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Majorization for certain classes of analytic functions defined by fractional derivatives
Applied Mathematics Letters 22 (2009) 1855–1858
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Majorization for certain classes of analytic functions defined by fractional derivatives S.P. Goyal a,∗ , Pranay Goswami b a
Department of Mathematics, University of Rajasthan, Jaipur-302055, India
b
Department of Mathematics, Amity University Rajasthan, Jaipur-302002, India
article
info
Article history: Received 12 June 2009 Accepted 13 July 2009
abstract Here we investigate a majorization problem involving starlike function of complex order belonging to a certain class defined by means of fractional derivatives. Relevant connections of the main results obtained in this paper with those given by earlier workers on the subject are also pointed out. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Multivalent functions Fractional calculus operators Majorization property Starlike functions
1. Introduction Let f (z ) and g (z ) be analytic in the open unit disk
∆ = {z : z ∈ C, |z | < 1} .
(1.1)
We say that f (z ) is majorized by g (z ) in ∆ (see [1]) and write f (z ) g (z )
(z ∈ ∆),
(1.2)
if there exists a function ϕ(z ), analytic in ∆ such that
|ϕ(z )| ≤ 1 and f (z ) = ϕ(z )g (z ) (z ∈ ∆).
(1.3)
It may be noted here that (1.2) is closely related to the concept of quasi-subordination between analytic functions. Let A(p) denote the class of functions of the form f (z ) = z p +
∞ X
ak z k ,
(p ∈ N = {1, 2, 3, . . .}),
(1.4)
k=p+1
which are analytic and multivalent in the open unit disk ∆. In particular if p = 1, then A(1) = A. For functions fj ∈ A(p) given by fj (z ) = z p +
∞ X
ak,j z k ,
(j = 1, 2; p ∈ N),
k=p+1
∗
Corresponding author. Tel.: +91 0141 2708392. E-mail addresses: [email protected] (S.P. Goyal), [email protected] (P. Goswami).
0893-9659/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2009.07.009
(1.5)
1856
S.P. Goyal, P. Goswami / Applied Mathematics Letters 22 (2009) 1855–1858
we define the Hadamard product (or convolution) of f1 and f2 by ∞ X
(f1 ∗ f2 )(z ) = z p +
ak,1 ak,2 z k = (f2 ∗ f1 )(z ).
k=p+1
Definition 1 ([2], See also [3]). The fractional integral of order λ (λ > 0) is defined, for a function f , analytic in a simplyconnected region of the complex plane containing the origin by D−λ z f (z ) =
Γ (λ)
f (ξ )
z
Z
1
0
(z − ξ )1−λ
dξ ,
(1.6)
where the multiplicity of (z − ξ )λ−1 is removed by requiring log(z − ξ ) to be real when z − ξ > 0. Definition 2 ([2], See also [3]). Under the hypothesis of Definition 1, the fractional derivative of f (z ) of order λ is defined by
Dλz f (z ) =
z
Z
1
Γ (1 − λ)
0
f (ξ )
( z − ξ )λ
dξ
(0 ≤ λ < 1), (1.7)
dn λ−n D f (z ) (n ≤ λ < n + 1; n ∈ N), n z dz
where the multiplicity of (z − ξ )λ−1 is removed by requiring log(z − ξ ) to be real when z − ξ > 0. (λ,p) Very recently, Patel and Mishra [4] defined the extended fractional differ-integral operator Ωz : Ap → Ap for a function f ∈ Ap and for a real number λ (−∞ < λ < p + 1) by
Ωz(λ,p) f (z ) =
Γ (p + 1 − λ) λ λ z Dz f (z ), Γ (p + 1)
(1.8)
where Dλz f (z ) is, respectively the fractional integral of f of order λ if 0 ≤ λ < p + 1. It is easy to see from (1.8) that for a function f of the form (1.1), we have
Ωz(λ,p) f (z ) = z p +
∞ X Γ (n + p + 1)Γ (p + 1 − λ) n an z ( z ∈ ∆) Γ (p + 1)Γ (n + p + 1 − λ) n=p+1
and z (Ωz(λ,p) f (z ))0 = (p − λ)Ωz(λ+1,p) f (z ) + λΩz(λ,p) f (z )
(−∞ < λ < p; z ∈ ∆).
(1.9)
λ,j
Definition 3. A function f (z ) ∈ A(p) is said to be in the class Sp (γ ) of p-valent functions of complex order γ 6= 0 in ∆ if and only if
( Re 1 +
1
γ
(λ,p)
z (Ωz
f (z ))(j+1)
(Ωz(λ,p) f (z ))(j)
!) −p+j
>0
(z ∈ ∆, p ∈ N, j ∈ N0 = N ∪ {0}, γ ∈ C − {0}, −∞ < λ < p).
(1.10)
Clearly, we have the following relationships:
(i) S10,0 (γ ) = S (γ ) (γ ∈ C − {0}), (ii) S11,0 (γ ) = K (γ ) (γ ∈ C − {0}), (iii) S10,0 (1 − α) = S ∗ (α) for 0 ≤ α < 1. The classes S (γ ) and K (γ ) are said to be classes of starlike and convex of complex order γ 6= 0 in ∆ which were considered by Naser and Aouf [5] and Wiatrowski [6], and S ∗ (α) denote the class of starlike functions of order α in ∆. A majorization problem for the class S (γ ) has recently been investigated by Altinas et al. [7]. Also, majorization problems for the class S ∗ = S ∗ (0) have been investigated by MacGregor [1]. Motivated by these works, in the present paper, we λ,j investigate a majorization problem for the class Sp (γ ). λ,j
2. Majorization problem for the class Sp (γ) We begin by proving λ,j
λ,p
λ,p
Theorem 2.1. Let the function f ∈ A(p) and suppose that g ∈ Sp (γ ). If (Ωz f (z ))(j) is majorized by (Ωz g (z ))(j) in ∆, then
|(Ωzλ+1,p f (z ))(j) | ≤ |(Ωzλ+1,p g (z ))(j) | for |z | ≤ r1 ,
(2.1)
S.P. Goyal, P. Goswami / Applied Mathematics Letters 22 (2009) 1855–1858
1857
where k−
r1 = r1 (p, γ , λ) =
p
k2 − 4(p − λ)|2γ − p + λ| 2|2γ − p + λ|
(k = 2 + p − λ + |2γ − p + λ|; p ∈ N , γ ∈ C − {0}).
(2.2)
λ,j
Proof. Since g ∈ Sp (γ ), we find from (1.10), if h( z ) = 1 +
1
(λ,p)
z ( Ωz
γ
g (z ))(j+1)
(Ωz(λ,p) g (z ))(j)
! −p+j
(γ ∈ C − {0}, j, p ∈ N and p > j),
(2.3)
then Re{h(z )} > 0 (z ∈ ∆) and h( z ) =
1 + w(z )
(w ∈ P ),
1 − w(z )
(2.4)
where w(z ) = c1 z + c2 z 2 +· · · and P denotes the well-known class of bounded analytic functions in ∆ (see [8]) and satisfies the conditions
w(0) = 0,
and |w(z )| ≤ |z |
(z ∈ ∆).
(2.5)
Making use of (2.3) and (2.4), we get (λ,p)
z ( Ωz
g (z ))(j+1)
(λ,p)
(Ωz
g (z ))(j)
=
(p − j) + (2γ − p + j)w(z ) . 1 − w(z )
(2.6)
Now using
(j)
z (Ωz(λ,p) g (z ))(j+1) = (p − λ) Ωz(λ+1,p) g (z )
+ (λ − j) Ωz(λ,p) g (z )
(j)
(1 ≤ j ≤ p; z ∈ ∆)
(2.7)
in (2.6) and making simple calculations, we get
(λ,p) (Ω g (z ))(j) ≤ z
(λ,p)
Next since (Ωz
(λ+1,p) (p − λ)[1 + |z |] (Ω g (z ))(j) . z (p − λ) − |2γ − p + λ||z | (λ,p)
f (z ))(j) is majorized by (Ωz
(2.8)
g (z ))(j) in the unit disk ∆, therefore from (1.3), we have
(Ωz(λ,p) f (z ))(j) = ϕ(z )(Ωz(λ,p) g (z ))(j) . Differentiating it with respect to ‘z’ and multiplying by ‘z’, we get z (Ωz(λ,p) f (z ))(j+1) = z ϕ 0 (z )(Ωz(λ,p) g (z ))(j) + ϕ(z )(Ωz(λ,p) g (z ))(j+1) . Using (2.7), in the above equation, it yields
(Ωz(λ+1,p) f (z ))(j) =
z ϕ 0 (z )
(p − λ)
(Ωz(λ,p) g (z ))(j) + ϕ(z )(Ωz(λ+1,p) g (z ))(j) .
(2.9)
Thus, noting that ϕ ∈ P satisfies the inequality (see, e.g. Nehari [8])
|ϕ 0 (z )| ≤
1 − |ϕ(z )|2 1 − | z |2
(z ∈ ∆)
(2.10)
and making use of (2.8) and (2.10) in (2.9), we get
(λ+1,p) (λ+1,p) 1 − |ϕ(z )|2 |z | (j) (Ω (Ω f ( z )) ≤ |ϕ( z )| + g (z ))(j) z z 1 − |z | [(p − λ) − |2γ − p + λ||z |] which upon setting
|z | = r and |ϕ(z )| = ρ (0 ≤ ρ ≤ 1) leads us to the inequality
(j) λ+1,p f (z ) ≤ Ωz
(j) Φ (ρ) λ+1,p g (z ) Ωz (1 − r )(p − λ − |2γ − p + λ|r )
(2.11)
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S.P. Goyal, P. Goswami / Applied Mathematics Letters 22 (2009) 1855–1858
where
Φ (ρ) = −r ρ 2 + (1 − r )(p − λ − |2γ − p + λ|r )ρ + r
(2.12)
takes its maximum value at ρ = 1, with r1 = r1 (p, γ , λ) where r1 (p, γ , λ) is given by Eq. (2.2). Furthermore, if 0 ≤ ρ ≤ r1 (p, γ , λ), then the function ψ(ρ) defined by
ψ(ρ) = −σ ρ 2 + (1 − σ )(p − λ − |2γ − p + λ|σ )ρ + σ
(2.13)
is seen to be an increasing function on the interval 0 ≤ ρ ≤ 1, so that
ψ(ρ) ≤ ψ(1) = (1 − σ )(p − λ − |2γ − p + λ|σ ) 0 ≤ ρ ≤ 1; (0 ≤ σ ≤ r1 (p, λ, γ )). Hence upon setting ρ = 1, in (2.13), we conclude that (2.1) of Theorem 2.1 holds true for |z | ≤ r1 (p, γ , λ), where r1 (p, γ , λ) is given by (2.2). This completes Theorem 2.1. Setting p = 1 and j = 0, in Theorem 2.1, we get λ,0
Corollary 2.1. Let the function f ∈ A and suppose that g ∈ S1 (γ ). If (Ωzλ,1 f (z )) is majorized by (Ωzλ,1 g (z )) in ∆, then
| Ωzλ+1,1 f (z ) | ≤ | Ωzλ+1,1 g (z ) | for |z | ≤ r2 ,
(2.14)
where r2 =
k−
k2 − 4(1 − λ)|2γ − 1 + λ|
p
2|2γ − 1 + λ|
(k = 3 − λ + |2γ − 1 + λ|; γ ∈ C − {0}).
(2.15)
Further putting λ = 0 in Corollary 2.1, we get Corollary 2.2. Let the function f (z ) ∈ A be analytic in the open unit disk ∆ and suppose that g ∈ S (γ ). If f (z ) is majorized by g (z ) in ∆, then
|f 0 (z )| ≤ |g 0 (z )| (|z | ≤ r1 ) where r1 = r1 (γ ) =
3 + |2γ − 1| −
9 + 2|2γ − 1| + |2γ − 1|2
p
2|2γ − 1|
which is a known result obtained by Altinas et al. [7]. For γ = 1, the above corollary reduces to the following result: Corollary 2.3. Let the function f (z ) ∈ A be an analytic and univalent in the open unit disk ∆ and suppose that g ∈ S ∗ = S ∗ (0). If f (z ) is majorized by g (z ) in ∆, then
|f 0 (z )| ≤ |g 0 (z )|
(|z | ≤ 2 −
√
3)
which is a known result obtained by MacGregor [1]. References [1] T.H. MacGreogor, Majorization by univalent functions, Duke Math. J. 34 (1967) 95–102. [2] S. Owa, On distortion theorems I, Kyungpook Math. J. 18 (1) (1978) 53–59. [3] H.M. Srivastava, S. Owa (Eds.), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, 1989, (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York. [4] J. Patel, A.K. Mishra, On certain subclasses of multivalent functions associated with an extended fractional differintegral operator, J. Math. Anal. Appl. 332 (2007) 109–122. [5] M.A. Naser, M.K. Aouf, Starlike function of complex order, J. Natur. Sci. Math. 25 (1985) 1–12. [6] P. Wiatrowski, On the coefficients of some family of holomorphic functions, Zeszyry Nauk. Uniw. Lddz. Nauk. Mat.-Przyrod. 39 (2) (1970) 75–85. [7] O. Altinas, Ö. Özkan, H.M. Srivastava, Majorization by starlike functions of complex order, Complex Var. 46 (2001) 207–218. [8] Z. Nehari, Conformal Mapping, Macgraw-Hill Book Company, New York, Toronto, London, 1955.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Majorization for certain classes of analytic functions defined by fractional derivatives S.P. Goyal a,∗ , Pranay Goswami b a
Department of Mathematics, University of Rajasthan, Jaipur-302055, India
b
Department of Mathematics, Amity University Rajasthan, Jaipur-302002, India
article
info
Article history: Received 12 June 2009 Accepted 13 July 2009
abstract Here we investigate a majorization problem involving starlike function of complex order belonging to a certain class defined by means of fractional derivatives. Relevant connections of the main results obtained in this paper with those given by earlier workers on the subject are also pointed out. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Multivalent functions Fractional calculus operators Majorization property Starlike functions
1. Introduction Let f (z ) and g (z ) be analytic in the open unit disk
∆ = {z : z ∈ C, |z | < 1} .
(1.1)
We say that f (z ) is majorized by g (z ) in ∆ (see [1]) and write f (z ) g (z )
(z ∈ ∆),
(1.2)
if there exists a function ϕ(z ), analytic in ∆ such that
|ϕ(z )| ≤ 1 and f (z ) = ϕ(z )g (z ) (z ∈ ∆).
(1.3)
It may be noted here that (1.2) is closely related to the concept of quasi-subordination between analytic functions. Let A(p) denote the class of functions of the form f (z ) = z p +
∞ X
ak z k ,
(p ∈ N = {1, 2, 3, . . .}),
(1.4)
k=p+1
which are analytic and multivalent in the open unit disk ∆. In particular if p = 1, then A(1) = A. For functions fj ∈ A(p) given by fj (z ) = z p +
∞ X
ak,j z k ,
(j = 1, 2; p ∈ N),
k=p+1
∗
Corresponding author. Tel.: +91 0141 2708392. E-mail addresses: [email protected] (S.P. Goyal), [email protected] (P. Goswami).
0893-9659/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2009.07.009
(1.5)
1856
S.P. Goyal, P. Goswami / Applied Mathematics Letters 22 (2009) 1855–1858
we define the Hadamard product (or convolution) of f1 and f2 by ∞ X
(f1 ∗ f2 )(z ) = z p +
ak,1 ak,2 z k = (f2 ∗ f1 )(z ).
k=p+1
Definition 1 ([2], See also [3]). The fractional integral of order λ (λ > 0) is defined, for a function f , analytic in a simplyconnected region of the complex plane containing the origin by D−λ z f (z ) =
Γ (λ)
f (ξ )
z
Z
1
0
(z − ξ )1−λ
dξ ,
(1.6)
where the multiplicity of (z − ξ )λ−1 is removed by requiring log(z − ξ ) to be real when z − ξ > 0. Definition 2 ([2], See also [3]). Under the hypothesis of Definition 1, the fractional derivative of f (z ) of order λ is defined by
Dλz f (z ) =
z
Z
1
Γ (1 − λ)
0
f (ξ )
( z − ξ )λ
dξ
(0 ≤ λ < 1), (1.7)
dn λ−n D f (z ) (n ≤ λ < n + 1; n ∈ N), n z dz
where the multiplicity of (z − ξ )λ−1 is removed by requiring log(z − ξ ) to be real when z − ξ > 0. (λ,p) Very recently, Patel and Mishra [4] defined the extended fractional differ-integral operator Ωz : Ap → Ap for a function f ∈ Ap and for a real number λ (−∞ < λ < p + 1) by
Ωz(λ,p) f (z ) =
Γ (p + 1 − λ) λ λ z Dz f (z ), Γ (p + 1)
(1.8)
where Dλz f (z ) is, respectively the fractional integral of f of order λ if 0 ≤ λ < p + 1. It is easy to see from (1.8) that for a function f of the form (1.1), we have
Ωz(λ,p) f (z ) = z p +
∞ X Γ (n + p + 1)Γ (p + 1 − λ) n an z ( z ∈ ∆) Γ (p + 1)Γ (n + p + 1 − λ) n=p+1
and z (Ωz(λ,p) f (z ))0 = (p − λ)Ωz(λ+1,p) f (z ) + λΩz(λ,p) f (z )
(−∞ < λ < p; z ∈ ∆).
(1.9)
λ,j
Definition 3. A function f (z ) ∈ A(p) is said to be in the class Sp (γ ) of p-valent functions of complex order γ 6= 0 in ∆ if and only if
( Re 1 +
1
γ
(λ,p)
z (Ωz
f (z ))(j+1)
(Ωz(λ,p) f (z ))(j)
!) −p+j
>0
(z ∈ ∆, p ∈ N, j ∈ N0 = N ∪ {0}, γ ∈ C − {0}, −∞ < λ < p).
(1.10)
Clearly, we have the following relationships:
(i) S10,0 (γ ) = S (γ ) (γ ∈ C − {0}), (ii) S11,0 (γ ) = K (γ ) (γ ∈ C − {0}), (iii) S10,0 (1 − α) = S ∗ (α) for 0 ≤ α < 1. The classes S (γ ) and K (γ ) are said to be classes of starlike and convex of complex order γ 6= 0 in ∆ which were considered by Naser and Aouf [5] and Wiatrowski [6], and S ∗ (α) denote the class of starlike functions of order α in ∆. A majorization problem for the class S (γ ) has recently been investigated by Altinas et al. [7]. Also, majorization problems for the class S ∗ = S ∗ (0) have been investigated by MacGregor [1]. Motivated by these works, in the present paper, we λ,j investigate a majorization problem for the class Sp (γ ). λ,j
2. Majorization problem for the class Sp (γ) We begin by proving λ,j
λ,p
λ,p
Theorem 2.1. Let the function f ∈ A(p) and suppose that g ∈ Sp (γ ). If (Ωz f (z ))(j) is majorized by (Ωz g (z ))(j) in ∆, then
|(Ωzλ+1,p f (z ))(j) | ≤ |(Ωzλ+1,p g (z ))(j) | for |z | ≤ r1 ,
(2.1)
S.P. Goyal, P. Goswami / Applied Mathematics Letters 22 (2009) 1855–1858
1857
where k−
r1 = r1 (p, γ , λ) =
p
k2 − 4(p − λ)|2γ − p + λ| 2|2γ − p + λ|
(k = 2 + p − λ + |2γ − p + λ|; p ∈ N , γ ∈ C − {0}).
(2.2)
λ,j
Proof. Since g ∈ Sp (γ ), we find from (1.10), if h( z ) = 1 +
1
(λ,p)
z ( Ωz
γ
g (z ))(j+1)
(Ωz(λ,p) g (z ))(j)
! −p+j
(γ ∈ C − {0}, j, p ∈ N and p > j),
(2.3)
then Re{h(z )} > 0 (z ∈ ∆) and h( z ) =
1 + w(z )
(w ∈ P ),
1 − w(z )
(2.4)
where w(z ) = c1 z + c2 z 2 +· · · and P denotes the well-known class of bounded analytic functions in ∆ (see [8]) and satisfies the conditions
w(0) = 0,
and |w(z )| ≤ |z |
(z ∈ ∆).
(2.5)
Making use of (2.3) and (2.4), we get (λ,p)
z ( Ωz
g (z ))(j+1)
(λ,p)
(Ωz
g (z ))(j)
=
(p − j) + (2γ − p + j)w(z ) . 1 − w(z )
(2.6)
Now using
(j)
z (Ωz(λ,p) g (z ))(j+1) = (p − λ) Ωz(λ+1,p) g (z )
+ (λ − j) Ωz(λ,p) g (z )
(j)
(1 ≤ j ≤ p; z ∈ ∆)
(2.7)
in (2.6) and making simple calculations, we get
(λ,p) (Ω g (z ))(j) ≤ z
(λ,p)
Next since (Ωz
(λ+1,p) (p − λ)[1 + |z |] (Ω g (z ))(j) . z (p − λ) − |2γ − p + λ||z | (λ,p)
f (z ))(j) is majorized by (Ωz
(2.8)
g (z ))(j) in the unit disk ∆, therefore from (1.3), we have
(Ωz(λ,p) f (z ))(j) = ϕ(z )(Ωz(λ,p) g (z ))(j) . Differentiating it with respect to ‘z’ and multiplying by ‘z’, we get z (Ωz(λ,p) f (z ))(j+1) = z ϕ 0 (z )(Ωz(λ,p) g (z ))(j) + ϕ(z )(Ωz(λ,p) g (z ))(j+1) . Using (2.7), in the above equation, it yields
(Ωz(λ+1,p) f (z ))(j) =
z ϕ 0 (z )
(p − λ)
(Ωz(λ,p) g (z ))(j) + ϕ(z )(Ωz(λ+1,p) g (z ))(j) .
(2.9)
Thus, noting that ϕ ∈ P satisfies the inequality (see, e.g. Nehari [8])
|ϕ 0 (z )| ≤
1 − |ϕ(z )|2 1 − | z |2
(z ∈ ∆)
(2.10)
and making use of (2.8) and (2.10) in (2.9), we get
(λ+1,p) (λ+1,p) 1 − |ϕ(z )|2 |z | (j) (Ω (Ω f ( z )) ≤ |ϕ( z )| + g (z ))(j) z z 1 − |z | [(p − λ) − |2γ − p + λ||z |] which upon setting
|z | = r and |ϕ(z )| = ρ (0 ≤ ρ ≤ 1) leads us to the inequality
(j) λ+1,p f (z ) ≤ Ωz
(j) Φ (ρ) λ+1,p g (z ) Ωz (1 − r )(p − λ − |2γ − p + λ|r )
(2.11)
1858
S.P. Goyal, P. Goswami / Applied Mathematics Letters 22 (2009) 1855–1858
where
Φ (ρ) = −r ρ 2 + (1 − r )(p − λ − |2γ − p + λ|r )ρ + r
(2.12)
takes its maximum value at ρ = 1, with r1 = r1 (p, γ , λ) where r1 (p, γ , λ) is given by Eq. (2.2). Furthermore, if 0 ≤ ρ ≤ r1 (p, γ , λ), then the function ψ(ρ) defined by
ψ(ρ) = −σ ρ 2 + (1 − σ )(p − λ − |2γ − p + λ|σ )ρ + σ
(2.13)
is seen to be an increasing function on the interval 0 ≤ ρ ≤ 1, so that
ψ(ρ) ≤ ψ(1) = (1 − σ )(p − λ − |2γ − p + λ|σ ) 0 ≤ ρ ≤ 1; (0 ≤ σ ≤ r1 (p, λ, γ )). Hence upon setting ρ = 1, in (2.13), we conclude that (2.1) of Theorem 2.1 holds true for |z | ≤ r1 (p, γ , λ), where r1 (p, γ , λ) is given by (2.2). This completes Theorem 2.1. Setting p = 1 and j = 0, in Theorem 2.1, we get λ,0
Corollary 2.1. Let the function f ∈ A and suppose that g ∈ S1 (γ ). If (Ωzλ,1 f (z )) is majorized by (Ωzλ,1 g (z )) in ∆, then
| Ωzλ+1,1 f (z ) | ≤ | Ωzλ+1,1 g (z ) | for |z | ≤ r2 ,
(2.14)
where r2 =
k−
k2 − 4(1 − λ)|2γ − 1 + λ|
p
2|2γ − 1 + λ|
(k = 3 − λ + |2γ − 1 + λ|; γ ∈ C − {0}).
(2.15)
Further putting λ = 0 in Corollary 2.1, we get Corollary 2.2. Let the function f (z ) ∈ A be analytic in the open unit disk ∆ and suppose that g ∈ S (γ ). If f (z ) is majorized by g (z ) in ∆, then
|f 0 (z )| ≤ |g 0 (z )| (|z | ≤ r1 ) where r1 = r1 (γ ) =
3 + |2γ − 1| −
9 + 2|2γ − 1| + |2γ − 1|2
p
2|2γ − 1|
which is a known result obtained by Altinas et al. [7]. For γ = 1, the above corollary reduces to the following result: Corollary 2.3. Let the function f (z ) ∈ A be an analytic and univalent in the open unit disk ∆ and suppose that g ∈ S ∗ = S ∗ (0). If f (z ) is majorized by g (z ) in ∆, then
|f 0 (z )| ≤ |g 0 (z )|
(|z | ≤ 2 −
√
3)
which is a known result obtained by MacGregor [1]. References [1] T.H. MacGreogor, Majorization by univalent functions, Duke Math. J. 34 (1967) 95–102. [2] S. Owa, On distortion theorems I, Kyungpook Math. J. 18 (1) (1978) 53–59. [3] H.M. Srivastava, S. Owa (Eds.), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, 1989, (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York. [4] J. Patel, A.K. Mishra, On certain subclasses of multivalent functions associated with an extended fractional differintegral operator, J. Math. Anal. Appl. 332 (2007) 109–122. [5] M.A. Naser, M.K. Aouf, Starlike function of complex order, J. Natur. Sci. Math. 25 (1985) 1–12. [6] P. Wiatrowski, On the coefficients of some family of holomorphic functions, Zeszyry Nauk. Uniw. Lddz. Nauk. Mat.-Przyrod. 39 (2) (1970) 75–85. [7] O. Altinas, Ö. Özkan, H.M. Srivastava, Majorization by starlike functions of complex order, Complex Var. 46 (2001) 207–218. [8] Z. Nehari, Conformal Mapping, Macgraw-Hill Book Company, New York, Toronto, London, 1955.