Gen. Math. Notes, Vol. 17, No. 2, August, 2013, pp.8-14
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
A Study of a New Family of Functions on the
Space of Analytic Functions
Abdul Rahman S. Juma1 and Hazha Zirar2
1
Department of Mathematics
University of Anbar, Ramadi, Iraq
E-mail: dr− juma@hotmail.com
2
Department of Mathematics, College of Science
University of Salahaddin Erbil, Kurdistan, Iraq
E-mail: hazhazirar@yahoo.com
(Received: 6-5-13 / Accepted: 15-6-13)
Abstract
By making use of a linear differential operator, we give some applications
of the new families of analytic functions on the same space associated with
quasi-Hadamard product in the unit disk U .
Keywords: Analytic functions, Differential operator, Quasi-Hadamard
product.
1
Introduction
Let H be the class of functions analytic in U = {z ∈ C : |z| < 1} and H[a, n]
be the subclass of H consisting of functions of the form f (z) = a + an z n +
an+1 z n+1 + an+2 z n+2 +P
.... Let Ap ⊆ H[a, n] denote the class of all functions of
p
k
the form f (z) = z + ∞
k=p+1 ak z , p ∈ N = {1, 2, ...}.
Let A denote
the class of functions of the form f (z) = z + a2 z 2 + a3 z 3 + ...
P∞
or f (z) = z + n=2 an z n which are analytic in the open unit disk U = {z ∈
C : |z| < 1} normalized by f (0) = f 0 (0) − 1 = 0.
P
P∞
n
n
For functions f (z) = z+ ∞
n=2 an z and g(z) = z+
n=2 bn z the Hadamard
9
A Study of a New Family of Functions on the...
product (or convolution) f ∗ g is defined by
(f ∗ g)(z) = z +
∞
X
an b n z n .
n=2
see [7], for f ∈ Ap we define the operator as follows:
Θ0p (β, γ)f (z) = f (z);
(p(γ + 1) + β)Θ1p (β, γ)f (z) = βf (z) + p(γ + 1)(
zf 0 (z)
);
p
.
.
.
Θm
p (β, γ)f (z)
= D(Θm−1
(β, γ)).
p
This gives rise to
Θm
p (β, γ)f (z)
∞
X
β + (p + n − 1)(γ + 1) m k
) ak z , β, γ ≥ 0, p ∈ N, (1)
(
=z +
β + p(γ + 1)
k=2
p
which was given for k = p+1 in [4]. This operator generalize certain differential
operators which already exist in literature as under.
• β = λ, γ = 0 we get Θm
p (m, λ, 0) of Aghalary et al. differential operator
[1].
• β = λ, γ = 0 and p = 1 we get Cho-Kim [2] and Cho-Srivastava [3]
differential operator.
• β = 1, γ = 0 and p = 1 we get Uralegaddi and Somanatha differential
operator [9].
• β = 0, γ = 0 and p = 1 we get Salagean differential operator [6].
• β = l, γ = 0 and p = 1 we get Kumar et al. differential operator [5] and
Srivastava et al. differential operator [8].
Note that
0
m+1
(γ + 1)z(Θm
(β, γ)f (z) − βΘm
p (β, γ)f (z)) = (p(γ + 1) + β)Θp
p (β, γ)f (z).
Throughout this paper, we consider the functions of the form as follow
f (z) = a1 z +
∞
X
n=2
an z n , (a1 > 0, an ≥ 0),
(2)
10
Abdul Rahman S. Juma et al.
fi (z) = a1,i z +
∞
X
an,i z n , (a1,i > 0, an,i ≥ 0),
(3)
n=2
g(z) = b1 z +
∞
X
bn z n , (b1 > 0, bn ≥ 0),
(4)
n=2
gj (z) = b1,j z +
∞
X
bn,j z n , (b1,j > 0, bn,j ≥ 0),
(5)
n=2
be regular and univalent in the unit disc U = {z ∈ C : |z| < 1}.
For 0 ≤ ρ < 1, 0 ≤ δ < 1 and η ≥ 0, we let U(k, ρ, δ, η) denote the class of
functions f defined by (2) and satisfying the analytic criterion
<{
0
0
z(Θm
z(Θm
p (β, γ))
p (β, γ))
−δ}
>
η{
−1}.
m
0
m
0
(1 − ρ)(Θm
(1 − ρ)(Θm
p (β, γ)) + ρz(Θp (β, γ))
p (β, γ)) + ρz(Θp (β, γ))
Also let E(k, ρ, δ, η) denote the class of functions f defined by (2) and satisfying
the analytic criterion
<{
0
m
00
0
m
00
(Θm
(Θm
p (β, γ)) + z(Θp (β, γ))
p (β, γ)) + z(Θp (β, γ))
−
δ}
>
η{
− 1}.
0
m
00
0
m
00
(Θm
(Θm
p (β, γ)) + ρz(Θp (β, γ))
p (β, γ)) + ρz(Θp (β, γ))
A function f ∈ U(k, ρ, δ, η)(0 ≤ ρ < 1, 0 ≤ δ < 1, η ≥ 0) if and only if
∞
X
β + (p + n − 1)(γ + 1) k
(
) [n(1 + η) − (δ + η)(1 + nρ − ρ)]an,i | ≤ (1 − δ)|a1,i |,
β + p(γ + 1)
n=2
and f ∈ E(k, ρ, δ, η)(0 ≤ ρ < 1, 0 ≤ δ < 1, η ≥ 0) if and only if
∞
X
β + (p + n − 1)(γ + 1) k+1
(
) [n(1+η)−(δ +η)(1+nρ−ρ)]an,i | ≤ (1−δ)|a1,i |.
β + p(γ + 1)
n=2
A function f which is analytic in U belonging to the class Ms (k, ρ, δ, η) if and
only if
∞
X
β + (p + n − 1)(γ + 1) s
(
) [n(1 + η) − (δ + η)(1 + nρ − ρ)]an,i | ≤ (1 − δ)|a1,i |,
β + p(γ + 1)
n=2
(6)
where 0 ≤ ρ < 1, 0 ≤ δ < 1, η ≥ 0 and s is any fixed nonnegative real number.
For s = k and s = k + 1, it is identical to the family of functions denoted
by U(k, ρ, δ, η) and E(k, ρ, δ, η) respectively. Further, for any positive integer
s > h > h − 1 > ... > k + 1 > k, we have the inclusion relation
U(k, ρ, δ, η) ⊆ E(k, ρ, δ, η) ⊆ ... ⊆ Mh (k, ρ, δ, η) ⊆ Ms (k, ρ, δ, η).
A Study of a New Family of Functions on the...
11
The class Ms (k, ρ, δ, η) is nonempty for any nonnegative real number s as the
functions of the form
∞
X
f (z) = a1 z+ (
n=2
(β + p(γ + 1))s (1 − δ)
λn z n ,
s
(β + (p + n − 1)(γ + 1)) [n(1 + η) − (δ + η)(1 + nρ − ρ)]
where a1 > 0, λn ≥ 0 and
2
P∞
n=2
λn ≤ 1; satisfy the inequality (6).
Main Results
Theorem 2.1: Let the functions fi defined by (3) belonging to the family
of functions E(k, ρ, δ, η) defined on space of analytic functions for all i =
1, 2, ..., r. Then quasi-Hadamard product of f1 ∗ f2 ∗ ... ∗ fr belongs to the
family Mr(k+2)−1 (n, ρ, δ, η) on same space of analytic functions.
Proof: Since fi ∈ E(k, ρ, δ, η), implies
∞
X
β + (p + n − 1)(γ + 1) k+1
) [n(1+η)−(δ +η)(1+nρ−ρ)]an,i | ≤ (1−δ)|a1,i |,
(
β + p(γ + 1)
n=2
(7)
implies
|an,i | ≤ (
β + (p + n − 1)(γ + 1) −k−2
)
|a1,i |, ∀i = 1, 2, ..., r.
β + p(γ + 1)
(8)
Using (7) as well as (8) for i = 1, 2, ..., r − 1, implies
∞
r−1
X
Y
β + (p + n − 1)(γ + 1) r(k+2)−1
(
)
[n(1 + η) − (δ + η)(1 + nρ − ρ)]
|an,i | ≤
β
+
p(γ
+
1)
n=2
i=1
∞
r
X
Y
β + (p + n − 1)(γ + 1) k+1
(
) [n(1 + η) − (δ + η)(1 + nρ − ρ)]|an,r |
|a1,i | =
β + p(γ + 1)
n=2
i=1
(1 − δ)
r
Y
|a1,i |.
i=1
Thus
f1 ∗ f2 ∗ ... ∗ fr ∈ Mr(k+2)−1 (k, ρ, δ, η).
Hence the proof is complete.
Theorem 2.2: Let the functions fi defined by (3) belonging to the family
of functions U(k, ρ, δ, η) defined on space of analytic functions for all i =
12
Abdul Rahman S. Juma et al.
1, 2, ..., r. Then quasi-Hadamard product of f1 ∗ f2 ∗ ... ∗ fr belongs to the
family Mr(k+1)−1 (n, ρ, δ, η) on the same space of analytic functions.
Proof: Using the same techniques of the proof of Theorem 2.1, we proved that
∞
r
X
Y
β + (p + n − 1)(γ + 1) r(k+1)−1
(
)
[n(1 + η) − (δ + η)(1 + nρ − ρ)]
|an,i | ≤
β + p(γ + 1)
n=2
n=1
∞
r−1
X
Y
β + (p + n − 1)(γ + 1) k+1
(
) [n(1 + η) − (δ + η)(1 + nρ − ρ)]|an,r |
|a1,i | =
β + p(γ + 1)
n=2
i=1
(1 − δ)
r
Y
|a1,i |.
i=1
Thus
f1 ∗ f2 ∗ ... ∗ fr ∈ Mr(k+1)−1 (k, ρ, δ, η).
Hence the proof is complete.
Theorem 2.3: Let the functions fi defined by (3) belonging to the family
E(k, ρ, δ, η) of functions on space of analytic functions for all i = 1, 2, ..., r
and let gi defined by (5) belonging to family U(k, ρ, δ, η) of functions on space
of analytic functions for all j = 1, 2, ..., q. Then quasi-Hadamard product of
f1 ∗ f2 ∗ ... ∗ fr ∗ g1 ∗ g2 ∗ ... ∗ gq belongs to the class Mr(k+2)+q(k+1)−1 (n, ρ, δ, η)
on the same space of analytic functions.
Proof: Let us denote f1 ∗ f2 ∗ ... ∗ fr ∗ g1 ∗ g2 ∗ ... ∗ gq by H. Then
H(z) = [
r
Y
i=1
q
q
∞ Y
r
Y
X
Y
|a1,i |][ |b1,j |]z +
[ |an,i |][ |bn,j |]z n .
j=1
n=2 i=1
j=1
Since fi ∈ E(k, ρ, δ, η) and gj ∈ U(k, ρ, δ, η) , implies
∞
X
β + (p + n − 1)(γ + 1) k+1
) [n(1+η)−(δ+η)(1+nρ−ρ)]an,i | ≤ (1−δ)|a1,i |, ∀i = 1, 2, ..., r.
(
β
+
p(γ
+
1)
n=2
|an,i | ≤
(β + p(γ + 1))k+1 (1 − δ)|a1,i |
,
(β + (p + n − 1)(γ + 1))k+1 [n(1 + η) − (δ + η)(1 + nρ − ρ)]
|an,i | ≤ (
β + (p + n − 1)(γ + 1) −k−2
)
|a1,i |, ∀i = 1, 2, ..., r.
β + p(γ + 1)
(9)
|bn,i | ≤ (
β + (p + n − 1)(γ + 1) −k−1
)
|ab,i |, ∀i = 1, 2, ..., q.
β + p(γ + 1)
(10)
A Study of a New Family of Functions on the...
13
Also
∞
X
β + (p + n − 1)(γ + 1) k
(
) [n(1 + η) − (δ + η)(1 + nρ − ρ)]|bn,j | ≤ (1 − δ)|b1,j |.
β
+
p(γ
+
1)
n=2
(11)
Using (9), (11) and (10) for i = 1, 2, ..., r, j = q and j = 1, 2, ..., q − 1 respectively. We have (consider t = r(k + 2) + q(k + 1) − 1)
q
r
∞
Y
Y
X
β + (p + n − 1)(γ + 1) t
) [n(1+η)−(δ+η)(1+nρ−ρ)][ |an,i |][ |bn,j |] ≤
(
β + p(γ + 1)
n=2
i=1
j=1
q−1
r
∞
Y
Y
X
β + (p + n − 1)(γ + 1) k
) [n(1+η)−(δ+η)(1+nρ−ρ)]|bn,q |[ |a1,i |][ |b1,j |] =
(
β + p(γ + 1)
n=2
i=1
j=1
q−1
r
∞
Y
X
Y
β + (p + n − 1)(γ + 1) k
(
) [n(1+η)−(δ+η)(1+nρ−ρ)]|bn,q |[ |a1,i |][ |b1,j |] ≤
β + p(γ + 1)
n=2
i=1
j=1
q
r
Y
Y
(1 − δ)[ |a1,i |][ |b1,j |].
i=1
j=1
Thus
f1 ∗ f2 ∗ ... ∗ fr ∗ g1 ∗ g2 ∗ ... ∗ gq ∈ Mr(k+2)+q(k+1)−1 (n, ρ, δ, η).
Hence the proof is complete.
References
[1] R. Aghalary, R.M. Ali, S.B. Joshi and V. Ravichandran, Inequalities for
analytic functions defined by certain linear operators, Int. J. Math. Sci.,
4(2005), 267-274.
[2] N.E. Cho and T.H. Kim, Multiplier transformations and strongly close to
convex functions, Bull. Korean Math. Soc., 40(2003), 399-410.
[3] N.E. Cho and H.M. Srivastava, Argument estimates of certain analytic
functions defined by a class of multiplier transformations, Math. Comput.
Modeling, 37(2003), 39-49.
[4] I. Faisal, Z. Shareef and M. Darus, A study of a new subclass of multivalent
analytic functions, Revista Notas de Mathematica, 7(1)(39) (2011), 101110.
14
Abdul Rahman S. Juma et al.
[5] S.S. Kumar, H.C. Taneja and V. Ravichandran, Classes multivalent functions defined by Dziok-Srivastava linear operator and multiplier transformations, Kyungpook Math. J., 46(2006), 97-109.
[6] G.S. Salagean, Subclasses of univalent functiond, Lecture Notes in Math.,
Springer-Verlag, Heideberg, 1013(1983), 362-372.
[7] R.K. Sharma and D. Singh, An inequality of subclasses of univalent functions related to complex order by convolution method, Gen. Math. Notes,
3(2) (2011), 59-65.
[8] H.M. Srivastava, K. Suchithra, B.A. Stephen and S. Sivasubramanian,
Inclusion and neighborgood properties of certain subclasses of multivalent
functions of complex order, J. Ineq. Pure Appl. Math., 7(5) (2006), 1-8.
[9] B.A. Uralegaddi and C. Somanatha, Certain classes of univalent functions,
In: Current Topics in Analytic Functions Theory, (eds.) H.M. Srivastava
and S. Owa, World Scientific Company, Singpore, (1992).