Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, 99-114
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2012
Harmonic multivalent meromorphic functions
defined by an integral operator
Amit Kumar Yadav1
Abstract
The object of this article is to study a class MH (p) of harmonic
multivalent meromorphic functions of the form f (z) = h(z) + g(z) , 0 <
|z| < 1, where h and g are meromorphic functions. An integral operator
is considered and is used to define a subclass MH (p, α, m, c) of MH (p).
Some properties of MH (p) are studied with the properties like coefficient
condition, bounds, extreme points, convolution condition and convex
combination for functions belongs to MH (p, α, m, c) class.
Mathematics Subject Classification: 30C45, 30C50, 30C55
Keywords: Meromorphic Function, Starlike functions, Integral Operator
1
Introduction and Prelimnaries
A function f = u + iv, which is continuos complex-valued harmonic in a
domain D ⊂ C, if both u and v are real harmonic in D. Cluine and Sheil1
Department of Information Technology Ibra College of Technology, PO Box : 327,
Postal Code: 400, Alsharqiyah, Ibra, Sultanate of Oman, e-mail: amitkumar@ict.edu.om
Article Info: Received : July 2, 2012. Revised : August 2, 2012
Published online : December 30, 2012
100
Harmonic multivalent meromorphic functions...
small [3] investigated the family of all complex valued harmonic mappings f
defined on the open unit disk U, which admits the representation f (z) = h(z)+
g(z) where h and g are analytic univalent in U. Hengartner and Schober [4]
considered the class of functions which are harmonic, meromorphic, orientation
e = {z : |z| > 1} so that f (∞) = ∞. Such
preserving and univalent in U
functions admit the representation
f (z) = h(z) + g(z) + A log |z| ,
where
∞
X
h(z) = αz +
an z −n ,
g(z) = βz +
n=1
∞
X
(1)
bn z −n
n=1
e = {z = |z| > 1} , α, β, A ∈ C with 0 ≤ |β| ≤ |α| and w(z) =
are analytic in U
e P0 denotes a class of functions
fz /fz is analytic with |w(z)| < 1 for z ∈ U.
PH0
of the form (1) with α = 1, β = 0. The class H has been studied in various
research papers such as [5], [6] and [7].
Theorem 1.1. [4] If f ∈
P0
H,
then the diameter Df of C \ f (U), satisfies
Df ≥ 2 |1 + b1 | .
This estimate is sharp for
f (z) = z + b1 /z + A log |z|
whenever |b1 | < 1 and |A| ≤ 1 − |b1 |2 / |1 + b1 | , |b1 | = 1 and A = 0, or
b1 = −1 and |A| ≤ 2.
A function is said to be meromorphic if poles are its only singularities in
the complex plane C.
Let Mp (p ∈ N0 = {1, 2, ...}) be a class of multivalent meromorphic functions of the form:
∞
h(z) =
a−p X
+
an+p−1 z n+p−1 , a−p ≥ 0, an+p−1 ∈ C, z ∈ U∗ = U \ {0}. (2)
zp
n=1
101
Amit Kumar Yadav
m
Definition 1.2. A Bernardi type integral operator Ip,c
(m ≥ 0, c > p) for
meromorphic multivalent function h ∈ Mp is defined as :
0
Ip,c
h(z) = h(z)
Z
c−p+1 z c 0
1
Ip,c h(z) =
t Ip,c h(t)dt
z c+1
Z0 z
c−p+1
m−1
m
tc Ip,c
h(t)dt, m ≥ 1.
Ip,c
h(z) =
z c+1
0
m
The Series expansion of Ip,c
h(z) for h(z) of the form (2) is given by
∞
m
Ip,c
h(z)
a−p X m
= p +
θ (n) an+p−1 z n+p−1 (c > p, m ≥ 0) ,
z
n=1
where
(3)
m
c−p+1
.
(4)
θ (n) =
n+p+c
m
c−p+1
m
m
Note that 0 < θ (n) < θ (1) = 1+p+c .
For fixed integer p ≥ 1, denote by MH (p), a family of harmonic multivalent
meromorphic functions of the form
m
f (z) = h(z) + g(z), z ∈ U∗ ,
(5)
where h ∈ Mp with a−p > 0 of the form (2) and g ∈ Mp of the form
g(z) =
∞
X
bn+p−1 z n+p−1 , bn+p−1 ∈ C, z ∈ U.
(6)
n=1
In the expression (5) , h is called a meromorphic part and g co-meromorphic
part of f . The class MH (p) with its subclass has been studied in [1] and [8] for
a−p = 1. A quite similar to the above mentioned integral operator was used
for harmonic analytic functions in [2].
In terms of the operator defined in Defininition 1.2, an operator f ∈ MH (p)
is defined as follows:
Definition 1.3. Let f = h + g be of the form (5) , the integral operator
is defined as
m
Ip,c
f (z)
m
m
m g(z), z ∈ U∗
Ip,c
f (z) = Ip,c
h(z) + (−1)m Ip,c
(7)
102
Harmonic multivalent meromorphic functions...
m
m
the series expansions for Ip,c
h(z) is given by (3) with a−p > 0 and for Ip,c
g(z)
with g(z) of the form (6) is given as:
m
Ip,c
g(z)
∞
X
=
θm (n)bn+p−1 z n+p−1 (c > p, m ≥ 0) .
(8)
n=1
m
Involving operator Ip,c
define by (7), a class MH (p, α, m, c) of functions
f ∈ MH (p) is defined as follows:
Definition 1.4. A function f ∈ MH (p) is said to be in MH (p, α, m, c) if
and only if it satisfies
Re
m
Ip,c
f (z)
m+1 f (z)
Ip,c
> α, z ∈ U∗ , 0 ≤ α < 1, c > p, m ∈ N.
(9)
Let MH (p, α, m, c) be a subclass of MH (p, α, m, c), consists of harmonic multivalent meromorphic functions fm = hm + gm , where hm and gm are of the
form
∞
a−p X
hm (z) = p −
|an+p−1 | z n+p−1 , a−p > 0
z
n=1
and
gm (z) = (−1)m+1
∞
X
(10)
|bn+p−1 | z n+p−1 .
n=1
2
Some results for the class MH (p)
In this section, some results for MH (p) class are derived.
Theorem 2.1. Let f ∈ MH (p) be of the form (5), then the diameter Df
of clf (U∗ ) satisfies
Df ≥ 2 |a−p | .
This estimate is sharp for f (z) = a−p z −p .
103
Amit Kumar Yadav
Proof. Let Df (r) denotes the diameter of clf (U∗r )(r), where
U∗r (r) = {z : 0 < |z| < r, 0 < r < 1} and
Df∗ (r) := max |f (z) − f (−z)| .
|z|=r
Since Df∗ (r) ≥ Df (r), it follows that
2
Df∗ (r)
1
≥
2π
"
= 4
"
Z2π
0
f (reiθ ) − f (−reiθ )2 dθ
2
∞
X
|a−p |
+
|a2n+p−2 |2 + |b2n+p−2 |
2p
r
n=1
2
#
r2(2n+p−2) , p is odd
#
∞
|a−p |2 X
+
= 4
|a2n+p−1 |2 + |b2n+p−1 |2 r2(2n+p−1) , p is even
2p
r
n=1
≥ 4
|a−p |2
,
r2p
noted that as r −→ 1, Df (r) decreases to Df , which concludes the result.
Remark 2.2. [4]Taking p = 1, a−p = 1 and w(z) := f (1/z) in Theorem
2.1, same result as Theorem 1.1 has been obtained for w(z), which is of the
form (1) with A = 0.
Theorem 2.3. If f ∈ MH (p) be of the form (5), then
∞
X
n=1
(n + p − 1) |an+p−1 |2 − |bn+p−1 |2 ≥ p |a−p |2 .
Equality occurs if and only if C\f (U∗ ) has zero area.
Proof. The area of the ommited set is
Z
Z
Z
1
1
1
0
0
f df = lim
hh dz +
lim
gg dz
r→1 2i |z|=r
r→1 2i |z|=r
2i |z|=r
"
#
∞
X
2
2
2
= π −p |a−p | +
(n + p − 1) |an+p−1 | − |bn+p−1 |
n=1
≥ 0.
104
Harmonic multivalent meromorphic functions...
Remark 2.4. [4]Taking p = 1, a−p = 1 and w(z) := f (1/z) in Theorem
P
2.3, then for w ∈ 0H ,with A = 0, it follows that
∞
X
n=1
3
n |an |2 − |bn |2 ≤ 1 + 2<b1 .
Coefficient Conditions
In this section, sufficient coefficient condition for a functiom f ∈ MH (p) to
be in MH (p, α, m, c) class is established and then it is shown that this coefficient
condition is necessary for its subclass MH (p, α, m, c).
Theorem 3.1. Let f (z) = h(z) + g(z) be the form (5) and θm (n) is given
by (4) if
∞
X
θm (n) 1 − αθ1 (n) |an+p−1 | + 1 + αθ1 (n) |bn+p−1 | ≤ (1 − α)a−p , (11)
n=1
holds for 0 ≤ α < 1 and m ∈ N, then f (z) is harmonic in U∗ and f ∈
MH (p, α, m, c).
Proof. Let the function f (z) = h(z) + g(z) be the form (5) satisfying (11). In
order to show f ∈ MH (p, α, m, c), it suffices to show that
m
Ip,c f (z)
Re
>α
(12)
m+1 f (z)
Ip,c
or,
Re
(
m
m g(z)
Ip,c
h(z) + (−1)m Ip,c
m+1 h(z) + (−1)m+1 I m+1 g(z)
Ip,c
p,c
)
>α
where z = reiθ , 0 < r ≤ 1, 0 ≤ θ ≤ 2π and 0 ≤ α < 1.
Let
m
m g(z)
A(z) := Ip,c
h(z) + (−1)m Ip,c
(13)
and
m+1
m+1 g(z).
B(z) := Ip,c
h(z) + (−1)m+1 Ip,c
(14)
105
Amit Kumar Yadav
It is observed that (12) holds if
|A(z) + (1 − α)B(z)| − |A(z) − (1 + α)B(z)| ≥ 0.
(15)
From (13) and (14) , it follows that
|A(z) + (1 − α)B(z)|
m
m g(z) + (1 − α) I m+1 h(z) + (−1)m+1 I m+1 g(z) h(z) + (−1)m Ip,c
= Ip,c
p,c
p,c
∞
a−p X m
θ (n) + (1 − α)θm+1 (n) an+p−1 z n+p−1
= (2 − α) p +
z
n=1
∞
X
+(−1)m
θm (n) − (1 − α)θm+1 (n) bn+p−1 z n+p−1 n=1
≥
(2 − α)a−p
|z|p
∞
X
−
θm (n) 1 + (1 − α)θ1 (n) |an+p−1 | |z|n+p−1
−
n=1
∞
X
m
n=1
and
θ (n) 1 − (1 − α)θ1 (n) |bn+p−1 | |z|n+p−1
|A(z) − (1 + α)B(z)|
m
m g(z) − (1 + α) I m+1 h(z) + (−1)m+1 I m+1 g(z) h(z) + (−1)m Ip,c
= Ip,c
p,c
p,c
∞
a−p X m
θ (n) − (1 + α)θm+1 (n) an+p−1 z n+p−1
= (−α) p +
z
n=1
∞
X
m
m
m+1
n+p−1
+(−1)
θ (n) + (1 + α)θ
(n) bn+p−1 z
n=1
∞
a
−p X m
= α p −
θ (n) − (1 + α)θm+1 (n) an+p−1 z n+p−1
z
n=1
∞
X
m
m
m+1
n+p−1
−(−1)
θ (n) + (1 + α)θ
(n) bn+p−1 z
n=1
∞
a−p X m
θ (n) 1 − (1 + α)θ1 (n) |an+p−1 | |z|n+p−1
≤ α p+
|z|
n=1
∞
X
+ θm (n) 1 + (1 + α)θ1 (n) |bn+p−1 | |z|n+p−1 .
n=1
106
Harmonic multivalent meromorphic functions...
Thus
|A(z) + (1 − α)B(z)|−|A(z) − (1 + α)B(z)|
∞
X
2(1 − α)a−p
≥
− 2 θm (n) 1 − αθ1 (n) |an+p−1 | |z|n+p−1
p
|z|
n=1
−2
2
≥
|z|p
(
n=1
θm (n) 1 + αθ1 (n) |bn+p−1 | |z|n+p−1
∞
X
a−p (1 − α) −
θm (n) 1 − αθ1 (n) |an+p−1 | |z|n−1
n=1
−
(
∞
X
∞
X
θm (n) 1 + αθ1 (n) |bn+p−1 | |z|n−1
n=1
≥ 2 (1 − α)a−p −
≥ 0,
∞
X
θm (n)
n=1
)
1 − αθ1 (n) |an+p−1 | + 1 + αθ1 (n) |bn+p−1 |
if (11) holds. This proves the Theorem.
Theorem 3.2. Let fm = hm +gm where hm and gm are of the form (10) then
fm ∈ MH (p, α, m, c) under the same parametric conditions taken in Theorem
3.1, if and only if the inequality (11) holds.
Proof. Since MH (p, α, m, c) ⊂ MH (p, α, m, c), if part is proved in Theorem
3.1. It only needs to prove the “only if” part of the Theorem. For this, it
suffices to show that fm ∈
/ MH (p, α, m, c) if the condition (11) does not hold.
If fm ∈ MH (p, α,nm, c),
o then writing corresponding series expansions in (9) , it
ξ(z)
follows that Re η(z) ≥ 0 for all values of z in U∗ where
∞
ξ(z) = (1 − α)
a−p X m
1
+
θ
(n)
1
−
αθ
(n)
|an+p−1 | z n+p−1
zp
n=1
−
∞
X
n=1
and
θm (n) 1 + αθ1 (n) |bn+p−1 | z n+p−1
∞
∞
X
a−p X m+1
n+p−1
θ
(n) |an+p−1 | z
−
θm+1 (n) |bn+p−1 | z n+p−1 .
η(z) = p +
z
n=1
n=1
)
107
Amit Kumar Yadav
Since
ξ(z) ξ(z)
η(z) ≥ Re η(z) ≥ 0,
n o
ξ(z)
hence the condition Re η(z)
≥ 0 holds if
(1 − α)a−p −
∞
P
θm (n) [(1 − αθ1 (n)) |an+p−1 | + (1 + αθ1 (n)) |bn+p−1 |] rn+2p−1
n=1
a−p +
∞
P
θm+1 (n) |a
n=1
n+p−1
| rn+2p−1
+
∞
P
≥ 0.
θm+1 (n) |b
n+p−1
| rn+2p−1
n=1
(16)
Now if the condition (11) does not holds then the numerator of above equation is non-positive for r sufficiently close to 1, which contradicts that fm ∈
MH (p, α, m, c) and this proves the required result.
4
Bounds and Extreme Points
In this section, bounds for functions belonging to the class MH (p, α, m, c)
are obtained and also provide extreme points for the same class.
Theorem 4.1. If fm = hm + gm ∈ MH (p, α, m, c) for 0 ≤ α < 1, 0 < |z| =
r < 1, and θm (n) is given by (4), under the same parametric conditions taken
in Theorem 3.1 then
m
a−p
p (1 − α)a−p
Ip,c fk (z) ≤ a−p + rp (1 − α)a−p .
≤
−
r
rp
[1 − αθ1 (1)]
rp
[1 − αθ1 (1)]
m
fm
Proof. Let fm = hm + gm ∈ MH (p, α, m, c). Taking the absolute value of Ip,c
from (7), it follows that
∞
∞
a
X
X
m
−p
m
n+p−1
2m+1
m
n+p−1
Ip,c fm (z) = +
θ (n) |an+p−1 | z
+ (−1)
θ (n) |bn+p−1 | z
p
z
n=1
n=1
∞
a
−p X m
n+p−1 θ (n) (|an+p−1 | − |bn+p−1 |) z
= p +
z
n=1
108
Harmonic multivalent meromorphic functions...
∞
X
m
[θm (n) − αθm+1 (n)] m
Ip,c fm (z) ≤ a−p + rp
θ (n) (|an+p−1 | + |bn+p−1 |)
rp
[θm (n) − αθm+1 (n)]
n=1
≤
∞
X
a−p
[θm (n) − αθm+1 (n)] m
p
+
r
θ (n) (|an+p−1 | + |bn+p−1 |)
m (n) [1 − αθ 1 (n)]
rp
θ
n=1
∞
X
m
a−p
rp
+
≤
θ (n) − αθm+1 (n) (|an+p−1 | + |bn+p−1 |)
p
1
r
[1 − αθ (1)] n=1
"∞
X
rp
a−p
θm (n) 1 − αθ1 (n) |an+p−1 |
+
≤
p
1
r
[1 − αθ (1)] n=1
#
∞
X
+ θm (n) 1 + αθ1 (n) |bn+p−1 |
n=1
≤
(1 − α)a−p
a−p
+ rp
p
r
[1 − αθ1 (1)]
and
∞
∞
a
X
X
m
m
n+p−1
2m+1
m
n+p−1
Ip,c fm (z) = −p +
θ (n) |an+p−1 | z
+ (−1)
θ (n) |bn+p−1 | z
p
z
n=1
n=1
∞
a
−p X m
n+p−1 θ (n) (|an+p−1 | − |bn+p−1 |) z
= p +
z
n=1
∞
X
m
[θm (n) − αθm+1 (n)] m
Ip,c fm (z) ≥ a−p − rp
θ (n) (|an+p−1 | + |bn+p−1 |)
m (n) − αθ m+1 (n)]
rp
[θ
n=1
∞
X
a−p
[θm (n) − αθm+1 (n)] m
p
≥
−
r
θ (n) (|an+p−1 | + |bn+p−1 |)
rp
θm (n) [1 − αθ1 (n)]
n=1
≥
∞
X
m
a−p
rp
m+1
−
θ
(n)
−
αθ
(n)
(|an+p−1 | + |bn+p−1 |)
rp
[1 − αθ1 (1)] n=1
p
≥
r
a−p
−
p
r
[1 − αθ1 (1)]
"∞
X
θm (n)
n=1
+
1 − αθ1 (n) |an+p−1 |
∞
X
n=1
≥
(1 − α)a−p
a−p
− rp
p
r
[1 − αθ1 (1)]
This proves the required result.
θm (n)
1 + αθ1 (n) |bn+p−1 |
#
109
Amit Kumar Yadav
Corollary 4.2. Let fm = hm + gm ∈ MH (p, α, m, c), for z ∈U∗ and θm (n)
is given by (4), under the same parametric conditions taken in Theorem 3.1
then
(1 − α)a−p
* f (U∗ ).
w : |w| < a−p −
1
[1 − αθ (1)]
Theorem 4.3. Let fm = hm + gm , where hm and gm are of the form (10)
then fm ∈ MH (p, α, m, c) and θm (n) is given by (4), under the same parametric
conditions taken in Theorem 3.1, if and only if fm can be expressed as:
fm (z) =
∞
X
n=0
xn+p−1 hmn+p−1 (z) + yn+p−1 gmn+p−1 (z) ,
(17)
where z ∈U∗ and
hmp−1 (z) =
gmp−1 (z) =
a−p
,
zp
a−p
,
zp
hmn+p−1 (z) =
gmn+p−1 (z) =
a−p
(1 − α)a−p
+ m
z n+p−1 ,
p
z
θ (n) − αθm+1 (n)
(18)
a−p
(1 − α)a−p
+(−1)m+1 m
z n+p−1 (19)
p
z
θ (n) + αθm+1 (n)
for n = 1, 2, 3, ......, and
∞
X
(xn+p−1 + yn+p−1 ) = 1,
xn+p−1 , yn+p−1 ≥ 0.
(20)
n=1
Proof. Let
fm (z) =
∞
X
n=0
xn+p−1 hmn+p−1 (z) + yn+p−1 gmn+p−1 (z)
∞
X
(1 − α)a−p
a−p
n+p−1
+ m
z
= xp−1 hmp−1 + yp−1 gmp−1 +
xn+p−1
p
m+1 (n)
z
θ
(n)
−
αθ
n=1
∞
X
(1 − α)a−p
a−p
m+1
n+p−1
z
+ yn+p−1
+ (−1)
zp
θm (n) + αθm+1 (n)
n=1
∞
∞ X
a−p X
(1 − α)a−p
fm (z) =
(xn+p−1 + yn+p−1 ) p +
xn+p−1
z
θm (n) − αθm+1 (n)
n=0
n=1
(1 − α)a−p
m+1
+(−1)
yn+p−1 z n+p−1 .
m
m+1
θ (n) + αθ
(n)
110
Harmonic multivalent meromorphic functions...
Thus by Theorem 3.2, fm ∈ MH (p, α, m, c), since,
∞ m
X
(1 − α)a−p
θ (n) − αθm+1 (n)
xn+p−1
(1 − α)a−p
θm (n) − αθm+1 (n)
n=1
(1 − α)a−p
θm (n) + αθm+1 (n)
yn+p−1
−
(1 − α)a−p
θm (n) + αθm+1 (n)
∞
X
=
(xn+p−1 + yn+p−1 ) = (1 − xp−1 − yp−1 ) ≤ 1.
n=1
Conversly, suppose that fm ∈ MH (p, α, m, c), then (11) holds.
Setting
θm (n) − αθm+1 (n)
|an+p−1 |
(1 − α)a−p
θm (n) + αθm+1 (n)
|bn+p−1 |
=
(1 − α)a−p
xn+p−1 =
yn+p−1
which satisfy (20), thus
∞
∞
X
a−p X
n+p−1
m+1
fm (z) =
+
|bn+p−1 | z n+p−1
|an+p−1 | z
+ (−1)
zp
n=1
n=1
∞
=
(1 − α)a−p
a−p X
+
x
z n+p−1
m (n) + αθ m+1 (n) n+p−1
zp
θ
n=1
+(−1)
m+1
∞
X
n=1
(1 − α)a−p
yn+p−1 z n+p−1
+ αθm+1 (n)
θm (n)
∞ h
∞
X
a−p X h
a−p i
a−p i
x
+
yn+p−1
+
g
−
h
−
n+p−1
mn+p−1
mn+p−1
p
p
zp
z
z
# n=1
" n=1 ∞
∞
∞
X
X
X
a−p
=
+
hmn+p−1 xn+p−1 + gmn+p−1 yn+p−1
1
−
x
−
y
n+p−1
n+p−1
zp
n=1
n=1
n=1
=
= xp−1 hmp−1 + yp−1 gmp−1
∞
∞
X
X
+
hmn+p−1 xn+p−1 +
gmn+p−1 yn+p−1
n=1
=
∞
X
xn+p−1 hmn+p−1 (z) + yn+p−1 gmn+p−1 (z)
n=1
This proves the Theorem.
n=1
111
Amit Kumar Yadav
Remark 4.4. The extreme points for the class MH (p, α, m, c) are given by
(18) and (19).
5
Convolution and Integral Convolution
In this section, convolution and integral convolution properties of the class
MH (p, α, m, c) are established.
Let fm , Fm ∈ MH (p) be defined as follows:
fm (z) =
∞
∞
X
a−p X
n+p−1
m+1
+
|bn+p−1 | z n+p−1 , z ∈ U∗ ,
|a
|
z
+
(−1)
n+p−1
zp
n=1
n=1
∞
∞
X
a−p X
n+p−1
m+1
Fm (z) = p +
|An+p−1 | z
+ (−1)
|Bn+p−1 | z n+p−1 , z ∈ U∗ .
z
n=1
n=1
The convolution of fm and Fm for m ∈ N, z ∈U∗ is defined as:
(fm ? Fm )(z) = fm (z) ? Fm (z)
∞
a−p X
+
|an+p−1 An+p−1 | z n+p−1 +
=
zp
n=1
+(−1)
m+1
∞
X
(21)
|bn+p−1 Bn+p−1 | z n+p−1 .
n=1
The integral convolution of fm and Fm for m ∈ N, z ∈U∗ is defined as:
(fm ♦Fm )(z) = fm (z)♦Fm (z)
∞
a−p X p |an+p−1 An+p−1 | n+p−1
+
z
=
zp
n
+
p
−
1
n=1
+(−1)
m+1
∞
X
p |bn+p−1 Bn+p−1 |
n=1
n+p−1
(22)
z n+p−1 .
Theorem 5.1. For 0 ≤ α < 1, m ∈ N. If fm , Fm ∈ MH (p, α, m, c) and
θ (n) is given by (4) then (fm ? Fm ) ∈ MH (p, α, m, c) .
m
112
Harmonic multivalent meromorphic functions...
Proof. Since Fm ∈ MH (p, α, m, c) , then by Theorem 3.2, |An+p−1 | ≤ 1 and
|Bn+p−1 | ≤ 1, hence,
∞
X
1 − αθ1 (n)
m
|An+p−1 an+p−1 |
θ (n)
(1 − α)a−p
n=1
1 + αθ1 (n)
+
|Bn+p−1 bn+p−1 |
(1 − α)a−p
∞
X
1 − αθ1 (n)
m
≤
|an+p−1 |
θ (n)
(1 − α)a−p
n=1
1 + αθ1 (n)
+
|bn+p−1 |
(1 − α)a−p
≤ 1
as fm ∈ MH (p, α, m, c). Thus by the Theorem 3.2, (fm ? Fm ) ∈ MH (p, α, m, c).
Theorem 5.2. For 0 ≤ α < 1, m ∈ N. If fm , Fm ∈ MH (p, α, m, c) and
θ (n) is given by (4) then (fm ♦Fm ) ∈ MH (p, α, m, c) .
m
Proof. Since Fm ∈ MH (p, α, m, c) , then by Theorem 3.2, |An+p−1 | ≤ 1,
p
≤ 1 hence,
|Bn+p−1 | ≤ 1 and n+p−1
∞
X
n=1
m
θ (n)
1 − αθ1 (n) p |an+p−1 An+p−1 |
(1 − α)a−p
n+p−1
1 + αθ1 (n) p |bn+p−1 Bn+p−1 |
+
(1 − α)a−p
n+p−1
∞
X
1 − αθ1 (n)
m
|an+p−1 |
≤
θ (n)
(1 − α)a−p
n=1
1 + αθ1 (n)
+
|bn+p−1 |
(1 − α)a−p
≤ 1
as fm ∈ MH (p, α, m, c). Thus by the Theorem 3.2, (fm ♦Fm ) ∈ MH (p, α, m, c).
113
Amit Kumar Yadav
6
Convex Combination
In this section, it is proved that the class MH (p, α, m, c) is closed under
convex linear combination of its members.
Theorem 6.1. Let the functions fmj (z) defined as:
∞
∞
X
a−p X
n+p−1
m+1
fmj (z) = p +
|an+p−1,j | z
+ (−1)
|bn+p−1,j | z n+p−1 , z ∈ U∗ .
z
n=1
n=1
(23)
be in the class MH (p, α, m, c) for every j = 1, 2, 3...., then the function
ψ(z) =
∞
X
cj fmj (z)
j=1
is also in the class MH (p, α, m, c), where
Proof. From the definition
∞
P
cj = 1 for cj ≥ 0 (j = 1, 2, 3....).
j=1
!
!
∞
∞
∞
∞
X
X
a−p X X
ψ(z) = p +
cj |an+p−1,j | z n+p−1 +(−1)m+1
cj |bn+p−1,j | z n+p−1 .
z
n=1
n=1
j=1
j=1
Since fmj (z) ∈ MH (p, α, m, c) for every j = 1, 2, 3..., then by Theorem 3.2, it
follows that
!
"
X
∞
∞
1
X
1
−
αθ
(n)
cj |an+p−1,j |
θm (n)
(1 − α)a−p
n=1
j=1
!#
X
∞
1 + αθ1 (n)
+
cj |bn+p−1,j |
(1 − α)a−p
j=1
=
∞
X
j=1
≤
∞
X
cj
∞ X
1 − αθ1 (n)
n=1
|an+p−1,j |
(1 − α)a−p
1 + αθ1 (n)
+
|bn+p−1,j |
(1 − α)a−p
cj .1 ≤ 1.
j=1
Hence ψ(z) ∈ MH (p, α, m, c), which is the desired result.
114
Harmonic multivalent meromorphic functions...
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763-770.
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