paper 19-17-2009. 109KB Jun 04 2011 12:03:07 AM
ACTA UNIVERSITATIS APULENSIS
No 17/2009
SOME PROPERTIES OF THE SUBCLASS OF P -VALENT
BAZILEVIC FUNCTIONS
Khalida Inayat Noor and Ali Muhammad
Abstract. In this paper, we define some new class, Bkλ (a, b, c, p, n, α, ρ)
λ
by using the integral operators Ip,n
(a, b, c)f (z). We also derive some interesting
properties of functions belonging to the class Bkλ (a, b, c, p, n, α, ρ). Our results
generalizing the works of Owa and Cho, see [2, 10].
2000 Mathematics Subject Classification: 30C45, 30C50.
Keywords and phrases: Analytic functions; Bazilevic functions, Functions
with positive real part, Integral operator.
1. Introduction
Let An (p) denote the class of functions f (z) normalized by
p
f (z) = z +
∞
X
ap+k z p+k
(p, n ∈ N = {1, 2, 3, ...})
(1.1)
k=n
which are analytic and p-valent in the unit disk
E = {z : z ∈ C and |z| < 1}.
Let Pk (ρ) be the class of functions h(z) analytic in E satisfying the properties h(0) = 1 and
Z2π
Re h(z) − ρ
dθ ≤ kπ
(1.2)
1−ρ
0
where z = reiθ , k ≥ 2 and 0 ≤ ρ < 1. This class has been introduced in [12].
We note, for ρ = 0 we obtain the class Pk defined and studied in [13], and for
189
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
ρ = 0, k = 2 we have the well known class P of functions with positive real
part. The case k = 2 gives the class P (ρ) of functions with positive real part
greater than ρ. From (1.4) we can easily deduce that h ∈ Pk (ρ) if, and only if,
there exists h1 , h2 ∈ P (ρ) such that for z ∈ E,
k 1
k 1
h(z) =
h1 (z) −
h2 (z).
(1.3)
+
−
4 2
4 2
For functions fj (z) ∈An (p), given by
fj (z) = z p +
∞
X
ap+k,j z p+k
(j = 1, 2),
(1.4)
k=n
we define the Hadmard product (or convolution) of f1 (z) and f2 (z) by
p
(f1 ∗ f2 )(z) = z +
∞
X
ap+k,1 ap+k,2 z p+k = (f2 ∗ f1 )(z)
(z ∈ E)
(1.5)
k=n
In our present investigation we shall make use of the Gauss hypergeometric
functions defined by
2 F1 (a, b; c; z) =
∞
X
(a)k (b)k z k
k=0
Z−
0
(c)k (1)k
,
(1.6)
where a, b, c ∈ C, a, b, c ∈
/
= {0, −1, −2, ...}, and (x)k denote the Pochhammer symbol (or the shifted factorial) given, in terms of the Gamma function
Γ, by
Γ(k + n)
x(x + 1)...(x + k − 1), k ∈ N
=
(x)k =
1,
k=0
Γ(k)
We note that the series defined by (1.6) converges absolutely for z ∈ E and
hence 2 F1 (a, b; z) represents an analytic function in the open unit disk E, see
[15].
We introduce a function (z p 2 F1 (a, b; c; z))(−1) given by
(z p 2 F1 (a, b; c; z))(z p 2 F1 (a, b; c; z))(−1) =
zp
, λ > −p,
(1 − z)λ+p
(1.7)
this leads us to a family of linear operators:
λ
Ip,n
(a, b, c)f (z) = (z p 2 F1 (a, b; c; z))(−1) ∗f (z), (a, b, c ∈ R\Z0− , λ > −p, z ∈ E).
(1.8)
190
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
λ
It is evident that I1,1
(a, n + 1, a)f (z) is the Noor integral operator, see [1,
4, 6, 7, 8, 9] which has fundament and significant applications in the geometric
λ
functions theory. The operatorIp,n
(a, 1, c) was introduced by Cho et al [3].
After some computations, we obtain
λ
Ip,n
(a, b, c)f (z) = z p +
∞
X
(c)k (λ + p)k
k=n
(a)k (b)k
ap+k z p+k
(1.9)
From equation (1.8) we deduce that
λ
λ
(a, p + λ, a)f (z) = f (z) and Ip,n
(a, p, a) =
Ip,n
zf ′ (z)
,
p
λ
λ+1
λ
z(Ip,n
(a, b; c)f (z))′ = (λ + p)Ip,n
(a, b; c)f (z) − λIp,n
(a, b; c)f (z),
(1.10)
λ
Using the operator Ip,n
(a, b; c)f (z) we now define a new subclass of An (p)
as follows:
Definition 1.1. Let f (z) ∈An (p). Then f (z) ∈ Bkλ (a, b, c, p, n, α, ρ) if and
only if
!
!α
λ
λ+1
Ip,n
(a, b; c)f (z)
Ip,n
(a, b; c)f (z)
∈ Pk (ρ),
λ (a, b; c)f (z)
Ip,n
zp
where k ≥ 2, α > 0, 0 ≤ ρ < p and z ∈ E.
Special Cases.
(i) B20 (a, 1 + λ, a, 1, n, α, ρ) = B(n, α, ρ) is the subclass of Bazileivic functions studied by [2, 10].
(ii) The class B20 (a, 1 + λ, a, 1, 1, α, 0) = B(α) is the subclass of Bazilevic
functions which has been studied by Singh [14], see also [11].
2. Preliminaries and main results
Lemma 2.1. [5] 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 ⊂ C 2 .
(ii) (1, 0) ∈ D and Ψ(1, 0) > 0.
−n
(1 + u22 ).
(iii) Re Ψ(iu2 , v1 ) ≤ 0 whenever (iu2 , v1 ) ∈ D and v1 ≤
2
191
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
If h(z) = 1 + cn z n + cn+1 z n+1 + ... is analytic in E such that (h(z), zh′ (z)) ∈
D and Re{Ψ(h(z), zh′ (z))} > 0 for z ∈ E, then Re h(z) > 0 in E.
Theorem 2.2. Let f (z) ∈ Bkλ (a, b, c, p, n, α, ρ). Then
(
)α
λ
Ip,n
(a, b, c)f (z)
∈ Pk (ρ1 ),
zp
where ρ1 is given by
ρ1 =
2α(λ + p)ρ + np
2α(λ + p)ρ + n
Proof. We begin by setting
(
)α
λ
Ip,n
(a, b, c)f (z)
= (p − ρ1 )h(z) + ρ1
zp
k 1
((p − ρ1 )h1 (z) + ρ1 )
+
=
4 2
k 1
((p − ρ1 )h2 (z) + ρ1 ) ,
−
−
4 2
(2.1)
(2.2)
so hi (z) is analytic in E, with hi (0) = 1, i = 1, 2.
Taking logarithmic differentiation of (2.2), and using the identity (1.10) in
the resulting equation we obtain
!
!α
λ+1
λ
Ip,n
(a, b, c)f (z)
Ip,n
(a, b, c)f (z)
=
λ (a, b, c)f (z)
Ip,n
zp
(p − ρ1 )zh′ (z)
= (p − ρ1 )h(z) + ρ1 +
∈ Pk (ρ), z ∈ E
α(λ + p)
This implies that
1
(p − ρ1 )zh′ (z)
(p − ρ1 )hi (z) + ρ1 − ρ +
∈ P, z ∈ E, i = 1, 2.
p−ρ
α(λ + p)
We form the functional Ψ(u, v) by choosing u = hi (z), v = zh′i (z).
(p − ρ1 )v
.
Ψ(u, v) = (p − ρ1 )u + ρ1 − ρ +
α(λ + p)
192
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
The first two conditions of Lemma 2.1 are clearly satisfied. We verify the
condition (iii) as follows:
(p − ρ1 )v1
Re{Ψ(iu2 , v1 )} = ρ1 − ρ + Re
α(λ + p)
n(p − ρ1 )(1 + u22 )
≤ ρ1 − ρ −
2α(λ + p)
2
A + Bu2
,
=
2C
where
A = 2α(λ + p)(ρ1 − ρ) − n(p − ρ1 ),
B = −n(p − ρ1 ),
C = 2α(λ + p) > 0.
We notice that Re{Ψ(iu2 , v1 )} ≤ 0 if and only if A ≤ 0, B ≤ 0 and this
gives us ρ1 as given by (2.1) and B ≤ 0 gives us 0 ≤ ρ1 < 1. Therefore
applying Lemma 2.1, hi ∈ P, i = 1, 2 and consequently h ∈ Pk (ρ1 ) for z ∈ E.
This completes the proof.
Corollary 2.3. If f (z) ∈ B20 (a, b, c, 1, n, α, ρ), then
(
)
λ
Ip,n
(a, b, c)f (z)
n + 2αρ
Re
>
, z ∈ E,
z
n + 2α
Corollary 2.4. If f (z) ∈ B20 (a, b, c, 1, n, 1, 0), then
(
)
λ
Ip,n
(a, b, c)f (z)
n
Re
>
, z ∈ E,
z
n+2
Corollary 2.5. If f (z) ∈ B20 (a, b, c, 1, n, 12 , ρ) then
)
(
λ
Ip,n
(a, b, c)f (z)
ρ+n
>
, z ∈ E,
Re
z
n+1
(2.3)
(2.4)
(2.5)
With certain choices of the parameters λ, k, a, b, c, p, α and ρ, we obtain the
corresponding works of [2, 10].
193
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
Theorem 2.6. Let f (z) ∈ Bkλ (a, b, c, p, n, α, ρ). Then
(
where
λ=
np +
λ
Ip,n
(a, b, c)f (z)
zp
p
) α2
∈ Pk (λ),
(np)2 + 4(α(λ + p) + n)(ρα(λ + p))
2(α(λ + p) + n)
(2.6)
Proof. Let
(
λ
Ip,n
(a, b, c)f (z)
zp
)α
= ((p − γ)h(z) + γ)2
k 1
=
((p − γ)h1 (z) + γ)2
+
4 2
k 1
−
−
((p − γ)h2 (z) + γ)2 ,
4 2
(2.7)
so hi (z) is analytic in E with hi (0) = 1, i = 1, 2.
Taking logarithmic differentiation of (2.7), and using the identity (1.10) in
the resulting equation we obtain that
!
!α
λ+1
Ip,n
(a, b, c)f (z)
Ipλ (a, b, c)f (z)
=
λ (a, b, c)f (z)
Ip,n
zp
= {(p − γ)h(z) + γ}2 +
2
′
{(p − γ)h(z) + γ}(p − γ)zh (z) ∈ Pk (ρ).
α(λ + p)
This implies that
1
2
2
′
{(p − γ)hi (z) + γ} +
{(p − γ)hi (z) + γ}(p − γ)zhi (z) − ρ ∈ P,
p−ρ
α(λ + p)
where z ∈ E, i = 1, 2. We form the functional Ψ(u, v) by choosing u =
hi (z), v = zh′i (z).
2
2
{(p − γ)u + γ}(p − γ)v − ρ .
Ψ(u, v) = {(p − γ)u + γ} +
α(λ + p)
194
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
2
γ(p − γ)v1 − ρ
Re{Ψ(iu2 , v1 )} = γ − (p −
+
α(λ + p)
nγ(p − γ)(1 + u22 )
2
2 2
≤ γ − ρ − (p − γ) u2 −
α(λ + p)
2
A + Bu2
,
=
2C
2
γ)2 u22
where
A = γ 2 α(λ + p) − ρα(λ + p) + nγ(p − γ),
B = −(p − γ)2 − nγ(p − γ)
α(λ + p)
C=
> 0.
2
We notice that Re{Ψ(iu2 , v1 )} ≤ 0 if and only if A ≤ 0, B ≤ 0 and this
gives us γ as given by (2.6) and B ≤ 0 gives us 0 ≤ γ < 1. Therefore applying
Lemma 2.1, hi ∈ P, i = 1, 2 and consequently h ∈ Pk (ρ1 ) for z ∈ E. This
completes the proof of Theorem 2.6.
Corollary 2.7. If f (z) ∈ B20 (a, b, c, 1, n, α, ρ) then
(
)
p
λ
Ip,n
(a, b, c)f (z)
n + n2 + 4(α + n)(ρα)
, z ∈ E.
Re
>
z
2(α + n)
Corollary 2.8. If f (z) ∈ B20 (a, b, c, 1, n, 1, ρ) then
(
)
p
λ
Ip,n
(a, b, c)f (z)
n + n2 + 4(1 + n)ρ
, z ∈ E.
Re
>
z
2(1 + n)
Corollary 2.9. If f (z) ∈ B20 (a, b, c, 1, n, 2, 0) then
(
)
λ
Ip,n
(a, b, c)f (z)
n
Re
, z ∈ E.
>
z
1+n
Again with certain choices of the parameters λ, k, a, b, c, p, α, ρ and we obtain the corresponding works of [2, 10].
Acknowledgement. We would like to thank Dr S. M. Junaid Zaidi, Rector CIIT for providing us excellent research facilities.
195
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
References
[1] N. E. Cho, The Noor integral operator and strongly close-to-convex functions, J. Math. Anal. Appl., 238 (2003), 202
[2] N. E. Cho, On certain subclasses of univalent functions, Bull. Korean
Math. Soc. 25 (1988), 215
[3] N. E. Cho, Oh Sang Kown and H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions
associated with a family of linear operators, J. Math. Anal. Appl., 292 (2004),
470
[4] J. L. Liu and K. I. Noor, Some properties of Noor integral operator, J.
Natur. Geom. 21 (2002), 8190.
[5] S. S. Miller and P. T. Mocanu, Second order differential inequalities in
the complex plane, J. Math. Anal. Appl., 65 (1978), 289
[6] K. I. Noor, Some classes of p-valent analytic functions defined by certain
integral operator, Appl. Math. Compute. 157 (2004), 835840.
[7] K. I. Noor, On new classes of integral operator, J. Nat. Geo, 16 (1999),
71
[8] K. I. Noor and M. A. Noor, On certain classes of analytic functions
defined by Noor integral operator, J. Math. Appl. 281 (2003), 244252.
[9] K. I. Noor and M. A. Noor, On integral operators, J. Math. Anal. Appl.,
238 (1999), 341
[10] S. Owa, On certain Bazilevic functions of order, Internat. J. Math.
and Math. Sci. 15 (1992), 613
[11] S. Owa and M. Obradovic, Certain subclasses of Bazilevic function of
type, Internat. J. Math. and Math. Sci., 9:2 (1986), 97105.
[12] B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math.,
10 (1971), 716.
[13] K. Padmanabhan and R. Parvatham, Properties of a class of functions
with bounded boundary rotation, Ann. Polon. Math., 31 (1975), 311323.
[14] R. Singh, On Bazilevic functions, Proc. Amer. Math. Sci., 38 (1973),
261267.
[15] E. T. Whittaker and G. N. Watson, A course on Modern Analysis:
An introduction to the general Theory of Infinite processes and of Analytic
Functions; With an account of the principle of Transcendental Functions, 4th
Edition, Cambridge University Press, Cambridge, 1927.
196
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
Authors:
Khalida Inayat Noor
Mathematics Department,
COMSATS Institute of Information Technology,
H-8/1 Islamabad, Pakistan.
e-mail: khalidanoor@hotmail.com
Ali Muhammad
Mathematics Department,
COMSATS Institute of Information Technology,
H-8/1 Islamabad, Pakistan.
e-mail: ali7887@gmail.com
197
No 17/2009
SOME PROPERTIES OF THE SUBCLASS OF P -VALENT
BAZILEVIC FUNCTIONS
Khalida Inayat Noor and Ali Muhammad
Abstract. In this paper, we define some new class, Bkλ (a, b, c, p, n, α, ρ)
λ
by using the integral operators Ip,n
(a, b, c)f (z). We also derive some interesting
properties of functions belonging to the class Bkλ (a, b, c, p, n, α, ρ). Our results
generalizing the works of Owa and Cho, see [2, 10].
2000 Mathematics Subject Classification: 30C45, 30C50.
Keywords and phrases: Analytic functions; Bazilevic functions, Functions
with positive real part, Integral operator.
1. Introduction
Let An (p) denote the class of functions f (z) normalized by
p
f (z) = z +
∞
X
ap+k z p+k
(p, n ∈ N = {1, 2, 3, ...})
(1.1)
k=n
which are analytic and p-valent in the unit disk
E = {z : z ∈ C and |z| < 1}.
Let Pk (ρ) be the class of functions h(z) analytic in E satisfying the properties h(0) = 1 and
Z2π
Re h(z) − ρ
dθ ≤ kπ
(1.2)
1−ρ
0
where z = reiθ , k ≥ 2 and 0 ≤ ρ < 1. This class has been introduced in [12].
We note, for ρ = 0 we obtain the class Pk defined and studied in [13], and for
189
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
ρ = 0, k = 2 we have the well known class P of functions with positive real
part. The case k = 2 gives the class P (ρ) of functions with positive real part
greater than ρ. From (1.4) we can easily deduce that h ∈ Pk (ρ) if, and only if,
there exists h1 , h2 ∈ P (ρ) such that for z ∈ E,
k 1
k 1
h(z) =
h1 (z) −
h2 (z).
(1.3)
+
−
4 2
4 2
For functions fj (z) ∈An (p), given by
fj (z) = z p +
∞
X
ap+k,j z p+k
(j = 1, 2),
(1.4)
k=n
we define the Hadmard product (or convolution) of f1 (z) and f2 (z) by
p
(f1 ∗ f2 )(z) = z +
∞
X
ap+k,1 ap+k,2 z p+k = (f2 ∗ f1 )(z)
(z ∈ E)
(1.5)
k=n
In our present investigation we shall make use of the Gauss hypergeometric
functions defined by
2 F1 (a, b; c; z) =
∞
X
(a)k (b)k z k
k=0
Z−
0
(c)k (1)k
,
(1.6)
where a, b, c ∈ C, a, b, c ∈
/
= {0, −1, −2, ...}, and (x)k denote the Pochhammer symbol (or the shifted factorial) given, in terms of the Gamma function
Γ, by
Γ(k + n)
x(x + 1)...(x + k − 1), k ∈ N
=
(x)k =
1,
k=0
Γ(k)
We note that the series defined by (1.6) converges absolutely for z ∈ E and
hence 2 F1 (a, b; z) represents an analytic function in the open unit disk E, see
[15].
We introduce a function (z p 2 F1 (a, b; c; z))(−1) given by
(z p 2 F1 (a, b; c; z))(z p 2 F1 (a, b; c; z))(−1) =
zp
, λ > −p,
(1 − z)λ+p
(1.7)
this leads us to a family of linear operators:
λ
Ip,n
(a, b, c)f (z) = (z p 2 F1 (a, b; c; z))(−1) ∗f (z), (a, b, c ∈ R\Z0− , λ > −p, z ∈ E).
(1.8)
190
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
λ
It is evident that I1,1
(a, n + 1, a)f (z) is the Noor integral operator, see [1,
4, 6, 7, 8, 9] which has fundament and significant applications in the geometric
λ
functions theory. The operatorIp,n
(a, 1, c) was introduced by Cho et al [3].
After some computations, we obtain
λ
Ip,n
(a, b, c)f (z) = z p +
∞
X
(c)k (λ + p)k
k=n
(a)k (b)k
ap+k z p+k
(1.9)
From equation (1.8) we deduce that
λ
λ
(a, p + λ, a)f (z) = f (z) and Ip,n
(a, p, a) =
Ip,n
zf ′ (z)
,
p
λ
λ+1
λ
z(Ip,n
(a, b; c)f (z))′ = (λ + p)Ip,n
(a, b; c)f (z) − λIp,n
(a, b; c)f (z),
(1.10)
λ
Using the operator Ip,n
(a, b; c)f (z) we now define a new subclass of An (p)
as follows:
Definition 1.1. Let f (z) ∈An (p). Then f (z) ∈ Bkλ (a, b, c, p, n, α, ρ) if and
only if
!
!α
λ
λ+1
Ip,n
(a, b; c)f (z)
Ip,n
(a, b; c)f (z)
∈ Pk (ρ),
λ (a, b; c)f (z)
Ip,n
zp
where k ≥ 2, α > 0, 0 ≤ ρ < p and z ∈ E.
Special Cases.
(i) B20 (a, 1 + λ, a, 1, n, α, ρ) = B(n, α, ρ) is the subclass of Bazileivic functions studied by [2, 10].
(ii) The class B20 (a, 1 + λ, a, 1, 1, α, 0) = B(α) is the subclass of Bazilevic
functions which has been studied by Singh [14], see also [11].
2. Preliminaries and main results
Lemma 2.1. [5] 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 ⊂ C 2 .
(ii) (1, 0) ∈ D and Ψ(1, 0) > 0.
−n
(1 + u22 ).
(iii) Re Ψ(iu2 , v1 ) ≤ 0 whenever (iu2 , v1 ) ∈ D and v1 ≤
2
191
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
If h(z) = 1 + cn z n + cn+1 z n+1 + ... is analytic in E such that (h(z), zh′ (z)) ∈
D and Re{Ψ(h(z), zh′ (z))} > 0 for z ∈ E, then Re h(z) > 0 in E.
Theorem 2.2. Let f (z) ∈ Bkλ (a, b, c, p, n, α, ρ). Then
(
)α
λ
Ip,n
(a, b, c)f (z)
∈ Pk (ρ1 ),
zp
where ρ1 is given by
ρ1 =
2α(λ + p)ρ + np
2α(λ + p)ρ + n
Proof. We begin by setting
(
)α
λ
Ip,n
(a, b, c)f (z)
= (p − ρ1 )h(z) + ρ1
zp
k 1
((p − ρ1 )h1 (z) + ρ1 )
+
=
4 2
k 1
((p − ρ1 )h2 (z) + ρ1 ) ,
−
−
4 2
(2.1)
(2.2)
so hi (z) is analytic in E, with hi (0) = 1, i = 1, 2.
Taking logarithmic differentiation of (2.2), and using the identity (1.10) in
the resulting equation we obtain
!
!α
λ+1
λ
Ip,n
(a, b, c)f (z)
Ip,n
(a, b, c)f (z)
=
λ (a, b, c)f (z)
Ip,n
zp
(p − ρ1 )zh′ (z)
= (p − ρ1 )h(z) + ρ1 +
∈ Pk (ρ), z ∈ E
α(λ + p)
This implies that
1
(p − ρ1 )zh′ (z)
(p − ρ1 )hi (z) + ρ1 − ρ +
∈ P, z ∈ E, i = 1, 2.
p−ρ
α(λ + p)
We form the functional Ψ(u, v) by choosing u = hi (z), v = zh′i (z).
(p − ρ1 )v
.
Ψ(u, v) = (p − ρ1 )u + ρ1 − ρ +
α(λ + p)
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K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
The first two conditions of Lemma 2.1 are clearly satisfied. We verify the
condition (iii) as follows:
(p − ρ1 )v1
Re{Ψ(iu2 , v1 )} = ρ1 − ρ + Re
α(λ + p)
n(p − ρ1 )(1 + u22 )
≤ ρ1 − ρ −
2α(λ + p)
2
A + Bu2
,
=
2C
where
A = 2α(λ + p)(ρ1 − ρ) − n(p − ρ1 ),
B = −n(p − ρ1 ),
C = 2α(λ + p) > 0.
We notice that Re{Ψ(iu2 , v1 )} ≤ 0 if and only if A ≤ 0, B ≤ 0 and this
gives us ρ1 as given by (2.1) and B ≤ 0 gives us 0 ≤ ρ1 < 1. Therefore
applying Lemma 2.1, hi ∈ P, i = 1, 2 and consequently h ∈ Pk (ρ1 ) for z ∈ E.
This completes the proof.
Corollary 2.3. If f (z) ∈ B20 (a, b, c, 1, n, α, ρ), then
(
)
λ
Ip,n
(a, b, c)f (z)
n + 2αρ
Re
>
, z ∈ E,
z
n + 2α
Corollary 2.4. If f (z) ∈ B20 (a, b, c, 1, n, 1, 0), then
(
)
λ
Ip,n
(a, b, c)f (z)
n
Re
>
, z ∈ E,
z
n+2
Corollary 2.5. If f (z) ∈ B20 (a, b, c, 1, n, 12 , ρ) then
)
(
λ
Ip,n
(a, b, c)f (z)
ρ+n
>
, z ∈ E,
Re
z
n+1
(2.3)
(2.4)
(2.5)
With certain choices of the parameters λ, k, a, b, c, p, α and ρ, we obtain the
corresponding works of [2, 10].
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K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
Theorem 2.6. Let f (z) ∈ Bkλ (a, b, c, p, n, α, ρ). Then
(
where
λ=
np +
λ
Ip,n
(a, b, c)f (z)
zp
p
) α2
∈ Pk (λ),
(np)2 + 4(α(λ + p) + n)(ρα(λ + p))
2(α(λ + p) + n)
(2.6)
Proof. Let
(
λ
Ip,n
(a, b, c)f (z)
zp
)α
= ((p − γ)h(z) + γ)2
k 1
=
((p − γ)h1 (z) + γ)2
+
4 2
k 1
−
−
((p − γ)h2 (z) + γ)2 ,
4 2
(2.7)
so hi (z) is analytic in E with hi (0) = 1, i = 1, 2.
Taking logarithmic differentiation of (2.7), and using the identity (1.10) in
the resulting equation we obtain that
!
!α
λ+1
Ip,n
(a, b, c)f (z)
Ipλ (a, b, c)f (z)
=
λ (a, b, c)f (z)
Ip,n
zp
= {(p − γ)h(z) + γ}2 +
2
′
{(p − γ)h(z) + γ}(p − γ)zh (z) ∈ Pk (ρ).
α(λ + p)
This implies that
1
2
2
′
{(p − γ)hi (z) + γ} +
{(p − γ)hi (z) + γ}(p − γ)zhi (z) − ρ ∈ P,
p−ρ
α(λ + p)
where z ∈ E, i = 1, 2. We form the functional Ψ(u, v) by choosing u =
hi (z), v = zh′i (z).
2
2
{(p − γ)u + γ}(p − γ)v − ρ .
Ψ(u, v) = {(p − γ)u + γ} +
α(λ + p)
194
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
2
γ(p − γ)v1 − ρ
Re{Ψ(iu2 , v1 )} = γ − (p −
+
α(λ + p)
nγ(p − γ)(1 + u22 )
2
2 2
≤ γ − ρ − (p − γ) u2 −
α(λ + p)
2
A + Bu2
,
=
2C
2
γ)2 u22
where
A = γ 2 α(λ + p) − ρα(λ + p) + nγ(p − γ),
B = −(p − γ)2 − nγ(p − γ)
α(λ + p)
C=
> 0.
2
We notice that Re{Ψ(iu2 , v1 )} ≤ 0 if and only if A ≤ 0, B ≤ 0 and this
gives us γ as given by (2.6) and B ≤ 0 gives us 0 ≤ γ < 1. Therefore applying
Lemma 2.1, hi ∈ P, i = 1, 2 and consequently h ∈ Pk (ρ1 ) for z ∈ E. This
completes the proof of Theorem 2.6.
Corollary 2.7. If f (z) ∈ B20 (a, b, c, 1, n, α, ρ) then
(
)
p
λ
Ip,n
(a, b, c)f (z)
n + n2 + 4(α + n)(ρα)
, z ∈ E.
Re
>
z
2(α + n)
Corollary 2.8. If f (z) ∈ B20 (a, b, c, 1, n, 1, ρ) then
(
)
p
λ
Ip,n
(a, b, c)f (z)
n + n2 + 4(1 + n)ρ
, z ∈ E.
Re
>
z
2(1 + n)
Corollary 2.9. If f (z) ∈ B20 (a, b, c, 1, n, 2, 0) then
(
)
λ
Ip,n
(a, b, c)f (z)
n
Re
, z ∈ E.
>
z
1+n
Again with certain choices of the parameters λ, k, a, b, c, p, α, ρ and we obtain the corresponding works of [2, 10].
Acknowledgement. We would like to thank Dr S. M. Junaid Zaidi, Rector CIIT for providing us excellent research facilities.
195
K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
References
[1] N. E. Cho, The Noor integral operator and strongly close-to-convex functions, J. Math. Anal. Appl., 238 (2003), 202
[2] N. E. Cho, On certain subclasses of univalent functions, Bull. Korean
Math. Soc. 25 (1988), 215
[3] N. E. Cho, Oh Sang Kown and H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions
associated with a family of linear operators, J. Math. Anal. Appl., 292 (2004),
470
[4] J. L. Liu and K. I. Noor, Some properties of Noor integral operator, J.
Natur. Geom. 21 (2002), 8190.
[5] S. S. Miller and P. T. Mocanu, Second order differential inequalities in
the complex plane, J. Math. Anal. Appl., 65 (1978), 289
[6] K. I. Noor, Some classes of p-valent analytic functions defined by certain
integral operator, Appl. Math. Compute. 157 (2004), 835840.
[7] K. I. Noor, On new classes of integral operator, J. Nat. Geo, 16 (1999),
71
[8] K. I. Noor and M. A. Noor, On certain classes of analytic functions
defined by Noor integral operator, J. Math. Appl. 281 (2003), 244252.
[9] K. I. Noor and M. A. Noor, On integral operators, J. Math. Anal. Appl.,
238 (1999), 341
[10] S. Owa, On certain Bazilevic functions of order, Internat. J. Math.
and Math. Sci. 15 (1992), 613
[11] S. Owa and M. Obradovic, Certain subclasses of Bazilevic function of
type, Internat. J. Math. and Math. Sci., 9:2 (1986), 97105.
[12] B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math.,
10 (1971), 716.
[13] K. Padmanabhan and R. Parvatham, Properties of a class of functions
with bounded boundary rotation, Ann. Polon. Math., 31 (1975), 311323.
[14] R. Singh, On Bazilevic functions, Proc. Amer. Math. Sci., 38 (1973),
261267.
[15] E. T. Whittaker and G. N. Watson, A course on Modern Analysis:
An introduction to the general Theory of Infinite processes and of Analytic
Functions; With an account of the principle of Transcendental Functions, 4th
Edition, Cambridge University Press, Cambridge, 1927.
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K.I. Noor, A. Muhammad - Some properties of the subclass of P -valent...
Authors:
Khalida Inayat Noor
Mathematics Department,
COMSATS Institute of Information Technology,
H-8/1 Islamabad, Pakistan.
e-mail: khalidanoor@hotmail.com
Ali Muhammad
Mathematics Department,
COMSATS Institute of Information Technology,
H-8/1 Islamabad, Pakistan.
e-mail: ali7887@gmail.com
197