Gen. Math. Notes, Vol. 28, No. 2, June 2015, pp. 30-41
ISSN 2219-7184; Copyright © ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
On a Subclass of Meromorphic Function with
Fixed Second Coefficient Involving FoxWright's Generalized Hypergeometic
Function
Abdul Rahman S. Juma1 and Husamaldin I. Dhayea2
1
Department of Mathematics, Alanbar University
Ramadi, Iraq
E-mail: dr_juma@hotmail.com
2
Department of Mathematics, Tikrit University
Tikrit, Iraq
E-mail: husamaddin@gmail.com
(Received: 13-4-15/ Accepted: 29-5-15)
Abstract
In this paper, we have introduced and studied new subclass of meromorphic
function with fixed second coefficient involving Fox-Wright's generalized
hypergeometric function. We have obtained coefficient estimates, extreme points,
growth and distortion theorems, radii of meromorphically starlikeness and
convexity for this new subclass and other interesting properties.
Keywords: Meromorphic functions, Hadamard product, Fixed second
coefficient, coefficient inequalities, radii of meromorphically starlikeness and
convexity.
On a Subclass of Meromorphic Function with…
1
31
Introduction
Let Ʃ denote the class of normalized meromorphic functions of the form
=
1
+
1
∞
defined on the punctured unit disk ∆∗ =
is meromorphic starlike of order
, 0≤
∈ ℂ ∶ 0 < | | < 1 . Afunction
< 1 if −ℜ
′
. A function
∆\ 0 . The class of all such functions is denoted by Σ∗
is$%&'$'&(ℎ*++',-*.' '&/%& , 0 ≤ < 1
If−ℜ 1 +
> ,
"
′ 1
>
∈Σ
∈ ∆∗ =
∈Σ
∈ ∆∗ = ∆\ 0 . Let Σ2 be the class of functions
∈ Σ 3*4ℎ
≥ 0. The subclass of Σ2 consisting of starlike functions of order
is denoted by Σ∗2
and convex functions of order by Σ62
.Various subclasses
of Σ have been defined and studied by various authors (see [1, 2, 3, 4,5, 6, 7, 8, 9,
10, 11, 12, 13]).
For functions
given by (1) and 7
= + ∑∞
Hadamard product or convolution of f and 7 by
∗ 7 = + ∑∞
9
1 + ∑?
>6 − ∑<6 :6 ≥ 0, ∈
6
hypergeometric function
,: ,…,
< , :<
we define the
.
For positive real parameters
1,2,3, … such that
l Ψ? E
9
,: ,…,
< , :< , =
∈ ℂ: 0 < | | < 1
, > , … , =? , >? @, $ ∈ ℕ =
the Wright's generalized
; = , > , … , =? , >? ; G =< Ψ? H
I , :I
,< ,
=I , >I
,? ;
J
P
Q
is defined by
l Ψ? H
I , :I
,< ,
=I , >I
,? ;
If :I = 1 4 = 1,2, … , @
Ω< Ψ? H
I , :I
<
J = ∑∞
6 NK∏I
NΓ
I
+ M:I O ∏?
I N Γ =I + M>I
6!
,/>I = 1 4 = 1,2, … , $ we have the relationship
,< ,
=I , >I
=
∞
6 N
,? ;
J ≡< T?
… <
= 6 … =?
6
6
6
,…,
6
M!
.
<; =
, … , =? ;
.
32
Abdul Rahman S. Juma et al.
@ ≤ $ + 1; @, $ ∈ ℕN = ℕ = 0,1,2, … ; ∈ ∆ .
This is the generalized hypergeometric function (see [6]). Here (
thePochammer symbol and
Ω = UV
<
P
W
I
Γ
I N
XV
?
) is
Γ =I Y.
I N
Using the generalized hypergeometric function, we define a linear operator
ZH
By
ZH
I , :I
,< ,
=I , >I
,? J
I , :I
=
For convenience, we denoteZH
(1) then,
ZE
G
= + ∑∞
,< ,
=I , >I
Ω< Ψ? H
P
I , :I
]
,< ,
,? J: Σ[
I , :I
=I , >I
→ Σ[ .
,< ,
=I , >I
,? Jby
ZE
,? ;
J∗
(2)
G. If f has the form
,
(3)
Where
]
=
a c…Γ^_d abd
ΩΓ^_` ab`
6a !Γ^e` af`
a c
a c…Γ^ed afd
a c
.
(4)
Now, we define a new subclass ofΣ[ by using the linear operator ZE
Gas follows.
For 0 ≤ η < 1 and 0 ≤ λ < 1 we let o λ, η denote a subclass of Σq consisting
functions of the form (1) satisfying the condition that
ℜ X rP
^ZE_` G
^ZE_` G
c
′
car ^ZE_` G
.
Where :I = 1 4 = 1,2, … , @
c′
Y>s
,/>I = 1 4 = 1,2, … , $ .
Now we prove the coefficient inequality for
2
5
∈ o u, s .
Coefficients Inequalities
Our first theorem gives a necessary and sufficient condition for a function f to be
in the class o u, s .
Theorem 1: v%4
∈ Σ[ 9% 7*-%, 9w 1 . xℎ%,
∈ o u, s *
,/ ',@w *
On a Subclass of Meromorphic Function with…
∞
, + s − su 1 + , ]
≤ 1−s
If
6
∈ Σz given by (1) is in the class o λ, s , Then by
Proof: At first suppose that
(5) we have
ℜ{
33
u − 1 ^ZE
^ZE
G
G
c
′
G
c + u ^ZE
−1 + ∑∞ ,]
ℜU
−1 + ∑∞ u − 1 + u, ]
.
P
→ 1 , we have
c
a
a
−1 + ∑∞ ,]
ℜU
−1 + ∑∞ u − 1 + u, ]
′|
>s
W>s
W > s.
This means that (6) holds, conversely suppose that the inequality (6) holds. Let
^ZE G
c
}=
u − 1 ^ZE G
c + u ^ZE
′
G
We have to prove that ℜ} > s It is enough to prove that
}−1
^Z E
~
~=•
} + 1 − 2s
^ZE G
=€
−2 1 − s + ∑∞
≤€
2 1 − s − ∑∞
.
Thus we have
|} − 1| < |} + 1 − 2s|
G
c − u − 1 ^Z E
′
G
c + 1 − 2s u − 1 ^ZE
′
G
c′
c + u ^ZE
c + u ^ZE
a
∑∞ 1 − u , + 1 ]
E, 1 + 1 − 2s u + 1 − 2s u − 1 G]
∑∞ 1 − u , + 1 ]
E, 1 + 1 − 2s u + 1 − 2s u − 1 G]
∈ o u, s . ∎
G
c
′
G
c
′
a
•
€
€≤1
From (6) we have
]
≤
1−s
1 + s − 2su
7
34
Abdul Rahman S. Juma et al.
]
≤
1−s +
, 0 < + < 1.
1 + s − 2su
8
Definition 1: The subclass o u, s, + of o u, s consists of all functions of the
form
= +
P„ …
a„P†„r
+ ∑∞
†]
,0 < + < 1
(9)
We now obtain the coefficient estimates, growth and distortion bounds, extreme
points, radii of mero-morphically starlikenss and convexity for the class o u, s
by fixing the second coefficient.
We now prove the coefficient inequality.
Theorem 2: Let f be defined by (9). Then f∈ o u, s, + if and only if
∑∞
†
, + s − su 1 + , ]
≤ 1−s 1−+ .
(10)
The result is sharp.
Proof: f∈ o u, s, + implies f∈ o u, s . Therefore by (6)
+
1 + s − 2su ]
Using (8)
1−s ++
∞
†
∞
†
, + s − su 1 + , ]
≤ 1−s
≤ 1−s .
, + s − su 1 + , ]
From which we get (10). The result is sharp for the function
= +
P„ …
a„P†„r
+E
P„
a„P„r a
P…
G‡ˆ _`
, , ≥ 2. ∎
(11)
Corollary 3: If f defined by (9) is in the class o u, s, + , then
≤E
P„
a„P„r
P…
a G‡ˆ _`
, ,≥2
(12)
The result is sharp for the function given by (11).
3
Growth and Distortion Theorems
A growth and distortion property for the function
follows:
∈ o u, s, + is given as
On a Subclass of Meromorphic Function with…
35
Theorem 4: If f given by (9) is in the class o u, s, + then for0 < | | = & < 1
|
and
|
|≥
|≤
1
1−s +
1−s 1−+ †
−
&−
&
& 1 + s − 2su
2 + s − 3su
1−s +
1−s 1−+ †
1
+
&+
& .
& 1 + s − 2su
2 + s − 3su
= +
The result is sharp for
P„ …
a„P†„r
Proof: Since f∈ o u, s, + by Theorem 2
]
=E
P„
P…
a„P„r a
For0 < | | = & < 1,
|
|≤
≤
≤
Similarly,
|
≥
P„
P…
†a„PŠ„r
†
.
.
14
(15)
1
1−s +
| |+
+
| | 1 + s − 2su
1
1−s +
+
& + &†
& 1 + s − 2su
]
∞
†
| |
]
∞
†
1
1−s +
1−s 1−+ †
+
&+
& .
& 1 + s − 2su
2 + s − 3su
|≥
≥
G
+
13
1
1−s +
| |−
−
| | 1 + s − 2su
1
1−s +
−
& − &†
& 1 + s − 2su
]
∞
†
∞
| |
]
†
1
1−s +
1−s 1−+ †
−
&−
& .∎
& 1 + s − 2su
2 + s − 3su
A distortion theorem for the function f to be in the class o u, s, + is given as
follow:
36
Abdul Rahman S. Juma et al.
Theorem 5: If given by (9) is in the class o u, s, + then for 0 < | | = & < 1
| ′
|≥
1
1−s +
1−s 1−+
−
−
&
†
&
1 + s − 2su
2 + s − 3su
and
| ′
|≤
1
1−s +
1−s 1−+
+
+
&.
†
&
1 + s − 2su
2 + s − 3su
The result is sharp for
4
16
=
1
+
1−s +
1−s 1−+
+
1 + s − 2su
2 + s − 3su
17
†
.
Extreme Points
In this section, we determine the extreme points for functions in the
classo u, s, + .
= +
Theorem 6: Let
=
Then
1
+
P„ …
a„P†„r
1−s +
+
1 + s − 2su
∞
, and
† ^,
1−s 1−+
∈ o u, s, + if and only if it can be expressed as
=
Proof: Suppose
=
1
+
= ∑∞
∞
•
•
1−s +
+
1 + s − 2su
, • ≥ 0,
∞
, • ≥ 0, ∑∞
∞
'&, ≥ 2.
+ s − su 1 + , c]
† ^,
• = 1.
• = 1. Then
1−s 1−+
+ s − su 1 + , c]
•
.
Now
∞
† ^,
1−s 1−+ •
+ s − su 1 + , c]
^, + s − su 1 + , c]
1−s 1−+
=
∞
This implies f∈ o u, s, + . Conversely, let f∈ o u, s, + . Then
≤
1−s 1−+
E, + s − su 1 + , G]
,
• =1−• ≤1.
†
, ≥ 2.
On a Subclass of Meromorphic Function with…
Set • =
^ a„P„r
P„
a
c‡ˆ _`
P…
=
37
, , ≥ 2 and • = 1 − ∑∞
•
∞
.∎
†•
. Then
Theorem 7: The class o u, s, + is closed under convex combination.
Proof: Let
, 7 ∈ o u, s, + such that
=
and
7
For 0 ≤ • ≤ 1,let
Then
Therefore
∞
†
ℎ
=
1
=
ℎ
+
1
1
+
+
1−s +
+
1 + s − 2su
1−s +
+
1 + s − 2su
=•
1−s +
+
1 + s − 2su
, + s − su 1 + , ]
E
†
9
∞
†
+ 1−• 7
∞
E
†
.
.
•+ 1−• 9 G
.
•+ 1−• 9 G≤ 1−s 1−+ .
This implies ℎ
=•
+ 1−• 7
closed under convex combination.∎
5
∞
∈ o u, s, + . Hence o u, s, +
is
Radii of Meromorphically Starlikeness and Convexity
The radii of starlikeness and convexity for the class o u, s, + is given by the
following theorem:
Theorem 8: Let f∈ o u, s, + .Then f is meromorphically starlike of order
– 0 ≤ – < 1 in the disk| | < & u, s, +, – ,where & u, s, +, – is the largest value
for which
U
3−– 1−s + †
,+2−– 1−s 1−+
W& + U
W&
1 + s − 2su
^, + s − su 1 + , c
a
≤ 1 − –, , ≥ 2.
18
38
Abdul Rahman S. Juma et al.
Proof: It is enough to show that
—
+ 1— ≤ 1 − –
′
€
(19)
+ 1€ = €
′
+
′
€=˜
† P„ … ™
a„P†„r
1+
Then we write (19) as
€
2 1−s + †
+
1 + s − 2su
†
≤ 1 − – €1 +
That is
3−– 1−s + †
& +
1 + s − 2su
P„ …
a„P†„r
,+1 ]
∞
+ ∑∞
a
1−s +
+
1 + s − 2su
∞
†
†
,+1 ]
+ ∑∞
€
,+2−–
†]
a
]
∞
a
†
&
a
a
˜
€.
≤ 1 − –.
From Theorem 1, we may take
=
1−s 1−+
E, + s − su 1 + , G]
• ,
, ≥ 2, • ≥ 0,
∞
• = 1.
†
For each fixed r, we choose the positive integer ,N = ,N & for which
a†Pš ‡ˆ _`
^ a„P„r
a
c
∞
†
&
a
is maximal. This implies
,+2−–
&
a
≤
,N + 2 − – 1 − s 1 − +
^,N + s − su 1 + , c
Then f is starlike of order – in0 < | | < & u, s, +, – If
,N + 2 − – 1 − s 1 − +
3−– 1−s + †
& +
&
1 + s − 2su
^,N + s − su 1 + , c
›a
&
›a
.
≤ 1 − –.
We have to find the value of &N = &N u, s, +, – and the corresponding integer
,N &N so that
ŠPš
P„ …
a„P†„r
&† +
› a†Pš
P„
^ › a„P„r
It is the value for which
a
P…
c
&
›a
= 1 − –.
is starlike of order – in 0 < | | < &N.∎
(20)
On a Subclass of Meromorphic Function with…
39
We now state a result for radius of meromorphic convexity for the class
o u, s, + for which the proof is similar to above.
Theorem 9: Let f∈ o u, s, + .Then f is meromorphically convex of order
– 0 ≤ – < 1 in the disk | | < &† u, s, +, – where&† u, s, +, – is the largestvalue
for , ≥ 2
ŠPš
P„ …
a„P†„r
6
&† + X
a†Pš
^ a„P„r
P„
a
P…
c
Y&
a
≤ 1 − –.
(21)
Integral Operators
In this section, we consider integral operators of functions in the class o u, s, + .
Theorem 10: Let f∈ o u, s, + .Thenthe integral operator
ℎ
= . œ •ž
N
is in o u, s, + , where
ٲ
. + , + 1 ^, + s − su 1 + , c − ., 1 − s 1 − +
.
. 1 − s 1 − + E1 − u 1 + , G + . + , + 1 ^, + s − su 1 + , c
= +
The result is sharp for
∈ o u, s, + . Then
Proof: Let
ℎ
• /• 0 < • ≤ 1, 0 < . < ∞
= .œ •
N
ž
• /• =
1
P„ …
a„P†„r
+
1−s +
+
+
1 + s − 2su
P„
P…
†a„PŠ„r
∞
†
.
.
]
†. + , + 1
.
It is sufficient to show that
∞
.E, + Ÿ − Ÿu 1 + , G]
† .+,+1 1−Ÿ 1−+
Since f∈ o u, s, + , we have
∞
Therefore (22) is true if
≤ 1.
^, + s − su 1 + , c]
1−s 1−+
†
22
≤ 1.
40
Abdul Rahman S. Juma et al.
.E, + Ÿ − Ÿu 1 + , G]
.+,+1 1−Ÿ 1−+
Solving for Ÿ, we have
ٲ
≤
^, + s − su 1 + , c]
1−s 1−+
.
. + , + 1 ^, + s − su 1 + , c − ., 1 − s 1 − +
=Ψ , .
. 1 − s 1 − + E1 − u 1 + , G + . + , + 1 ^, + s − su 1 + , c
A simple computation will show that Ψ , is increasing and Ψ , ≥ Ψ 1 .∎
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