Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 2, Article 58, 2005
MEROMORPHIC FUNCTIONS WITH POSITIVE COEFFICIENTS DEFINED
USING CONVOLUTION
S. SIVAPRASAD KUMAR, V. RAVICHANDRAN, AND H.C. TANEJA
D EPARTMENT OF A PPLIED M ATHEMATICS
D ELHI C OLLEGE OF E NGINEERING
D ELHI 110042, I NDIA
sivpk71@yahoo.com
URL: http://sivapk.topcities.com
S CHOOL OF M ATHEMATICAL S CIENCES
U NIVERSITI S AINS M ALAYSIA
11800 USM P ENANG , M ALAYSIA
vravi@cs.usm.my
URL: http://cs.usm.my/∼vravi
D EPARTMENT OF A PPLIED M ATHEMATICS
D ELHI C OLLEGE OF E NGINEERING
D ELHI 110042, I NDIA
hctaneja47@rediffmail.com
Received 25 May, 2005; accepted 31 May, 2005
Communicated by H. Silverman
A BSTRACT
P∞. For certain meromorphic function g and h, we study a class of functions f (z) =
z −1 + n=1 fn z n , (fn ≥ 0), defined in the punctured unit disk ∆∗ , satisfying
(f ∗ g)(z)
<
> α (z ∈ ∆; 0 ≤ α < 1) .
(f ∗ h)(z)
Coefficient inequalities, growth and distortion inequalities, as well as closure results are obtained. Properties of an integral operator and its inverse defined on the new class is also discussed. In addition, we apply the concepts of neighborhoods of analytic functions to this class.
Key words and phrases: Meromorphic functions, Starlike function, Convolution, Positive coefficients, Coefficient inequalities, Growth and distortion theorems, Closure theorems, Integral operator.
2000 Mathematics Subject Classification. 30C45.
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
164-05
2
S. S IVAPRASAD K UMAR , V. R AVICHANDRAN , AND H.C. TANEJA
1. I NTRODUCTION
Let Σ denote the class of normalized meromorphic functions f of the form
∞
1 X
(1.1)
f (z) = +
fn z n ,
z n=1
defined on the punctured unit disk ∆∗ := {z ∈ C : 0 < |z| < 1}. A function f ∈ Σ is
meromorphic starlike of order α (0 ≤ α < 1) if
zf 0 (z)
−<
> α (z ∈ ∆ := ∆∗ ∪ {0}).
f (z)
The class of all such functions is denoted by Σ∗ (α). Similarly the class of convex functions of
order α is defined. Let Σp be the class of functions f ∈ Σ with fn ≥ 0. The subclass of Σp
consisting of starlike functions of order α is denoted by Σ∗p (α). The following class M Rp (α) is
related to the class of functions with positive real part:
M Rp (α) := f |<{−z 2 f 0 (z)} > α, (0 ≤ α < 1) .
In Definition 1.1 below, we unify these classes by using convolution. The Hadamard product
or convolution of two functions f (z) given by (1.1) and
∞
1 X
(1.2)
g(z) = +
gn z n
z n=1
is defined by
∞
(f ∗ g)(z) :=
1 X
+
fn gn z n .
z n=1
Definition 1.1. Let 0 ≤ α < 1. Let f (z) ∈ Σp be given by (1.1) and g(z) ∈ Σp be given by
(1.2) and
∞
1 X
(1.3)
h(z) = +
hn z n .
z n=1
Let hn , gn be real and gn + (1 − 2α)hn ≤ 0 ≤ αhn − gn . The class Mp (g, h, α) is defined by
(f
∗
g)(z)
Mp (g, h, α) = f ∈ Σp <
>α .
(f ∗ h)(z)
Of course, one can consider a more general class of functions satisfying the subordination:
(f ∗ g)(z)
≺ h(z) (z ∈ ∆).
(f ∗ h)(z)
However the results for this class will follow from the corresponding results of the class Mp (g, h, α).
See [5] for details.
When
1
z
1
g(z) = −
and h(z) =
,
2
z (1 − z)
z(1 − z)
we have gn = −n and hn = 1 and therefore Mp (g, h, α) reduces to the class Σ∗p (α). Similarly
when
1
z
1
g(z) = −
and
h(z)
=
,
z (1 − z)2
z
we have
Mp (g, h, α) = {f | − <{z 2 f 0 (z)} > α} =: M Rp (α).
J. Inequal. Pure and Appl. Math., 6(2) Art. 58, 2005
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M EROMORPHIC F UNCTIONS WITH P OSITIVE C OEFFICIENTS
3
In this paper, coefficient inequalities, growth and distortion inequalities, as well as closure
results for the class Mp (g, h, α) are obtained. Properties of an integral operator and its inverse
defined on the new class Mp (g, h, α) is also discussed.
2. C OEFFICIENTS I NEQUALITIES
Our first theorem gives a necessary and sufficient condition for a function f to be in the class
Mp (g, h, α).
Theorem 2.1. Let f (z) ∈ Σp be given by (1.1). Then f ∈ Mp (g, h, α) if and only if
∞
X
(2.1)
(αhn − gn )fn ≤ 1 − α.
n=1
Proof. If f ∈ Mp (g, h, α), then
P
1+ ∞
fn gn z n+1
(f ∗ g)(z)
n=1
P
<
=<
> α.
n+1
(f ∗ h)(z)
1+ ∞
n=1 fn hn z
By letting z → 1− , we have
P
1+ ∞
fn gn
n=1
P
> α.
1+ ∞
n=1 fn hn
This shows that (2.1) holds.
Conversely, assume that (2.1) holds. Since
<w > α
if and only if
|w − 1| < |w + 1 − 2α|,
it is sufficient to show that
(f
∗
g)(z)
−
(f
∗
h)(z)
(f ∗ g)(z) + (1 − 2α)(f ∗ h)(z) < 1
(z ∈ ∆).
Using (2.1), we see that
P∞
n+1
f
(g
−
h
)z
(f
∗
g)(z)
−
(f
∗
h)(z)
n
n
n
n=1
=
P
∞
(f ∗ g)(z) + (1 − 2α)(f ∗ h)(z) 2(1 − α) +
n+1 [g
+
(1
−
2α)h
]f
z
n
n
n
P∞ n=1
n=1 fn (hn − gn )
P
≤
≤ 1.
2(1 − α) − ∞
n=1 [(2α − 1)hn − gn ]fn
Thus we have f ∈ Mp (g, h, α).
Corollary 2.2. Let f (z) ∈ Σp be given by (1.1). Then f ∈ Σ∗p (α) if and only if
∞
X
(n + α)fn ≤ 1 − α.
n=1
Corollary 2.3. Let f (z) ∈ Σp be given by (1.1). Then f ∈ M Rp (α) if and only if
≤ 1 − α.
P∞
n=1
nfn
Our next result gives the coefficient estimates for functions in Mp (g, h, α).
Theorem 2.4. If f ∈ Mp (g, h, α), then
1−α
fn ≤
,
n = 1, 2, 3, . . . .
αhn − gn
The result is sharp for the functions Fn (z) given by
1
1−α n
Fn (z) = +
z ,
n = 1, 2, 3, . . . .
z αhn − gn
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4
S. S IVAPRASAD K UMAR , V. R AVICHANDRAN , AND H.C. TANEJA
Proof. If f ∈ Mp (g, h, α), then we have, for each n,
(αhn − gn )fn ≤
∞
X
(αhn − gn )fn ≤ 1 − α.
n=1
Therefore we have
fn ≤
1−α
.
αhn − gn
Since
1
1−α n
+
z
z αhn − gn
satisfies the conditions of Theorem 2.1, Fn (z) ∈ Mp (g, h, α) and the inequality is attained for
this function.
Fn (z) =
Corollary 2.5. If f ∈ Σ∗p (α), then
fn ≤
1−α
,
n+α
n = 1, 2, 3, . . . .
Corollary 2.6. If f ∈ M Rp (α), then
1−α
,
n = 1, 2, 3, . . . .
n
Theorem 2.7. Let αh1 − g1 ≤ αhn − gn . If f ∈ Mp (g, h, α), then
fn ≤
1
1−α
1
1−α
−
r ≤ |f (z)| ≤ +
r
r αh1 − g1
r αh1 − g1
The result is sharp for
1
1−α
(2.2)
f (z) = +
z.
z αh1 − g1
P
n
Proof. Since f (z) = z1 + ∞
n=1 fn z , we have
|f (z)| ≤
(|z| = r).
∞
∞
X
1 X
1
+
fn rn ≤ + r
fn .
r n=1
r
n=1
Since αh1 − g1 ≤ αhn − gn , we have
∞
∞
X
X
(αh1 − g1 )
fn ≤
(αhn − gn )fn ≤ 1 − α,
n=1
and therefore
∞
X
n=1
fn ≤
n=1
1−α
.
αh1 − g1
Using this, we have
|f (z)| ≤
1
1−α
+
r.
r αh1 − g1
|f (z)| ≥
1
1−α
−
r.
r αh1 − g1
Similarly
The result is sharp for f (z) =
1
z
+
1−α
z.
αh1 −g1
Similarly we have the following:
J. Inequal. Pure and Appl. Math., 6(2) Art. 58, 2005
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M EROMORPHIC F UNCTIONS WITH P OSITIVE C OEFFICIENTS
5
αh1 − g1 ≤ (αhn − gn )/n. If f ∈ Mp (g, h, α), then
1
1−α
1
1−α
0
−
≤
|f
(z)|
≤
+
(|z| = r).
r2 αh1 − g1
r2 αh1 − g1
The result is sharp for the function given by (2.2).
Theorem 2.8. Let
3. C LOSURE T HEOREMS
Let the functions Fk (z) be given by
∞
1 X
Fk (z) = +
fn,k z n ,
z n=1
(3.1)
k = 1, 2, . . . , m.
We shall prove the following closure theorems for the class Mp (g, h, α).
Theorem 3.1. Let the function Fk (z) defined by (3.1) be in the class Mp (g, h, α) for every
k = 1, 2, . . . , m. Then the function f (z) defined by
∞
1 X
an z n
(an ≥ 0)
f (z) = +
z n=1
P
belongs to the class Mp (g, h, α), where an = m1 m
k=1 fn,k (n = 1, 2, . . . )
Proof. Since Fn (z) ∈ Mp (g, h, α), it follows from Theorem 2.1 that
∞
X
(3.2)
(αhn − gn )fn,k ≤ 1 − α
n=1
for every k = 1, 2, . . . , m. Hence
∞
X
(αhn − gn )an =
n=1
∞
X
(αhn − gn )
n=1
!
m
1 X
fn,k
m k=1
!
m
∞
1 X X
(αhn − gn )fn,k
=
m k=1 n=1
≤ 1 − α.
By Theorem 2.1, it follows that f (z) ∈ Mp (g, h, α).
Theorem 3.2. The class Mp (g, h, α) is closed under convex linear combination.
Proof. Let the function Fk (z) given by (3.1) be in the class Mp (g, h, α). Then it is enough to
show that the function
H(z) = λF1 (z) + (1 − λ)F2 (z)
(0 ≤ λ ≤ 1)
is also in the class Mp (g, h, α). Since for 0 ≤ λ ≤ 1,
∞
1 X
H(z) = +
[λfn,1 + (1 − λ)fn,2 ]z n ,
z n=1
we observe that
∞
∞
∞
X
X
X
(αhn − gn )[λfn,1 + (1 − λ)fn,2 ] = λ
(αhn − gn )fn,1 + (1 − λ)
(αhn − gn )fn,2
n=1
n=1
n=1
≤ 1 − α.
By Theorem 2.1, we have H(z) ∈ Mp (g, h, α).
J. Inequal. Pure and Appl. Math., 6(2) Art. 58, 2005
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6
S. S IVAPRASAD K UMAR , V. R AVICHANDRAN , AND H.C. TANEJA
Theorem 3.3. Let F0 (z) = z1 and Fn (z) = z1 + αh1−α
z n for n = 1, 2, . . .. Then f (z) ∈
n −gn
P∞
Mp (g, h, α) P
if and only if f (z) can be expressed in the form f (z) =
n=0 λn Fn (z), where
∞
λn ≥ 0 and n=0 λn = 1.
Proof. Let
∞
X
∞
1 X λn (1 − α) n
f (z) =
λn Fn (z) = +
z .
z n=1 αhn − gn
n=0
Then
∞
X
λn (1 − α) αhn − gn
n=1
αhn − gn
1−α
=
∞
X
λn = 1 − λ0 ≤ 1.
n=1
By Theorem 2.1, we have f (z) ∈ Mp (g, h, α).
Conversely, let f (z) ∈ Mp (g, h, α). From Theorem 2.4, we have
1−α
for n = 1, 2, . . .
fn ≤
αhn − gn
we may take
αhn − gn
λn =
fn for n = 1, 2, . . .
1−α
and
∞
X
λ0 = 1 −
λn .
n=1
Then
f (z) =
∞
X
λn Fn (z).
n=0
4. I NTEGRAL O PERATORS
In this section, we consider integral transforms of functions in the class Mp (g, h, α).
Theorem 4.1. Let the function f (z) given by (1.1) be in Mp (g, h, α). Then the integral operator
Z 1
F (z) = c
uc f (uz)du
(0 < u ≤ 1, 0 < c < ∞)
0
is in Mp (g, h, δ), where
δ=
(c + 2)(αh1 − g1 ) + (1 − α)cg1
.
(c + 2)(αh1 − g1 ) + (1 − α)ch1
The result is sharp for the function f (z) =
Proof. Let f (z) ∈ Mp (g, h, α). Then
Z
F (z) = c
1
z
+
1−α
z.
αh1 −g1
1
uc f (uz)du
0
Z
!
∞
uc−1 X
+
fn un+c z n du
z
n=1
1
=c
0
∞
1 X
c
= +
fn z n .
z n=1 c + n + 1
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M EROMORPHIC F UNCTIONS WITH P OSITIVE C OEFFICIENTS
7
It is sufficient to show that
∞
X
(4.1)
n=1
c(δhn − gn )
fn ≤ 1.
(c + n + 1)(1 − δ)
Since f ∈ Mp (g, h, α), we have
∞
X
αhn − gn
n=1
(1 − α)
fn ≤ 1.
Note that (4.1) is satisfied if
c(δhn − gn )
αhn − gn
≤
.
(c + n + 1)(1 − δ)
(1 − α)
Rewriting the inequality, we have
c(δhn − gn )(1 − α) ≤ (c + n + 1)(1 − δ)(αhn − gn ).
Solving for δ, we have
δ≤
(αhn − gn )(c + n + 1) + cgn (1 − α)
= F (n).
chn (1 − α) + (αhn − gn )(c + n + 1)
A computation shows that
F (n + 1) − F (n)
=
(1 − α)c[(1 − α)(n + 1)gn hn+1 + (hn − gn )(αhn+1 − gn+1 )]
>0
[chn (1 − α) + (αhn − gn )(c + n + 1)][chn+1 (1 − α) + (αhn+1 − gn+1 )(c + n + 2)]
for all n. This means that F (n) is increasing and F (n) ≥ F (1). Using this, the results follows.
In particular, we have the following result of Uralegaddi and Ganigi [4]:
Corollary 4.2. Let the function f (z) defined by (1.1) be in Σ∗p (α). Then the integral operator
Z 1
F (z) = c
uc f (uz)du
(0 < u ≤ 1, 0 < c < ∞)
0
is in
Σ∗p (δ),
where δ =
1+α+cα
.
1+α+c
The result is sharp for the function
f (z) =
1 1−α
+
z.
z 1+α
Also we have the following:
Corollary 4.3. Let the function f (z) defined by (1.1) be in M Rp (α). Then the integral operator
Z 1
F (z) = c
uc f (uz)du
(0 < u ≤ 1, 0 < c < ∞)
0
is in
M Rp ( 2+cα
).
c+2
The result is sharp for the function f (z) =
1
z
+ (1 − α)z.
Theorem 4.4. Let f (z), given by (1.1), be in Mp (g, h, α),
∞
(4.2)
1
1 Xc+n+1
F (z) = [(c + 1)f (z) + zf 0 (z)] = +
fn z n ,
c
z n=1
c
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c > 0.
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8
S. S IVAPRASAD K UMAR , V. R AVICHANDRAN , AND H.C. TANEJA
Then F (z) is in Mp (g, h, β) for |z| ≤ r(α, β), where
1
n+1
c(1 − β)(αhn − gn )
r(α, β) = inf
,
n
(1 − α)(c + n + 1)(βhn − gn )
The result is sharp for the function fn (z) =
Proof. Let w =
(f ∗g)(z)
.
(f ∗h)(z)
1
z
+
1−α
zn,
αhn −gn
n = 1, 2, 3, . . . .
n = 1, 2, 3, . . . .
Then it is sufficient to show that
w−1 w + 1 − 2β < 1.
A computation shows that this is satisfied if
∞
X
(βhn − gn )(c + n + 1)
(4.3)
fn |z|n+1 ≤ 1.
(1
−
β)c
n=1
Since f ∈ Mp (g, h, α), by Theorem 2.1, we have
∞
X
(αhn − gn )fn ≤ 1 − α.
n=1
The equation (4.3) is satisfied if
(αhn − gn )fn
(βhn − gn )(c + n + 1)
fn |z|n+1 ≤
.
(1 − β)c
1−α
Solving for |z|, we get the result.
In particular, we have the following result of Uralegaddi and Ganigi [4]:
Corollary 4.5. Let the function f (z) defined by (1.1) be in Σ∗p (α) and F (z) given by (4.2). Then
F (z) is in Σ∗p (α) for |z| ≤ r(α, β), where
1
n+1
c(1 − β)(n + α)
, n = 1, 2, 3, . . . .
r(α, β) = inf
n
(1 − α)(c + n + 1)(n + β)
The result is sharp for the function fn (z) =
1
z
+
1−α n
z ,
n+α
n = 1, 2, 3, . . . .
Corollary 4.6. Let the function f (z) defined by (1.1) be in M Rp (α) and F (z) given by (4.2).
Then F (z) is in M Rp (α) for |z| ≤ r(α, β), where
1
n+1
c(1 − β)
r(α, β) = inf
, n = 1, 2, 3, . . . .
n
(1 − α)(c + n + 1)
The result is sharp for the function fn (z) =
1
z
+
1−α n
z ,
n
n = 1, 2, 3, . . . .
(γ)
5. N EIGHBORHOODS FOR THE C LASS Mp (g, h, α)
(γ)
In this section, we determine the neighborhood for the class Mp (g, h, α), which we define
as follows:
(γ)
Definition 5.1. A function f ∈ Σp is said to be in the class Mp (g, h, α) if there exists a
function g ∈ Mp (g, h, α) such that
f (z)
< 1 − γ,
(5.1)
−
1
(z ∈ ∆, 0 ≤ γ < 1).
g(z)
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M EROMORPHIC F UNCTIONS WITH P OSITIVE C OEFFICIENTS
9
Following the earlier works on neighborhoods of analytic functions by Goodman [1] and
Ruscheweyh [3], we define the δ-neighborhood of a function f ∈ Σp by
(
)
∞
∞
X
1 X
(5.2)
Nδ (f ) := g ∈ Σp : g(z) = +
bn z n and
n|an − bn | ≤ δ .
z n=1
n=1
Theorem 5.1. If g ∈ Mp (g, h, α) and
(5.3)
γ =1−
δ(αh1 − g1 )
,
α(h1 + 1) − (g1 + 1)
then
Nδ (g) ⊂ Mp(γ) (g, h, α).
Proof. Let f ∈ Nδ (g). Then we find from (5.2) that
∞
X
(5.4)
n|an − bn | ≤ δ,
n=1
which implies the coefficient inequality
∞
X
(5.5)
|an − bn | ≤ δ,
(n ∈ N).
n=1
Since g ∈ Mp (g, h, α), we have [cf. equation (2.1)]
∞
X
1−α
,
(5.6)
fn ≤
(αh1 − g1 )
n=1
so that
P∞
f (z)
|an − bn |
< n=1P
−
1
g(z)
1− ∞
n=1 bn
δ(αh1 − g1 )
=
α(h1 + 1) − (g1 + 1)
= 1 − γ,
(γ)
provided γ is given by (5.3). Hence, by definition, f ∈ Mp (g, h, α) for γ given by (5.3),
which completes the proof.
R EFERENCES
[1] A.W. GOODMAN, Univalent functions and nonanalytic curve, Proc. Amer. Math. Soc., 8 (1957),
598–601.
[2] M.L. MOGRA, T.R. REDDY AND O.P. JUNEJA, Meromorphic univalent functions with positive
coefficients, Bull. Austral. Math. Soc., 32 (1985), 161–176.
[3] S. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., 81 (1981), 521–
527.
[4] B.A. URALEGADDI AND M.D. GANIGI, A certain class of meromorphically starlike functions
with positive coefficients, Pure Appl. Math. Sci., 26 (1987), 75–81.
[5] V. RAVICHANDRAN, On starlike functions with negative coefficients, Far. East. J. Math. Sci., 8(3)
(2003), 359–364.
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