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Some properties of certain subclasses of p -valent Bazilevic functions associated with the generalized operator
Applied Mathematics Letters 24 (2011) 1953–1958
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Some properties of certain subclasses of p-valent Bazilevic functions associated with the generalized operator M.K. Aouf a , T.M. Seoudy b,∗ a
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
b
Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt
article
abstract
info
Article history: Received 7 July 2010 Received in revised form 14 May 2011 Accepted 22 May 2011
The purpose of the present paper is to introduce two subclasses of p-valent Bazilevic functions by using the generalized operator and to investigate various properties for these subclasses. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Analytic function Bazilevic function Integral operator Hadamard product
1. Introduction Let A(p, n) denote the class of functions of the form: f (z ) = z p +
∞ −
ap+j z p+j
(p, n ∈ N = {1, 2, 3, . . .}),
(1.1)
j =n
which are analytic and p-valent in the open unit disc U = {z ∈ C : |z | < 1}. Let Pk (ρ) be the class of functions g (z ) analytic in U satisfying the properties g (0) = 1 and 2π
∫ 0
ℜ{g (z )} − ρ dθ ≤ kπ , 1−ρ
(1.2)
where z = reiθ , k ≥ 2 and 0 ≤ α < 1. This class was introduced by Padmanabhan and Parvatham [1]. For ρ = 0, the class Pk (0) = Pk was introduced by Pinchuk [2]. Also we note that P2 (ρ) = P (ρ), where P (ρ) is the class of functions with positive real part greater than ρ and P2 (0) = P , where P is the class of functions with positive real part. From (1.2), we have g (z ) ∈ Pk (ρ) if and only if there exist g1 , g2 ∈ P (ρ) such that g (z ) =
k 4
+
1 2
g1 (z ) −
k 4
−
1 2
g2 (z )
(z ∈ U ) .
It is known that [3] the class Pk (ρ) is a convex set.
∗
Corresponding author. E-mail addresses: [email protected] (M.K. Aouf), [email protected] (T.M. Seoudy).
0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.05.029
(1.3)
1954
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958
Also let the Hadamard product or convolution f ∗ g of two analytic functions f (z ) =
∞ −
ak z k ,
g (z ) =
∞ −
k=0
bk z k
(1.4)
k=0
be given (as usual) by
(f ∗ g ) (z ) =
∞ −
ak bk z k = (g ∗ f ) (z ).
(1.5)
k=0
Making use of the Hadamard product (or convolution) given by (1.5), we now define the Dziok–Srivastava operator H a1 , . . . , aq ; b1 , . . . , bs : A(p, n) → A(p, n),
(1.6)
which was introduced and studied in a series of recent papers by Dziok and Srivastava [4–6]; see also [7]. Indeed, for complex parameters a1 , . . . , aq ; b1 , . . . , bs (bj ̸∈ Z− 0 = {0, −1, −2, . . .} ; j = 1, . . . , s), the generalized hypergeometric function q Fs a1 , . . . , aq ; b1 , . . . , bs ; z is given by
∞ (a ) · · · a − 1 j q j zj q Fs a1 , . . . , aq ; b1 , . . . , bs ; z = (b1 )j · · · (bs )j j! j =0
(q ≤ s + 1; q, s ∈ N0 = N ∪ {0}; z ∈ U ) ,
(1.7)
where (x)j is the Pochhammer symbol defined (in terms of the Gamma function) by
Γ (x + j) 1 (j = 0) , = x (x + 1) · · · (x + j − 1) (j ∈ N) . Γ (x) Corresponding to a function hp a1 , . . . , aq ; b1 , . . . , bs ; z , defined by (x)j =
(1.8)
h p a1 , . . . , aq ; b 1 , . . . , b s ; z = z p q F s a1 , . . . , aq ; b 1 , . . . , b s ; z ,
(1.9)
Dziok and Srivastava [4] (see also [5]) considered a linear operator defined by the following Hadamard product: Hp a1 , . . . , aq ; b1 , . . . , bs ; z f (z ) = hp a1 , . . . , aq ; b1 , . . . , bs ; z ∗ f (z ).
(1.10)
Corresponding to the function hp a1 , . . . , aq ; b1 , . . . , bs ; z , defined by (1.9), we introduce a function hp
−1
a1 , . . . , aq ;
b1 , . . . , bs ; z given by
1 h p a1 , . . . , aq ; b 1 , . . . , b s ; z ∗ h − a1 , . . . , aq ; b1 , . . . , bs ; z = p
zp
(µ > 0) .
(1 − z )µ µ
(1.11)
Analogous to Hp a1 , . . . , aq ; b1 , . . . , bs , we now define the linear operator Hp a1 , . . . , aq ; b1 , . . . , bs ; z on A(p, n) as follows:
1 Hpµ a1 , . . . , aq ; b1 , . . . , bs f (z ) = h− a1 , . . . , aq ; b1 , . . . , bs ; z ∗ f (z ) p ai , b j ∈ C \ Z − 0 , i = 1, . . . , s, j = 1, . . . , q; µ > 0; z ∈ U ; f ∈ A(p, n) .
(1.12)
For convenience, we write Hpµ,q,s (a1 ) = Hpµ a1 , . . . , aq ; b1 , . . . , bs .
µ
µ
(1.13) µ
We note that H1,q,s (a1 ) = Hq,s (a1 ), where the linear operator Hq,s (a1 ) is introduced by Kwon and Cho [8]. It is easily verified from definition (1.12) that
′
z Hpµ,q,s (a1 + 1) f (z )
= a1 Hpµ,q,s (a1 ) f (z ) − (a1 − p) Hpµ,q,s (a1 + 1) f (z ) ,
(1.14)
and
′
z Hpµ,q,s (a1 ) f (z )
1 µ = µHpµ+ ,q,s (a1 ) f (z ) − (µ − p) Hp,q,s (a1 ) f (z ).
µ
(1.15)
Next, by using the linear operator Hp,q,s (a1 ), we introduce the two subclasses of p-valent Bazilevic function of A(p, n) as follows:
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958 n,µ
Definition 1. A function f ∈ A(p, n) is said to be in the class Bk if it satisfies the following condition:
(1 − γ )
µ
Hp,q,s (a1 ) f (z ) zp
α
+γ
µ+1
Hp,q,s (a1 ) f (z ) µ
1955 n,µ
a1 , . . . , aq ; b1 , . . . , bs ; γ , α, ρ = Bk,q,s (a1 ; p, γ , α, ρ)
µ
Hp,q,s (a1 ) f (z )
Hp,q,s (a1 ) f (z )
α
zp
∈ Pk (ρ)
(k ≥ 2; α, γ , µ > 0; 0 ≤ ρ < 1; z ∈ U ) . Remarks. (i) For γ = 1, q = 2, s = 1, a1 = a, a2 = b, b1 = c a, b, c ∈ C \ Z− and µ = λ + p (λ > −p), we have 0
n,λ+p,
n,λ+p
(a, b; c ; p, 0, α, ρ) = Bkn,λ (a, b; c ; p, α, ρ), where α 1 Ipλ+ Ipλ,n (a, b; c ) f (z ) , n ( a, b ; c ) f ( z ) n,λ Bk (a, b; c ; p, α, ρ) = f ∈ A(p, n) : ∈ Pk (ρ) Ipλ,n (a, b; c ) f (z ) zp
Bk,2,1 (a; p, 0, α, ρ) = Bk
was introduced by Noor and Muhammad [9] and Ipλ,n (a, b; c ) is a linear operator introduced by Fu and Liu [10];
A = 2, q = 2, s = 1, a1 = b1 , a2 = µ > 0, and ρ = 11− 1 ≤ B < A ≤ 1), we have −B (− 1−A n b1 , µ; b1 ; γ , α, 1−B = B (γ , α, p, A, B), where α α ′ f (z ) zf (z ) f (z ) 1 + Az n B (γ , α, p, A, B) = f ∈ A(p, n) : (1 − γ ) +γ ≺ zp pf (z ) zp 1 + Bz
(ii) For k n,µ B2
is p-valent Bazilevic functions which introduced and studied by Liu [11]. n,µ
Definition 2. A function f ∈ A(p, n) is said to be in the class Nk if it satisfies the following condition:
[
(1 − γ )
µ
Hp,q,s (a1 + 1) f (z )
α
zp
+γ
µ
Hp,q,s (a1 ) f (z )
n,µ
a1 , . . . , aq ; b1 , . . . , bs ; γ , α, ρ = Nk,q,s (a1 ; p, γ , α, ρ)
µ
µ
Hp,q,s (a1 + 1) f (z )
Hp,q,s (a1 + 1) f (z )
α ]
zp
∈ Pk (ρ)
(k ≥ 2; α, γ , µ > 0; 0 ≤ ρ < 1; z ∈ U ) . n,µ
n,µ
In this paper, we investigate several properties of the classes Bk,q,s (a1 ; p, γ , α, ρ) and Nk,q,s (a1 ; p, γ , α, ρ) associated with the operator
µ Hp,q,s
(a1 ).
2. Main results The following lemma will be required in our investigation. Lemma 1 ([12]). Let u = u1 + iu2 and v = v1 + iv2 and Ψ (u, v) be a complex-valued function satisfying the conditions: (i) Ψ (u, v) is continuous in a domain D ∈ C2 . (ii) (0, 1) ∈ D and Ψ (1, 0) > 0. (iii) ℜ {Ψ (iu2 , v1 )} > 0 whenever (iu2 , v1 ) ∈ D and v1 ≤ − 2n 1 + u22 .
′
If h(z ) = 1 + cn z n + cn+1 z n+1 + · · · is analytic in U such that h(z ), zh (z ) ∈ D and ℜ Ψ
′
h(z ), zh (z )
> 0 for z ∈ U,
′ then ℜ Ψ h(z ), zh (z ) > 0 in U. Unless otherwise mentioned, we assume throughout this paper that k ≥ 2, α, γ , µ > 0, 0 ≤ ρ < 1 and all powers are understood as principle values. n,µ
n,µ
Theorem 1. If f ∈ Bk,q,s (a1 ; p, γ , α, ρ), then f ∈ Bk,q,s (a1 ; p, 0, α, ρ1 ) and
µ
Hp,q,s (a1 ) f (z )
α
zp
∈ Pk (ρ1 ) ,
(2.1)
where ρ1 is given by
ρ1 =
2αρµ + nγ 2αµ + nγ
.
(2.2)
1956
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958
Proof. We begin by setting
µ
Hp,q,s (a1 ) f (z )
α
=
zp
k 4
+
1
2
{(1 − ρ1 ) h1 (z ) + ρ1 } −
k 4
+
1
2
{(1 − ρ1 ) h2 (z ) + ρ1 } ,
= (1 − ρ1 ) h(z ) + ρ1 ,
(2.3)
where hi (z ) is analytic in U with hi (0) = 1, i = 1, 2. Differentiating of (2.3) with respect to z, and using identity (1.14) in the resulting equation, we obtain
(1 − γ )
µ
Hp,q,s (a1 ) f (z )
α
zp
+γ
µ+1
µ
µ
Hp,q,s (a1 ) f (z )
Hp,q,s (a1 + 1) f (z ) ′
= (1 − ρ1 ) h(z ) + ρ1 +
Hp,q,s (a1 ) f (z )
γ (1 − ρ1 ) zh (z ) αµ
α
zp
∈ Pk (ρ)
(z ∈ U ) ,
this implies that
1 1−ρ
′
γ (1 − ρ1 ) zhi (z ) (1 − ρ1 ) hi (z ) + ρ1 − ρ + αµ
∈P
(z ∈ U ; i = 1, 2) .
′
We form the functional Ψ (u, v) by choosing u = hi (z ), v = zhi (z ),
Ψ (u, v) = (1 − ρ1 ) u + ρ1 − ρ +
γ (1 − ρ1 ) v . αµ
Clearly, the first two conditions of Lemma 1 are satisfied. Now, we verify condition (iii) as follows:
γ ( 1 − ρ 1 ) v1 αµ nγ (1 − ρ1 ) 1 + u22 ≤ ρ1 − ρ − 2αµ A + Bu22 = ,
ℜ {Ψ (iu2 , v1 )} = ρ1 − ρ + ℜ
2C
where A = 2αµ (ρ1 − ρ) − nγ (1 − ρ1 ) ,
B = −nγ (1 − ρ1 ) < 0,
C = 2αµ > 0.
We note that ℜ {Ψ (iu2 , v1 )} < 0 if and only if A = 0, B < 0, C > 0 and this gives us ρ1 as given by (2.1) and B < 0 gives us 0 ≤ ρ1 < 1. Therefore, applying Lemma 1, hi (z ) ∈ P (i = 1, 2) and consequently h(z ) ∈ Pk (ρ1 ) for z ∈ U. This completes the proof of Theorem 1. Taking γ = 0, q = 2, s = 1, a1 = a, a2 = b, b1 = c a, b, c ∈ C \ Z− 0 and µ = λ + p (λ > −p) in Theorem 1, we obtain the following result which improves the result of Noor and Muhammad [9, Theorem 2.2].
n,λ
(a, b; c ; p, α, ρ), then α Ipλ,n (a, b; c ) f (z ) ∈ Pk (ρ2 ) , p
Corollary 1. If f ∈ Bk
(2.4)
z
where ρ2 is given by
ρ2 =
2αρ (λ + p) + nγ 2α (λ + p) + nγ
.
(2.5) n,µ
Similarly, we can prove the following theorem for the class Nk,q,s (a1 ; p, γ , α, ρ). n,µ
n,µ
Theorem 2. If f ∈ Nk,q,s (a1 ; p, γ , α, ρ), then f ∈ Nk,q,s (a1 ; p, 0, α, ρ3 ) and
µ
Hp,q,s (a1 + 1) f (z ) zp
α
∈ Pk (ρ3 ) ,
(2.6)
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958
1957
where ρ3 is given by
ρ3 =
2αρ a1 + nγ 2α a1 + nγ
.
(2.7)
µ
µ
Theorem 3. If f ∈ Bk,p (a1 ; γ , α, ρ), then f ∈ Bk,p (a1 ; 0, α, ρ4 ) and
µ
Hp,q,s (a1 ) f (z )
α2
zp
∈ Pk (ρ4 ) ,
(2.8)
where ρ4 is given by
ρ4 =
nγ +
n2 γ 2 + 4 (αµ + nγ ) αµρ 2 (αµ + nγ )
.
(2.9)
µ
Proof. Let f ∈ Bk,p (a1 ; γ , α, ρ) and let
µ
Hp,q,s (a1 ) f (z )
α
=
zp
k 4
+
1
2
[(1 − ρ4 ) h1 (z ) + ρ4 ]2 −
k 4
+
1
2
[(1 − ρ4 ) h2 (z ) + ρ4 ]2
= [(1 − ρ4 ) h(z ) + ρ4 ]2 ,
(2.10)
where hi (z ) is analytic in U with hi (0) = 1, i = 1, 2. Differentiating both sides (2.10) with respect to z, and using identity (1.15) in the resulting equation, we obtain
(1 − γ )
µ
Hp,q,s (a1 ) f (z )
α
zp
+γ
µ+1
Hp,q,s (a1 ) f (z )
µ
µ
Hp,q,s (a1 ) f (z )
Hp,q,s (a1 ) f (z )
α
zp
′
= [(1 − ρ4 ) h(z ) + ρ4 ] + [(1 − ρ4 ) h(z ) + ρ4 ] 2
2γ (1 − ρ4 ) zh (z )
αµ
∈ Pk (ρ)
(z ∈ U ) ,
this implies that 1 1−ρ
′
[(1 − ρ4 ) hi (z ) + ρ4 ] + [(1 − ρ4 ) h(z ) + ρ4 ] 2
2γ (1 − ρ3 ) zhi (z )
αµ
−ρ ∈P
(z ∈ U ; i = 1, 2) .
′
We form the functional Ψ (u, v) by choosing u = hi (z ), v = zhi (z ),
Ψ (u, v) = [(1 − ρ4 ) u + ρ4 ]2 + [(1 − ρ4 ) u + ρ4 ]
2γ (1 − ρ3 ) v
αµ
− ρ.
Clearly, conditions (i) and (ii) of Lemma 1 are satisfied. Now, we verify condition (iii) as follows:
ℜ {Ψ (iu2 , v1 )} = ρ42 − (1 − ρ4 )2 u22 +
2γ ρ4 (1 − ρ4 ) v1
−ρ αµ nγ ρ4 (1 − ρ4 ) 1 + u22 ≤ ρ42 − ρ − (1 − ρ4 )2 u22 − αµ A + Bu22 = , 2C
where A = αµ ρ42 − ρ − nγ ρ4 (1 − ρ4 ) ,
B = − (1 − ρ4 )2 + nγ ρ4 (1 − ρ4 ) < 0,
C =
αµ 2
> 0.
We note that ℜ {Ψ (iu2 , v1 )} < 0 if and only if A = 0, B < 0, C > 0 and this gives us ρ4 as given by (2.9) and B < 0 gives us 0 ≤ ρ4 < 1. Therefore, applying Lemma 1, hi (z ) ∈ P (i = 1, 2) and consequently h(z ) ∈ Pk (ρ4 ) for z ∈ U. This completes the proof of Theorem 4. Taking γ = 0, q = 2, s = 1, a1 = a, a2 = b, b1 = c a, b, c ∈ C \ Z− 0 and µ = λ + p (λ > −p) in Theorem 3, we obtain the following result which improves the result of Noor and Muhammad [9, Theorem 2.6].
1958
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958 n,λ
Corollary 2. If f ∈ Bk
(a, b; c ; p, α, ρ), then
Ipλ,n (a, b; c ) f (z )
α2 ∈ Pk (ρ5 ) ,
zp
(2.11)
where ρ5 is given by
ρ5 =
nγ +
n2 γ 2 + 4 [α (λ + p) + nγ ] α (λ + p) ρ
2 [α (λ + p) + nγ ]
.
(2.12) n,µ
Similarly, we can prove the following theorem for the class Nk,q,s (a1 ; p, γ , α, ρ).
n,µ n,µ Theorem 4. If f ∈ Nk,q,s (a1 ; p, γ , α, ρ), then f ∈ Nk,q,s a1 ; p, 0, α2 , ρ6 and
µ
Hp,q,s (a1 + 1) f (z ) zp
α2
∈ Pk (ρ6 ) ,
(2.13)
where ρ6 is given by
ρ6 =
nγ +
n2 γ 2 + 4 (α a1 + nγ ) α a1 ρ
2 (α a1 + nγ )
.
(2.14)
Acknowledgments The authors are grateful to the referees for their valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
K.S. Padmanabhan, R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math. 31 (1975) 311–323. B. Pinchuk, Functions with bounded boundary rotation, Israel J. Math. 10 (1971) 7–16. K.I. Noor, On subclasses of close-to-convex functions of higher order, Int. J. Math. Math. Sci. 15 (1992) 279–290. J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1) (1999) 1–13. J. Dziok, H.M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math. 5 (2) (2002) 115–125. J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (1) (2003) 7–18. J.-L. Liu, H.M. Srivastava, Certain properties of the Dziok–Srivastava operator, Appl. Math. Comput. 159 (2) (2004) 485–493. O.S. Kwon, N.E. Cho, Inclusion properties for subclasses of analytic functions associated with the Dziok–Srivastava operator, J. Inequal. Appl. (2007) 1–10. Art. ID 51079. K.I. Noor, A. Muhammad, Some properties of the subclass of p-valent Bazilevic functions, Acta Univ. Apulensis 17 (2009) 189–197. X.-L. Fu, M.-S. Liu, Some subclasses of analytic functions involving the generalized Noor integral operator, J. Math. Anal. Appl. 323 (1) (2006) 190–208. M. Liu, On certain subclass of p-valent functions, Soochow J. Math. 20 (2) (2000) 163–171. S.S. Miller, P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (2) (1978) 289–305.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Some properties of certain subclasses of p-valent Bazilevic functions associated with the generalized operator M.K. Aouf a , T.M. Seoudy b,∗ a
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
b
Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt
article
abstract
info
Article history: Received 7 July 2010 Received in revised form 14 May 2011 Accepted 22 May 2011
The purpose of the present paper is to introduce two subclasses of p-valent Bazilevic functions by using the generalized operator and to investigate various properties for these subclasses. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Analytic function Bazilevic function Integral operator Hadamard product
1. Introduction Let A(p, n) denote the class of functions of the form: f (z ) = z p +
∞ −
ap+j z p+j
(p, n ∈ N = {1, 2, 3, . . .}),
(1.1)
j =n
which are analytic and p-valent in the open unit disc U = {z ∈ C : |z | < 1}. Let Pk (ρ) be the class of functions g (z ) analytic in U satisfying the properties g (0) = 1 and 2π
∫ 0
ℜ{g (z )} − ρ dθ ≤ kπ , 1−ρ
(1.2)
where z = reiθ , k ≥ 2 and 0 ≤ α < 1. This class was introduced by Padmanabhan and Parvatham [1]. For ρ = 0, the class Pk (0) = Pk was introduced by Pinchuk [2]. Also we note that P2 (ρ) = P (ρ), where P (ρ) is the class of functions with positive real part greater than ρ and P2 (0) = P , where P is the class of functions with positive real part. From (1.2), we have g (z ) ∈ Pk (ρ) if and only if there exist g1 , g2 ∈ P (ρ) such that g (z ) =
k 4
+
1 2
g1 (z ) −
k 4
−
1 2
g2 (z )
(z ∈ U ) .
It is known that [3] the class Pk (ρ) is a convex set.
∗
Corresponding author. E-mail addresses: [email protected] (M.K. Aouf), [email protected] (T.M. Seoudy).
0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.05.029
(1.3)
1954
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958
Also let the Hadamard product or convolution f ∗ g of two analytic functions f (z ) =
∞ −
ak z k ,
g (z ) =
∞ −
k=0
bk z k
(1.4)
k=0
be given (as usual) by
(f ∗ g ) (z ) =
∞ −
ak bk z k = (g ∗ f ) (z ).
(1.5)
k=0
Making use of the Hadamard product (or convolution) given by (1.5), we now define the Dziok–Srivastava operator H a1 , . . . , aq ; b1 , . . . , bs : A(p, n) → A(p, n),
(1.6)
which was introduced and studied in a series of recent papers by Dziok and Srivastava [4–6]; see also [7]. Indeed, for complex parameters a1 , . . . , aq ; b1 , . . . , bs (bj ̸∈ Z− 0 = {0, −1, −2, . . .} ; j = 1, . . . , s), the generalized hypergeometric function q Fs a1 , . . . , aq ; b1 , . . . , bs ; z is given by
∞ (a ) · · · a − 1 j q j zj q Fs a1 , . . . , aq ; b1 , . . . , bs ; z = (b1 )j · · · (bs )j j! j =0
(q ≤ s + 1; q, s ∈ N0 = N ∪ {0}; z ∈ U ) ,
(1.7)
where (x)j is the Pochhammer symbol defined (in terms of the Gamma function) by
Γ (x + j) 1 (j = 0) , = x (x + 1) · · · (x + j − 1) (j ∈ N) . Γ (x) Corresponding to a function hp a1 , . . . , aq ; b1 , . . . , bs ; z , defined by (x)j =
(1.8)
h p a1 , . . . , aq ; b 1 , . . . , b s ; z = z p q F s a1 , . . . , aq ; b 1 , . . . , b s ; z ,
(1.9)
Dziok and Srivastava [4] (see also [5]) considered a linear operator defined by the following Hadamard product: Hp a1 , . . . , aq ; b1 , . . . , bs ; z f (z ) = hp a1 , . . . , aq ; b1 , . . . , bs ; z ∗ f (z ).
(1.10)
Corresponding to the function hp a1 , . . . , aq ; b1 , . . . , bs ; z , defined by (1.9), we introduce a function hp
−1
a1 , . . . , aq ;
b1 , . . . , bs ; z given by
1 h p a1 , . . . , aq ; b 1 , . . . , b s ; z ∗ h − a1 , . . . , aq ; b1 , . . . , bs ; z = p
zp
(µ > 0) .
(1 − z )µ µ
(1.11)
Analogous to Hp a1 , . . . , aq ; b1 , . . . , bs , we now define the linear operator Hp a1 , . . . , aq ; b1 , . . . , bs ; z on A(p, n) as follows:
1 Hpµ a1 , . . . , aq ; b1 , . . . , bs f (z ) = h− a1 , . . . , aq ; b1 , . . . , bs ; z ∗ f (z ) p ai , b j ∈ C \ Z − 0 , i = 1, . . . , s, j = 1, . . . , q; µ > 0; z ∈ U ; f ∈ A(p, n) .
(1.12)
For convenience, we write Hpµ,q,s (a1 ) = Hpµ a1 , . . . , aq ; b1 , . . . , bs .
µ
µ
(1.13) µ
We note that H1,q,s (a1 ) = Hq,s (a1 ), where the linear operator Hq,s (a1 ) is introduced by Kwon and Cho [8]. It is easily verified from definition (1.12) that
′
z Hpµ,q,s (a1 + 1) f (z )
= a1 Hpµ,q,s (a1 ) f (z ) − (a1 − p) Hpµ,q,s (a1 + 1) f (z ) ,
(1.14)
and
′
z Hpµ,q,s (a1 ) f (z )
1 µ = µHpµ+ ,q,s (a1 ) f (z ) − (µ − p) Hp,q,s (a1 ) f (z ).
µ
(1.15)
Next, by using the linear operator Hp,q,s (a1 ), we introduce the two subclasses of p-valent Bazilevic function of A(p, n) as follows:
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958 n,µ
Definition 1. A function f ∈ A(p, n) is said to be in the class Bk if it satisfies the following condition:
(1 − γ )
µ
Hp,q,s (a1 ) f (z ) zp
α
+γ
µ+1
Hp,q,s (a1 ) f (z ) µ
1955 n,µ
a1 , . . . , aq ; b1 , . . . , bs ; γ , α, ρ = Bk,q,s (a1 ; p, γ , α, ρ)
µ
Hp,q,s (a1 ) f (z )
Hp,q,s (a1 ) f (z )
α
zp
∈ Pk (ρ)
(k ≥ 2; α, γ , µ > 0; 0 ≤ ρ < 1; z ∈ U ) . Remarks. (i) For γ = 1, q = 2, s = 1, a1 = a, a2 = b, b1 = c a, b, c ∈ C \ Z− and µ = λ + p (λ > −p), we have 0
n,λ+p,
n,λ+p
(a, b; c ; p, 0, α, ρ) = Bkn,λ (a, b; c ; p, α, ρ), where α 1 Ipλ+ Ipλ,n (a, b; c ) f (z ) , n ( a, b ; c ) f ( z ) n,λ Bk (a, b; c ; p, α, ρ) = f ∈ A(p, n) : ∈ Pk (ρ) Ipλ,n (a, b; c ) f (z ) zp
Bk,2,1 (a; p, 0, α, ρ) = Bk
was introduced by Noor and Muhammad [9] and Ipλ,n (a, b; c ) is a linear operator introduced by Fu and Liu [10];
A = 2, q = 2, s = 1, a1 = b1 , a2 = µ > 0, and ρ = 11− 1 ≤ B < A ≤ 1), we have −B (− 1−A n b1 , µ; b1 ; γ , α, 1−B = B (γ , α, p, A, B), where α α ′ f (z ) zf (z ) f (z ) 1 + Az n B (γ , α, p, A, B) = f ∈ A(p, n) : (1 − γ ) +γ ≺ zp pf (z ) zp 1 + Bz
(ii) For k n,µ B2
is p-valent Bazilevic functions which introduced and studied by Liu [11]. n,µ
Definition 2. A function f ∈ A(p, n) is said to be in the class Nk if it satisfies the following condition:
[
(1 − γ )
µ
Hp,q,s (a1 + 1) f (z )
α
zp
+γ
µ
Hp,q,s (a1 ) f (z )
n,µ
a1 , . . . , aq ; b1 , . . . , bs ; γ , α, ρ = Nk,q,s (a1 ; p, γ , α, ρ)
µ
µ
Hp,q,s (a1 + 1) f (z )
Hp,q,s (a1 + 1) f (z )
α ]
zp
∈ Pk (ρ)
(k ≥ 2; α, γ , µ > 0; 0 ≤ ρ < 1; z ∈ U ) . n,µ
n,µ
In this paper, we investigate several properties of the classes Bk,q,s (a1 ; p, γ , α, ρ) and Nk,q,s (a1 ; p, γ , α, ρ) associated with the operator
µ Hp,q,s
(a1 ).
2. Main results The following lemma will be required in our investigation. Lemma 1 ([12]). Let u = u1 + iu2 and v = v1 + iv2 and Ψ (u, v) be a complex-valued function satisfying the conditions: (i) Ψ (u, v) is continuous in a domain D ∈ C2 . (ii) (0, 1) ∈ D and Ψ (1, 0) > 0. (iii) ℜ {Ψ (iu2 , v1 )} > 0 whenever (iu2 , v1 ) ∈ D and v1 ≤ − 2n 1 + u22 .
′
If h(z ) = 1 + cn z n + cn+1 z n+1 + · · · is analytic in U such that h(z ), zh (z ) ∈ D and ℜ Ψ
′
h(z ), zh (z )
> 0 for z ∈ U,
′ then ℜ Ψ h(z ), zh (z ) > 0 in U. Unless otherwise mentioned, we assume throughout this paper that k ≥ 2, α, γ , µ > 0, 0 ≤ ρ < 1 and all powers are understood as principle values. n,µ
n,µ
Theorem 1. If f ∈ Bk,q,s (a1 ; p, γ , α, ρ), then f ∈ Bk,q,s (a1 ; p, 0, α, ρ1 ) and
µ
Hp,q,s (a1 ) f (z )
α
zp
∈ Pk (ρ1 ) ,
(2.1)
where ρ1 is given by
ρ1 =
2αρµ + nγ 2αµ + nγ
.
(2.2)
1956
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958
Proof. We begin by setting
µ
Hp,q,s (a1 ) f (z )
α
=
zp
k 4
+
1
2
{(1 − ρ1 ) h1 (z ) + ρ1 } −
k 4
+
1
2
{(1 − ρ1 ) h2 (z ) + ρ1 } ,
= (1 − ρ1 ) h(z ) + ρ1 ,
(2.3)
where hi (z ) is analytic in U with hi (0) = 1, i = 1, 2. Differentiating of (2.3) with respect to z, and using identity (1.14) in the resulting equation, we obtain
(1 − γ )
µ
Hp,q,s (a1 ) f (z )
α
zp
+γ
µ+1
µ
µ
Hp,q,s (a1 ) f (z )
Hp,q,s (a1 + 1) f (z ) ′
= (1 − ρ1 ) h(z ) + ρ1 +
Hp,q,s (a1 ) f (z )
γ (1 − ρ1 ) zh (z ) αµ
α
zp
∈ Pk (ρ)
(z ∈ U ) ,
this implies that
1 1−ρ
′
γ (1 − ρ1 ) zhi (z ) (1 − ρ1 ) hi (z ) + ρ1 − ρ + αµ
∈P
(z ∈ U ; i = 1, 2) .
′
We form the functional Ψ (u, v) by choosing u = hi (z ), v = zhi (z ),
Ψ (u, v) = (1 − ρ1 ) u + ρ1 − ρ +
γ (1 − ρ1 ) v . αµ
Clearly, the first two conditions of Lemma 1 are satisfied. Now, we verify condition (iii) as follows:
γ ( 1 − ρ 1 ) v1 αµ nγ (1 − ρ1 ) 1 + u22 ≤ ρ1 − ρ − 2αµ A + Bu22 = ,
ℜ {Ψ (iu2 , v1 )} = ρ1 − ρ + ℜ
2C
where A = 2αµ (ρ1 − ρ) − nγ (1 − ρ1 ) ,
B = −nγ (1 − ρ1 ) < 0,
C = 2αµ > 0.
We note that ℜ {Ψ (iu2 , v1 )} < 0 if and only if A = 0, B < 0, C > 0 and this gives us ρ1 as given by (2.1) and B < 0 gives us 0 ≤ ρ1 < 1. Therefore, applying Lemma 1, hi (z ) ∈ P (i = 1, 2) and consequently h(z ) ∈ Pk (ρ1 ) for z ∈ U. This completes the proof of Theorem 1. Taking γ = 0, q = 2, s = 1, a1 = a, a2 = b, b1 = c a, b, c ∈ C \ Z− 0 and µ = λ + p (λ > −p) in Theorem 1, we obtain the following result which improves the result of Noor and Muhammad [9, Theorem 2.2].
n,λ
(a, b; c ; p, α, ρ), then α Ipλ,n (a, b; c ) f (z ) ∈ Pk (ρ2 ) , p
Corollary 1. If f ∈ Bk
(2.4)
z
where ρ2 is given by
ρ2 =
2αρ (λ + p) + nγ 2α (λ + p) + nγ
.
(2.5) n,µ
Similarly, we can prove the following theorem for the class Nk,q,s (a1 ; p, γ , α, ρ). n,µ
n,µ
Theorem 2. If f ∈ Nk,q,s (a1 ; p, γ , α, ρ), then f ∈ Nk,q,s (a1 ; p, 0, α, ρ3 ) and
µ
Hp,q,s (a1 + 1) f (z ) zp
α
∈ Pk (ρ3 ) ,
(2.6)
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958
1957
where ρ3 is given by
ρ3 =
2αρ a1 + nγ 2α a1 + nγ
.
(2.7)
µ
µ
Theorem 3. If f ∈ Bk,p (a1 ; γ , α, ρ), then f ∈ Bk,p (a1 ; 0, α, ρ4 ) and
µ
Hp,q,s (a1 ) f (z )
α2
zp
∈ Pk (ρ4 ) ,
(2.8)
where ρ4 is given by
ρ4 =
nγ +
n2 γ 2 + 4 (αµ + nγ ) αµρ 2 (αµ + nγ )
.
(2.9)
µ
Proof. Let f ∈ Bk,p (a1 ; γ , α, ρ) and let
µ
Hp,q,s (a1 ) f (z )
α
=
zp
k 4
+
1
2
[(1 − ρ4 ) h1 (z ) + ρ4 ]2 −
k 4
+
1
2
[(1 − ρ4 ) h2 (z ) + ρ4 ]2
= [(1 − ρ4 ) h(z ) + ρ4 ]2 ,
(2.10)
where hi (z ) is analytic in U with hi (0) = 1, i = 1, 2. Differentiating both sides (2.10) with respect to z, and using identity (1.15) in the resulting equation, we obtain
(1 − γ )
µ
Hp,q,s (a1 ) f (z )
α
zp
+γ
µ+1
Hp,q,s (a1 ) f (z )
µ
µ
Hp,q,s (a1 ) f (z )
Hp,q,s (a1 ) f (z )
α
zp
′
= [(1 − ρ4 ) h(z ) + ρ4 ] + [(1 − ρ4 ) h(z ) + ρ4 ] 2
2γ (1 − ρ4 ) zh (z )
αµ
∈ Pk (ρ)
(z ∈ U ) ,
this implies that 1 1−ρ
′
[(1 − ρ4 ) hi (z ) + ρ4 ] + [(1 − ρ4 ) h(z ) + ρ4 ] 2
2γ (1 − ρ3 ) zhi (z )
αµ
−ρ ∈P
(z ∈ U ; i = 1, 2) .
′
We form the functional Ψ (u, v) by choosing u = hi (z ), v = zhi (z ),
Ψ (u, v) = [(1 − ρ4 ) u + ρ4 ]2 + [(1 − ρ4 ) u + ρ4 ]
2γ (1 − ρ3 ) v
αµ
− ρ.
Clearly, conditions (i) and (ii) of Lemma 1 are satisfied. Now, we verify condition (iii) as follows:
ℜ {Ψ (iu2 , v1 )} = ρ42 − (1 − ρ4 )2 u22 +
2γ ρ4 (1 − ρ4 ) v1
−ρ αµ nγ ρ4 (1 − ρ4 ) 1 + u22 ≤ ρ42 − ρ − (1 − ρ4 )2 u22 − αµ A + Bu22 = , 2C
where A = αµ ρ42 − ρ − nγ ρ4 (1 − ρ4 ) ,
B = − (1 − ρ4 )2 + nγ ρ4 (1 − ρ4 ) < 0,
C =
αµ 2
> 0.
We note that ℜ {Ψ (iu2 , v1 )} < 0 if and only if A = 0, B < 0, C > 0 and this gives us ρ4 as given by (2.9) and B < 0 gives us 0 ≤ ρ4 < 1. Therefore, applying Lemma 1, hi (z ) ∈ P (i = 1, 2) and consequently h(z ) ∈ Pk (ρ4 ) for z ∈ U. This completes the proof of Theorem 4. Taking γ = 0, q = 2, s = 1, a1 = a, a2 = b, b1 = c a, b, c ∈ C \ Z− 0 and µ = λ + p (λ > −p) in Theorem 3, we obtain the following result which improves the result of Noor and Muhammad [9, Theorem 2.6].
1958
M.K. Aouf, T.M. Seoudy / Applied Mathematics Letters 24 (2011) 1953–1958 n,λ
Corollary 2. If f ∈ Bk
(a, b; c ; p, α, ρ), then
Ipλ,n (a, b; c ) f (z )
α2 ∈ Pk (ρ5 ) ,
zp
(2.11)
where ρ5 is given by
ρ5 =
nγ +
n2 γ 2 + 4 [α (λ + p) + nγ ] α (λ + p) ρ
2 [α (λ + p) + nγ ]
.
(2.12) n,µ
Similarly, we can prove the following theorem for the class Nk,q,s (a1 ; p, γ , α, ρ).
n,µ n,µ Theorem 4. If f ∈ Nk,q,s (a1 ; p, γ , α, ρ), then f ∈ Nk,q,s a1 ; p, 0, α2 , ρ6 and
µ
Hp,q,s (a1 + 1) f (z ) zp
α2
∈ Pk (ρ6 ) ,
(2.13)
where ρ6 is given by
ρ6 =
nγ +
n2 γ 2 + 4 (α a1 + nγ ) α a1 ρ
2 (α a1 + nγ )
.
(2.14)
Acknowledgments The authors are grateful to the referees for their valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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