General Mathematics Vol. 17, No. 2 (2009), 23–40
On Certain Subclasses of Meromorphic
Functions With Positive Coefficients 1
M. K. Aouf
Abstract
In this paper, we obtain coefficient estimates, distortion theorem, radii
P∗
of starlikeness and convexity for the class
p (A, B, β) of meromorphic
functions with positive coefficients. Several interesting results involving
P
Hadamard product of functions belonging to the classes ∗p (α, β) and
P∗
p (A, B, β) are also derived.
2000 Mathematics Subject Classification: 30C45.
Key words and phrases: Meromorphic, distortion theorem, Hadamard
product.
1
Introduction
Let
P
p
denote the class of functions of the form
∞
(1.1)
f (z) =
1 X
an z n ( p ∈ N ∗ = {2i − 1 : i ∈ N = {1, 2, 3, ....})
+
z n=p
1
Received 19 September, 2008
Accepted for publication (in revised form) 25 October, 2008
23
24
M. K. Aouf
which are analytic in the punctured disc U ∗ = {z : 0 < |z| < 1} with a
simple pole at the origin with residue one there.
P
A function f (z) ∈ p is said to be meromorphically starlike of order α
if it satisfies
(1.2)
Re
0
z f (z)
−
f (z)
>α
for some α(0 ≤ α < 1) and for all z ∈ U ∗ .
P
Further a function f (z) ∈ p is said to be meromorphically convex of
order α if it satisfies
(1.3)
Re
00
z f (z)
)
−(1 + 0
f (z)
>α
for some α(0 ≤ α < 1) and for all z ∈ U ∗ .
P
Some subclasses of 1 when p = 1 were recently introduced and studied
by Pommerenke [8], Miller [6], Mogra. et al. [7], Cho [3], Cho et al. [4] and
Aouf ( [1] and [2] ) .
P
P
Let ∗p be the subclass of p consisting of functions of the form
∞
(1.4)
1 X
an z n
f (z) = +
z n=p
We denote by
the condition
P∗
(1.5)
−z p+1 f (p) (z) ≺ p!
p (A, B)
(an ≥ 0, p ∈ N ∗ ).
the class of functions f (z) ∈
1 + Az
1 + Bz
P∗
p,
that satisfy
(z ∈ U ∗ ),
where ≺ denotes subordination, −1 ≤ A < B ≤ 1 and −1 ≤ A < 0 .
It is easy to see that (1.5) is equivalent to
On Certain Subclasses of Meromorphic...
25
z p+1 f (p) (z) + p! ∗
(1.6)
B z p+1 f (p) (z) + p! A < 1 (z ∈ U ).
P
Definition 1 A function f (z) ∈ ∗p is said to be a member of the class
P∗
p (A, B, β) if it satisfies
(1.7)
z p+1 f (p) (z) + p! B z p+1 f (p) (z) + p! A < β
for some β(0 < β ≤ 1), −1 ≤ A < B ≤ 1 , −1 ≤ A < 0 and for all z ∈ U ∗ .
We note that:
P
(i) ∗p (−1, 1, β) = Tp (β)(Kim et al. [5]);
P
P
(ii) ∗1 (−A, −B, 1) = d (A, B)(Cho [3]);
P
P
1
(iii) ∗1 (2αγ − 1, 2γ − 1, β) = d (α, β, γ)(0 ≤ α < 1, 0 ≤ γ ≤ ,
2
0 < β ≤ 1) (Cho et al. [4] );
P
P
(iv) ∗p ( 2α
− 1, 1, β) = ∗p (α, β)
p!
P∗ z p+1 f (p) (z) + p! = f (z) ∈ p : p+1 (p)
< β, 0 ≤ α < p!, 0 < β ≤ 1 ;
z f (z) + 2α − p!
P 2α
P
(v) ∗p
− 1, 1, 1 = ∗p (α)
p!
n
o
p+1 (p) P∗
∗
= f (z) ∈ p : Re −z f (z) > α , 0 ≤ α < p!, z ∈ U .
In this paper, we obtain coefficient estimates, distortion theorem and
P
radii of starlikeness and convexity for the class ∗p (A, B, β) of meromorphic functions with positive coefficients. Several interesting results involvP
ing Hadamard product of functions belonging to the classes ∗p (α, β) and
P∗
p (A, B, β) are also derived.
2
Coefficient estimates
The following theorem gives a necessary and sufficient condition for a funcP
tion to be in the class ∗p (A, B, β).
26
M. K. Aouf
Theorem 1 A function f (z) ∈
if
∞
X
(2.1)
n=p
P∗
p
n
p
!
n
p
!
is in the class
an ≤
P∗
p (A, B, β)if
and only
(B − A)β
,
(1 + βB)
where
=
n!
.
p!(n − p)!
Proof. Suppose (2.1) holds for all admissible values of A, B and β. Then
we have
p+1 (p)
z f (z) + p! − β B z p+1 f (p) (z) + p! A
(2.2)
∞
∞
X
X
n!
n!
n+1
n+1 =
a z − β p!(B − A) − B
an z n=p (n − p)! n
(n − p)!
n=p
≤
∞
X
n=p
n!
(1 + βB)an |z|n+1 − p!(B − A)β .
(n − p)!
Since the above inequality holds for all r = |z|, 0 < r < 1, letting r → 1, we
have
(2.3)
∞
X
n=p
n!
(1 + βB)an − p!(B − A)β ≤ 0,
(n − p)!
P
which shows that f (z) ∈ ∗p (A, B, β).
P
Conversely, if f (z) ∈ ∗p (A, B, β), then
27
On Certain Subclasses of Meromorphic...
z p+1 f (p) (z) + p! B z p+1 f (p) (z) + p! A (2.4)
∞
P
n!
n+1
a z
(n−p)! n
n=p
< β (z ∈ U ∗ ).
=
∞
P
n!
n+1 p!(B − A) − B
a
z
n
(n−p)!
n=p
Since Re(z) ≤ |z| for all z, (2.4) gives
(2.5)
Re
∞
P




n=p
n!
(n−p)!




an z n+1
∞
P


 p!(B − A) − B
n=p
n!
(n−p)!

an z n+1 

< β (z ∈ U ∗ ).
Choose values of z on the real axis so that z p+1 f (p) (z) is real. Upon clearing
the denominator in (2.5) and letting z → 1− , we have
∞
X
(2.6)
n=p
n!
p!(B − A)β
an ≤
.
(n − p)!
(1 + βB)
Hence the result follows.
Corollary 1 If the function f (z) =
P∗
p (A, B, β), then
(2.7)
an ≤
1
z
(B − A)β
!
n
(1 + βB)
p
The result is sharp for the function
+
∞
P
n=p
an z n (an ≥ 0) is in the class
(n ≥ p; p ∈ N ∗ ).
28
M. K. Aouf
(2.8)
f (z) =
1
+
z
(B − A)β
!
z n (n ≥ p; p ∈ N ∗ ).
n
(1 + βB)
p
Putting p = 1 in Theorem 1, we have
Corollary 2 f (z) ∈
P∗
1 (A, B, β)
∞
X
(2.9)
n=1
if and only if
nan ≤
(B − A)β
.
(1 + βB)
2α
− 1(0 ≤ α < p!) and B = 1 in Theorem 1, we obtain
p!
P
the following necessary and sufficient condition for functions in ∗p (α, β).
Putting A =
Corollary 3 Let a function f (z) defined by (1.4). Then f (z) ∈
if and only if
(2.10)
∞
X
n=p
3
n
p
!
an ≤
α
2β(1 − p!
)
1+β
P∗
p (α, β)
.
Distortion Theorem
∞
1 X
Theorem 2 If the function f (z) = +
an z n (an ≥ 0) is in the class
z n=p
P∗
p (A, B, β), then
(3.1)
(j) f (z) ≥
p!(B − A)β
j!
−
|z|p−j
j+1
|z|
(p − j)!(1 + βB)
29
On Certain Subclasses of Meromorphic...
and
(3.2)
(j) f (z) ≤
p!(B − A)β
j!
|z|p−j
+
j+1
|z|
(p − j)!(1 + βB)
j!(p − j)!
.
p!(B − A) − j!(p − j)!B
Equalities in (3.1) and (3.2) are attained for the function f (z) given by
for z ∈ U ∗ , where 0 ≤ j ≤ p and 0 < β ≤
(3.3)
f (z) =
1 (B − A)β p
+
z (p ∈ N ∗ ) .
z
1 + βB
Proof. It follows from Theorem 1, that
(3.4)
∞
∞
X
n!
(p − j)!(1 + βB) X
an ≤
p!
(n
−
j)!
n=p
n=p
n
p
!
(1 + βB)an ≤ (B − A)β .
Therefore, we have
(3.5)
∞
X
n=p
p!(B − A)β
n!
an ≤
.
(n − j)!
(p − j)!(1 + βB)
Thus we have
∞
(3.6)
X n!
j!
−
an |z|n−j
|z|j+1 n=p (n − j)!
(j) f (z) ≥
≥
and
j!
p!(B − A)β
−
|z|p−j ,
j+1
|z|
(p − j)!(1 + βB)
∞
(3.7)
(j) f (z) ≤
X n!
j!
+
an |z|n−j
|z|j+1 n=p (n − j)!
30
M. K. Aouf
≤
Hence we have Theorem 2.
p!(B − A)β
j!
+
|z|p−j .
j+1
|z|
(p − j)!(1 + βB)
Putting j = 0 in Theorem 2, we have
Corollary 4 If f (z) ∈
(3.8)
P∗
p (A, B, β),
then
(B − A)β p
1
(B − A)β
1
−
|z| ≤ |f (z)| ≤
+
|z|p
|z|
1 + βB
|z|
1 + βB
for z ∈ U ∗ . Equalities in (3.8) are attained for the function f (z) given by
(3.3).
Putting j = 1 in Theorem 2, we have
Corollary 5 If f (z) ∈
(3.9)
P∗
p (A, B, β),
then
1
p(B − A)β p−1
1
p(B − A)β
0
−
|z|
≤ |f (z)| ≤ 2 +
|z|p−1
2
|z|
1 + βB
|z|
1 + βB
1
. Equalities in (3.9) are attained
p(B − A) − B
for the function f (z) given by (3.3).
for z ∈ U ∗ , where 0 < β ≤
Putting (i) p = 1 and j = 0
Corollary 6 If f (z) ∈
(3.10)
P∗
(ii) p = j = 1
1 (A, B, β),
in Theorem 2, we obtain
then
1
(B − A)β
1
(B − A)β
−
|z| ≤ |f (z)| ≤
+
|z| ,
|z|
1 + βB
|z|
1 + βB
31
On Certain Subclasses of Meromorphic...
and
(3.11)
1
(B − A)β
1
(B − A)β
−
≤ |f 0 (z)| ≤ 2 +
2
|z|
1 + βB
|z|
1 + βB
for z ∈ U ∗ . Equalities in (3.10)and (3.11) are attained for the function f (z)
given by
(3.12)
4
f (z) =
1 (B − A)β
+
z.
z
1 + βB
Radii of starlikeness and convexity
P
Theorem 3 Let the function f (z) defined by (1.4) be in the class ∗p (A, B, β),
then f (z) is meromorphically starlike of order δ(0 ≤ δ < 1) in |z| < r 1 ,
where
(4.1)
r1 = inf
n≥p






n
p
!
1
 n+1


(1 + βB)(1 − δ) 



(B − A)β(n + 2 − δ)









The result is sharp for the function f (z) given by (2.8).
P
Proof. Let f (z) ∈ ∗p (A, B, β). Then, by Theorem 1
(4.2)
∞
X
n=p
n
p
!
(1 + βB)
an ≤ 1 .
(B − A)β
It is sufficient to show that
(4.3)
0
z f (z)
≤1−δ
+
1
f (z)
.
32
M. K. Aouf
for |z| < r1 . We note that
(4.4)
P
∞
P
∞
(n + 1)an |z|n+1
0
(n + 1)an z n n=p
z f (z)
n=p
≤
∞
∞
f (z) + 1 = 1
P
+ P an z n an |z|n+1
1
−
z
n=p
n=p
This will be bounded by 1 − δ if
∞
X
(n + 2 − δ)
an |z|n+1 ≤ 1 .
(1
−
δ)
n=p
(4.5)
In view of (4.3), it follows that (4.5) is true if
(4.6)
or
(4.7)
(n + 2 − δ) n+1
|z|
≤
(1 − δ)
|z| ≤






n
p
!
n
p
!
(1 + βB)
(n ≥ p),
(B − A)β
1
 n+1


(1 + βB)(1 − δ) 



(B − A)β(n + 2 − δ)









(n ≥ p; p ∈ N ∗ ).
Setting |z| = r1 in (4.7), the result follows.
P
Theorem 4 Let the function f (z) defined by (1.4) be in the class ∗p (A, B, β),
then f (z) is meromorphically convex of order δ in |z| < r2 , where
(4.9)
r2 = inf
n≥p






n
p
!
1
 n+1


(1 + βB)(1 − δ) 



(B − A)β n(n + 2 − δ) 








The result is sharp for the function f (z) defined by (2.8).
.
33
On Certain Subclasses of Meromorphic...
5
Convolution properties
For functions
∞
(5.1)
1 X
an,j z n (an,j ≥ 0, j = 1, 2; p ∈ N ∗ )
fj (z) = +
z n=p
P∗
belonging to the class
p , we denote by (f1 ∗ f2 )(z) the convolution (or
Hadamard product) of the functions f1 (z) and f2 (z), that is,
∞
1 X
(f1 ∗ f2 )(z) = +
an,1 an,2 z n .
z n=p
(5.2)
Theorem 5 Let the functions fj (z) (j = 1, 2) defined by (5.1)be in the class
P∗
P∗
p (α, β), then (f1 ∗ f2 )(z) ∈
p (γ, β), where
(5.3)
γ = p! −
2β(p! − α)2
.
p!(1 + β)
The result is sharp for the functions
(5.4)
fj (z) =
1 2β(p! − α) p
+
z (j = 1, 2, p ∈ N ∗ ).
z
p!(1 + β)
Proof. Employing the technique used earlier Schild and Silverman [9] , we
need to find the largest γ such that
∞
X
(5.5)
n=p
for fj (z) ∈
see that
P∗
p (α, β)(j
n
p
!
(1 + β)
an,1 an,2 ≤ 1 .
2β(1 − p!γ )
= 1, 2). Since fj (z) ∈
P∗
p (α, β)(j
= 1, 2), we readily
34
(5.6)
M. K. Aouf
∞
X
n=p
n
p
!
(1 + β)
α an,j ≤ 1 (j = 1, 2).
2β(1 − p!
)
Therefore, by the Cauchy-Schwarz inequality, we obtain
(5.7)
∞
X
n
p
n=p
!
(1 + β) √
an,1 an,2 ≤ 1 ,
α
)
2β(1 − p!
this implies that, we need only show that
an,1 an,2
≤
1 − p!γ
(5.8)
√
an,1 an,2
(n ≥ p)
α
1 − p!
or , equivalently, that
(5.9)
√
an,1 an,2 ≤
1−
1−
γ
p!
α
p!
(n ≥ p).
Hence, by the inequality (5.7), it is sufficient to prove that
(5.10)
α
1−
2β(1 − p!
)
!
≤
1−
n
(1 + β)
p
γ
p!
α
p!
(n ≥ p).
It follows from (5.10) that
(5.11)
γ ≤ p! −
2β(p! − α)2
!
(n ≥ p).
n
p!
(1 + β)
p
Now, defining the function φ(n) by
On Certain Subclasses of Meromorphic...
(5.12)
φ(n) = p! −
35
2β(p! − α)2
!
(n ≥ p).
n
p!
(1 + β)
p
We see that ϕ(n) is an increasing function of n. Therefore, we conclude
that
(5.13)
γ ≤ φ(p) = p! −
2β(p! − α)2
,
p!(1 + β)
which evidently completes the proof of Theorem 5.
Using similar arguments as in the proof of Theorem 5, we can obtain
the following result.
∞
∗
X
X
1
1
n
Theorem 6 If f (z) =
+
+
an,1 z ∈
(α, β) and f2 (z) =
z
z
n=p
p
∞
∗
X
X
P
n
an,2 z ∈
(γ, β), then (f1 ∗ f2 ) ∈ ∗p (=, β), where
n=p
(5.14)
p
= = p! −
2β(p! − α)(p! − γ)
.
p!(1 + β)
The result is possible for the functions fj (z)(j = 1, 2) given by
(5.15)
f1 (z) =
1 2β(p! − α) p
+
z (p ∈ N ∗ )
z
p!(1 + β)
f2 (z) =
1 2β(p! − γ) p
+
z (p ∈ N ∗ ).
z
p!(1 + β)
and
(5.16)
36
M. K. Aouf
Theorem 7 Let the functions fj (z)(j = 1, 2) defined by (5.1) be in the class
P∗
p (α, β). Then the function h(z) defined by
∞
1 X 2
(a + a2n,2 ) z n
h(z) = +
z n=p n,1
(5.17)
P∗
p (δ, β),
also belongs to the class
(5.18)
δ = p! −
where
4β(p! − α)2
.
p!(1 + β)
The result is sharp for the functions fj (z)(j = 1, 2) defined by (5.4).
Proof. Noting that
∞
X
(5.19)
n=p
≤
for fj (z) ∈
(5.20)
"
P∗
∞
X
n=p
p (α, β)
∞
X
1
n=p
n
p
2
"
!
n
p
!
1+β
α
)
2β(1 − p!
1+β
α an,j
)
2β(1 − p!
#2
#2
a2n,j
≤ 1 (j = 1, 2)
(j = 1, 2), we have
"
n
p
!
1+β
α
)
2β(1 − p!
#2
(a2n,1 + a2n,2 ) ≤ 1 .
Therefore, we have to find the largest δ such that
(5.21)
n
p
!
(1 + β)
2β(1 − p!δ )
1
≤
2
n
p
!2
(1 + β)2
α 2 (n ≥ p),
4β 2 (1 − p!
)
37
On Certain Subclasses of Meromorphic...
that is, that
(5.22)
δ ≤ p! −
4β(p! − α)2
!
(n ≥ p).
n
p!(1 + β)
p
Now, defining a function Ψ(n) by
(5.23)
4β(p! − α)2
!
n
p!(1 + β)
p
Ψ(n) = p! −
(n ≥ p).
We observe that Ψ(n) is an increasing function of n. Thus, we conclude
that
(5.24)
δ ≤ Ψ(p) = p! −
4β(p! − α)2
,
p!(1 + β)
which completes the proof of Theorem 7.
In the remaining theorems, we consider the functions in the class
Theorem 8 If f (z) =
∞
P
n=p
1
z
+
∞
P
n=p
an,1 z n ∈
P∗
p (A, B, β)
an,2 z n with |an,2 | ≤ 1(n ≥ p), then (f1 ∗ f2 )(z) ∈
Proof. Since
∞
X
n=p
n
p
!
∞
X
1 + βB
|an,1 an,2 | =
(B − A)β
n=p
≤
∞
X
n=p
by Theorem 1, it follows that (f1 ∗ f2 )(z) ∈
P∗
p (A, B, β).
and g(z) =
P∗
p (A, B, β).
n
p
!
1 + βB
an,1 |an,2 |
(B − A)β
n
p
!
1 + βB
an,1 ≤ 1,
(B − A)β
P∗
p (A, B, β).
1
z
+
38
M. K. Aouf
∞
∗
X
1 X
n
Corollary 7 If f1 (z) = +
an,1 z ∈
(A, B, β) and f2 (z) =
z
n=p
p
∞
P
P
n
an,2 z with 0 ≤ an,2 ≤ 1, n ≥ p, then (f1 ∗ f2 )(z) ∈ ∗p (A, B, β).
1
z
+
n=p
Theorem 9 Let the functions fj (z)(j = 1, 2) defined by (5.1) be in the class
P∗
p (A, B, β) and 1−βB +2βA ≥ 0, then the function h(z) defined by (5.17)
P
also belongs to ∗p (A, B, β).
Proof. Since f1 (z) ∈
P∗
p (A, B, β),
∞
X
n
p
n=p
and so
∞
X
n=p
Similarly, since f2 (z) ∈
∞
X
n=p
Hence
∞
X
1
n=p
2
"
"
n
p
P∗
!
!
we have
1 + βB
an,1 ≤ 1
(B − A)β
(1 + βB)
(B − A)β
p (A, B, β),
"
n
p
n
p
!
!
#2
a2n,1 ≤ 1 .
we have
(1 + βB)
(B − A)β
1 + βB
(B − A)β
#2
#2
a2n,2 ≤ 1 .
(a2n,1 + a2n,2 ) ≤ 1 .
In view of Theorem 1, it is sufficient to show that
(5.25)
∞
X
n=p
n
p
!
(1 + βB) 2
( a + a2n,2 ) ≤ 1 .
(B − A)β n,1
Thus the inequality (5.25) will be satisfied if for n ≥ p,
39
On Certain Subclasses of Meromorphic...
(5.26)
n
p
!
(1 + βB)
1
≤
(B − A)β
2
n
p
!2
(1 + βB)2
,
(B − A)2 β 2
or if
(5.27)
n
p
!
(1 + βB) + 2β(A − B) ≥ 0
for n ≥ p (p ∈ N ∗ ). The left hand side of (5.27) is an increasing function
of n , hence (5.27) is satisfied for all n if
1 − βB + 2βA ≥ 0 ,
which is true by our assumption . Hence the theorem.
References
[1] M. K. Aouf, A certain subclass of meromorphically starlike functions
with positive coefficients, Rend. Mat. 9(1989), 225-235.
[2] M. K. Aouf, On a certain class of meromorphic univalent functions with
positive coefficients, Rend. Mat. 11(1991), 209-219.
[3] N. E. Cho, On certain class of meromorphic functions with positive coefficients, J. Inst. Math. Comput. Sci. 3(1990), no.2, 119-125.
[4] N. E. Cho, S. H. Lee, S. Owa, A class of meromorphic univalent functions with positive coefficients, Kobe J. Math. 4(1987), 43-50.
[5] Y. G. Kim, S. H. Lee, S. Owa, On certain meromorphic functions with
positive coefficients, Internat. J. Math. Math. Sci. 16(1993), no.2, 409412.
40
M. K. Aouf
[6] J. E. Miller, Convex meromorphic mapping and related functions, Proc.
Amer. Math. Soc. 25(1970), 220-228.
[7] M. L. Morga, T. Reddy, O. P. Juneja, Meromorphic univalent functions
with positive coefficients, Bull. Austral. Math. Soc. 32(1985), 161-176.
[8] Ch. Pommerenke, On meromorphic starlike functions, Pacific J. Math.
13(1963), 221-235.
[9] A. Schild,H.Silverman , Convolution of univalent functions with negative
coefficients, Ann. Univ Mariae Curre-Sklodowska sect. A29 (1975), 99107.
M. K. Aouf
Department of Mathematics
Faculty of Science
Mansoura University, Mansoura 35516, Egypt.
e-mail: mkaouf127@yahoo.com