysia, 2010

  Saibah Siregar : Faculty of Science and Biotechnology, University of Selangor, Bestari Jaya 45600, Selangor Darul Ehsan, Malaysia

  E-mail: saibahmath@yahoo.com

  Siregar, S. and Joan, C.P. – New Subclasses of Bi-Univalent involving Becker Christina Pushpa Joan : Faculty of Science and Biotechnology, University of Selangor, Bestari Jaya 45600, Selangor Darul Ehsan, Malaysia

  E-mail:leighann8928@yahoo.com