Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 1, Article 22, 2005
A CLASS OF MULTIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS
DEFINED BY CONVOLUTION
ROSIHAN M. ALI, M. HUSSAIN KHAN, V. RAVICHANDRAN, AND K.G. SUBRAMANIAN
S CHOOL OF M ATHEMATICAL S CIENCES
U NIVERSITI S AINS M ALAYSIA
11800 USM, P ENANG , M ALAYSIA
rosihan@cs.usm.my
D EPARTMENT OF M ATHEMATICS
I SLAMIAH C OLLEGE
VANIAMBADI 635 751, I NDIA
khanhussaff@yahoo.co.in
S CHOOL OF M ATHEMATICAL S CIENCES
U NIVERSITI S AINS M ALAYSIA
11800 USM, P ENANG , M ALAYSIA
vravi@cs.usm.my
D EPARTMENT OF M ATHEMATICS
M ADRAS C HRISTIAN C OLLEGE
TAMBARAM , C HENNAI - 600 059, I NDIA
kgsmani@vsnl.net
Received 12 December, 2004; accepted 27 January, 2005
Communicated by H. Silverman
A BSTRACT. For a given p-valent analytic functionP
g with positive coefficients in the open unit
∞
disk ∆, we study a class of functions f (z) = z p + n=m an z n , an ≥ 0 satisfying
1
z(f ∗ g)0 (z)
m+p
<
<α
z ∈ ∆; 1 < α <
.
p
(f ∗ g)(z)
2p
Coefficient inequalities, distortion and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special cases.
Key words and phrases: Starlike function, Ruscheweyh derivative, Convolution, Positive coefficients, Coefficient inequalities,
Growth and distortion theorems.
2000 Mathematics Subject Classification. 30C45.
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
The authors R. M. Ali and V. Ravichandran respectively acknowledged support from an IRPA grant 09-02-05-00020 EAR and a postdoctoral research fellowship from Universiti Sains Malaysia.
237-04
2
ROSIHAN M. A LI , M. H USSAIN K HAN , V. R AVICHANDRAN , AND K.G. S UBRAMANIAN
1. I NTRODUCTION
Let A denote the class of all analytic functions f (z) in the unit disk ∆ := {z ∈ C : |z| < 1}
with f (0) = 0 = f 0 (0) − 1. The class M (α) defined by
0 zf (z)
3
M (α) := f ∈ A : <
<α
1<α< ; z∈∆
f (z)
2
was investigated by Uralegaddi et al. [6]. A subclass of M (α) was recently investigated by Owa
and Srivastava [3]. Motivated by M (α), we introduce a more general class P Mg (p, m, α) of
analytic functions with positive coefficients. For two analytic functions
∞
∞
X
X
f (z) = z p +
an z n and g(z) = z p +
bn z n ,
n=m
n=m
the convolution (or Hadamard product) of f and g, denoted by f ∗ g or (f ∗ g)(z), is defined by
∞
X
p
(f ∗ g)(z) := z +
an b n z n .
n=m
P
n
Let T (p, m) be the class of all analytic p-valent functions f (z) = z p − ∞
n=m an z (an ≥ 0),
defined on the unit disk ∆ and let T := T (1, 2). A function f (z) ∈ T (p, m) is called a
function with negative coefficients. The subclass of T consisting of starlike functions of order
α, denoted by T S ∗ (α), was studied by Silverman [5]. Several other classes of starlike functions
with negative coefficients were studied; for e.g. see [2].
Let P (p, m) be the class of all analytic functions
∞
X
(1.1)
f (z) = z p +
an z n (an ≥ 0)
n=m
and P := P (1, 2).
Definition 1.1. Let
(1.2)
p
g(z) = z +
∞
X
bn z n
(bn > 0)
n=m
be a fixed analytic function in ∆. Define the class P Mg (p, m, α) by
1
z(f ∗ g)0 (z)
m+p
P Mg (p, m, α) := f ∈ P (p, m) : <
< α,
1<α<
;z ∈ ∆
.
p
(f ∗ g)(z)
2p
When g(z) = z/(1 − z), p = 1 and m = 2, the class P Mg (p, m, α) reduces to the subclass
P M (α) := P ∩ M (α). When g(z) = z/(1 − z)λ+1 , p = 1 and m = 2, the class P Mg (p, m, α)
reduces to the class:
z(Dλ f (z))0
3
Pλ (α) = f ∈ P : <
< α,
λ > −1, 1 < α < ; z ∈ ∆
,
Dλ f (z)
2
where Dλ denotes the Ruscheweyh derivative of order λ. When
∞
X
g(z) = z +
nl z n ,
n=2
the class of functions P Mg (1, 2, α) reduces to the class P Ml (α) where
z(Dl f (z))0
3
P Ml (α) = f ∈ P : <
< α,
1 < α < ; l ≥ 0; z ∈ ∆
,
Dl f (z)
2
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F UNCTIONS WITH P OSITIVE C OEFFICIENTS
3
where Dl denotes the Salagean derivative of order l. Also we have
P M (α) ≡ P0 (α) ≡ P M0 (α).
A function f ∈ A(p, m) is in P P C(p, m, α, β) if
!
(1 − β)zf 0 (z) + βp z(zf 0 )0 (z)
1
<α
<
p
(1 − β)f (z) + βp zf 0 (z)
m+p
β ≥ 0; 0 ≤ α <
2p
This class is similar to the class of β-Pascu convex functions of order α and it unifies the class
of P M (α) and the corresponding convex class.
For the newly defined class P Mg (p, m, α), we obtain coefficient inequalities, distortion and
covering theorems, as well as closure theorems. As special cases, we obtain results for the
classes Pλ (α), and P Ml (α). Similar results for the class P P C(p, m, α, β) also follow from our
results, the details of which are omitted here.
2. C OEFFICIENT I NEQUALITIES
Throughout the paper, we assume that the function f (z) is given by the equation (1.1) and
g(z) is given by by (1.2). We first prove a necessary and sufficient condition for functions to be
in the class P Mg (p, m, α) in the following:
Theorem 2.1. A function f ∈ P Mg (p, m, α) if and only if
∞
X
m+p
(2.1)
(n − pα)an bn ≤ p(α − 1)
1<α<
.
2p
n=m
Proof. If f ∈ P Mg (p, m, α), then (2.1) follows from
1
z(f ∗ g)0 (z)
<
<α
p
(f ∗ g)(z)
by letting z → 1− through real values. To prove the converse, assume that (2.1) holds. Then
by making use of (2.1), we obtain
P∞
z(f ∗ g)0 (z) − p(f ∗ g)(z)
n=m (n − p)an bn
≤
z(f ∗ g)0 (z) − (2α − 1)p(f ∗ g)(z) 2(α − 1)p − P∞ [n − (2α − 1)p]an bn ≤ 1
n=m
or equivalently f ∈ P Mg (p, m, α).
Corollary 2.2. A function f ∈ Pλ (α) if and only if
∞
X
(n − α)an Bn (λ) ≤ α − 1
n=2
3
1<α<
2
,
where
(2.2)
Bn (λ) =
(λ + 1)(λ + 2) · · · (λ + n − 1)
.
(n − 1)!
Corollary 2.3. A function f ∈ P Mm (α) if and only if
∞
X
m
(n − α)an n ≤ α − 1
n=2
3
1<α<
2
.
Our next theorem gives an estimate for the coefficient of functions in the class P Mg (p, m, α).
J. Inequal. Pure and Appl. Math., 6(1) Art. 22, 2005
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4
ROSIHAN M. A LI , M. H USSAIN K HAN , V. R AVICHANDRAN , AND K.G. S UBRAMANIAN
Theorem 2.4. If f ∈ P Mg (p, m, α), then
p(α − 1)
(n − pα)bn
an ≤
with equality only for functions of the form
fn (z) = z p +
p(α − 1) n
z .
(n − pα)bn
Proof. Let f ∈ P Mg (p, m, α). By making use of the inequality (2.1), we have
∞
X
(n − pα)an bn ≤
(n − pα)an bn ≤ p(α − 1)
n=m
or
p(α − 1)
.
(n − pα)bn
an ≤
Clearly for
fn (z) = z p +
p(α − 1) n
z ∈ P Mg (p, m, α),
(n − pα)bn
we have
p(α − 1)
.
(n − pα)bn
an =
Corollary 2.5. If f ∈ Pλ (α), then
an ≤
α−1
(n − α)Bn (λ)
with equality only for functions of the form
fn (z) = z +
α−1
zn,
(n − α)Bn (λ)
where Bn (λ) is given by (2.2).
Corollary 2.6. If f ∈ P Mm (α), then
an ≤
α−1
(n − α)nm
with equality only for functions of the form
fn (z) = z +
3. G ROWTH
AND
α−1
zn.
(n − α)nm
D ISTORTION T HEOREMS
We now prove the growth theorem for the functions in the class P Mg (p, m, α).
Theorem 3.1. If f ∈ P Mg (p, m, α), then
p(α − 1) m
p(α − 1) m
r ≤ |f (z)| ≤ rp +
r ,
(m − pα)bm
(m − pα)bm
provided bn ≥ bm ≥ 1. The result is sharp for
p(α − 1) m
(3.1)
f (z) = z p +
z .
(m − pα)bm
rp −
J. Inequal. Pure and Appl. Math., 6(1) Art. 22, 2005
|z| = r < 1,
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F UNCTIONS WITH P OSITIVE C OEFFICIENTS
5
Proof. By making use of the inequality (2.1) for f ∈ P Mg (p, m, α) together with
(m − pα)bm ≤ (n − pα)bn ,
we obtain
bm (m − pα)
∞
X
an ≤
n=m
∞
X
(n − pα)an bn ≤ p(α − 1)
n=m
or
∞
X
p(α − 1)
.
(m
−
pα)b
m
n=m
P
n
By using (3.2) for the function f (z) = z p + ∞
n=m an z ∈ P Mg (p, m, α), we have for |z| = r,
(3.2)
an ≤
p
|f (z)| ≤ r +
∞
X
an r n
n=m
≤ rp + rm
∞
X
an
n=m
p(α − 1) m
≤ rp +
r ,
(m − pα)bm
and similarly,
|f (z)| ≥ rp −
p(α − 1) m
r .
(m − pα)bm
Theorem 3.1 also shows that f (∆) for every f ∈ P Mg (p, m, α) contains the disk of radius
p(α−1)
1 − (m−pα)b
.
m
Corollary 3.2. If f ∈ Pλ (α), then
α−1
α−1
r2 ≤ |f (z)| ≤ r +
r2
r−
(2 − α)(λ + 1)
(2 − α)(λ + 1)
(|z| = r).
The result is sharp for
f (z) = z +
(3.3)
α−1
z2.
(2 − α)(λ + 1)
Corollary 3.3. If f ∈ P Mm (α), then
α−1
α−1
2
r−
r
≤
|f
(z)|
≤
r
+
r2
(2 − α)2m
(2 − α)2m
(|z| = r).
The result is sharp for
f (z) = z +
(3.4)
α−1
z2.
(2 − α)2m
The distortion estimates for the functions in the class P Mg (p, m, α) is given in the following:
Theorem 3.4. If f ∈ P Mg (p, m, α), then
prp−1 −
mp(α − 1) m−1
mp(α − 1) m−1
r
≤ |f 0 (z)| ≤ prp−1 +
r
,
(m − pα)bm
(m − pα)bm
|z| = r < 1,
provided bn ≥ bm . The result is sharp for the function given by (3.1).
J. Inequal. Pure and Appl. Math., 6(1) Art. 22, 2005
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6
ROSIHAN M. A LI , M. H USSAIN K HAN , V. R AVICHANDRAN , AND K.G. S UBRAMANIAN
Proof. By making use of the inequality (2.1) for f ∈ P Mg (p, m, α), we obtain
∞
X
an b n ≤
n=m
p(α − 1)
(m − pα)
and therefore, again using the inequality (2.1), we get
∞
X
nan ≤
n=m
For the function f (z) = z p +
P∞
n=m
mp(α − 1)
.
(m − pα)bm
an z n ∈ P Mg (p, m, α), we now have
|f 0 (z)| ≤ prp−1 +
∞
X
nan rn−1
(|z| = r)
n=m
≤ pr
p−1
+r
m−1
∞
X
nan
n=m
mp(α − 1) m−1
r
≤ prp−1 +
(m − pα)bm
and similarly we have
|f 0 (z)| ≥ prp−1 −
mp(α − 1) m−1
r
.
(m − pα)bm
Corollary 3.5. If f ∈ Pλ (α), then
1−
2(α − 1)
2(α − 1)
r ≤ |f 0 (z)| ≤ 1 +
r
(2 − α)(λ + 1)
(2 − α)(λ + 1)
(|z| = r).
The result is sharp for the function given by (3.3)
Corollary 3.6. If f ∈ P Mm (α), then
1−
2(α − 1)
2(α − 1)
r ≤ |f 0 (z)| ≤ 1 +
r
m
(2 − α)2
(2 − α)2m
(|z| = r).
The result is sharp for the function given by (3.4)
4. C LOSURE T HEOREMS
We shall now prove the following closure theorems for the class P Mg (p, m, α). Let the
functions Fk (z) be given by
(4.1)
p
Fk (z) = z +
∞
X
fn,k z n ,
(k = 1, 2, . . . , M ).
n=m
P
Theorem 4.1. Let λk ≥ 0 for k = 1, 2, . . . , M and M
k=1 λk ≤ 1. Let the function Fk (z)
defined by (4.1) be in the class P Mg (p, m, α) for every k = 1, 2, . . . , M . Then the function
f (z) defined by
!
∞
M
X
X
f (z) = z p +
λk fn,k z n
n=m
k=1
belongs to the class P Mg (p, m, α).
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F UNCTIONS WITH P OSITIVE C OEFFICIENTS
7
Proof. Since Fk (z) ∈ P Mg (p, m, α), it follows from Theorem 2.1 that
∞
X
(4.2)
(n − pα)bn fn,k ≤ p(α − 1)
n=m
for every k = 1, 2, . . . , M. Hence
∞
X
(n − pα)bn
n=m
M
X
!
λk fn,k
M
X
=
k=1
λk
≤
!
(n − pα)bn fn,k
n=m
k=1
M
X
∞
X
λk p(α − 1)
k=1
≤ p(α − 1).
By Theorem 2.1, it follows that f (z) ∈ P Mg (p, m, α).
Corollary 4.2. The class P Mg (p, m, α) is closed under convex linear combinations.
Theorem 4.3. Let
p(α − 1) n
z
(n − pα)bn
for n = m, m + 1, . . .. Then f (z) ∈ P Mg (p, m, α) if and only if f (z) can be expressed in the
form
∞
X
p
(4.3)
f (z) = λp z +
λn Fn (z),
Fp (z) = z p and Fn (z) = z p +
n=m
where each λj ≥ 0 and λp +
P∞
n=m
λn = 1.
Proof. Let f (z) be of the form (4.3). Then
∞
X
λn p(α − 1) n
f (z) = z +
z
(n − pα)bn
n=m
p
and therefore
∞
∞
X
X
λn p(α − 1) (n − pα)bn
=
λn = 1 − λp ≤ 1.
(n
−
pα)b
p(α
−
1)
n
n=m
n=m
By Theorem 2.1, we have f (z) ∈ P Mg (p, m, α).
Conversely, let f (z) ∈ P Mg (p, m, α). From Theorem 2.4, we have
an ≤
p(α − 1)
(n − pα)bn
λn =
(n − pα)bn an
p(α − 1)
n = m, m + 1, . . . .
for
Therefore we may take
for
and
λp = 1 −
∞
X
n = m, m + 1, . . .
λn .
n=m
Then
p
f (z) = λp z +
∞
X
λn Fn (z).
n=m
J. Inequal. Pure and Appl. Math., 6(1) Art. 22, 2005
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ROSIHAN M. A LI , M. H USSAIN K HAN , V. R AVICHANDRAN , AND K.G. S UBRAMANIAN
We now prove that the class P Mg (p, m, α) is closed under convolution with certain functions
and give an application of this result to show that the class P Mg (p, m, α) is closed under the
familiar Bernardi integral operator.
P
n
Theorem 4.4. Let h(z) = z p + ∞
n=m hn z be analytic in ∆ with 0 ≤ hn ≤ 1. If f (z) ∈
P Mg (p, m, α), then (f ∗ h)(z) ∈ P Mg (p, m, α).
Proof. The result follows directly from Theorem 2.1.
The generalized Bernardi integral operator is defined by the following integral:
Z
c + p z c−1
(4.4)
F (z) =
t f (t)dt (c > −1; z ∈ ∆).
zc 0
Since
!
∞
X
c+p n
p
F (z) = f (z) ∗ z +
z ,
c+n
n=m
we have the following:
Corollary 4.5. If f (z) ∈ P Mg (p, m, α), then F (z) given by (4.4) is also in P Mg (p, m, α).
5. O RDER AND R ADIUS R ESULTS
Let P Sh∗ (p, m, β) be the subclass of P (m, p) consisting of functions f for which f ∗ h is
starlike of order β.
P
n
Theorem 5.1. Let h(z) = z p + ∞
n=m hn z with hn > 0. Let (α − 1)nhn ≤ (n − pα)bn . If
∗
f ∈ P Mg (p, m, α), then f ∈ P Sh (p, m, β), where
(n − pα)bn − (α − 1)nhn
.
β := inf
n≥m (n − pα)bn − (α − 1)phn
Proof. Let us first note that the condition (α − 1)nhn ≤ (n − pα)bn implies f ∈ P Sh∗ (p, m, 0).
From the definition of β, it follows that
(n − pα)bn − (α − 1)nhn
β≤
(n − pα)bn − (α − 1)phn
or
(n − pβ)hn
(n − pα)bn
≤
1−β
α−1
and therefore, in view of (2.1),
∞
∞
X
X
(n − pβ)
(n − pα)
an hn ≤
an bn ≤ 1.
p(1
−
β)
p(α
−
1)
n=m
n=m
Thus
P∞
1 z(f ∗ h)0 (z)
− 1)an hn
·
≤ n=m (n/p
P∞
−
1
≤1−β
p (f ∗ h)(z)
1 − n=m an hn
and therefore f ∈ P Sh∗ (p, m, β).
Similarly we can prove the following:
Theorem 5.2. If f ∈ P Mg (p, m, α), then f ∈ P Mh (p, m, β) in |z| < r(α, β) where
(
1 )
(n − pα) (β − 1) bn n−p
r(α, β) := min 1; inf
.
n≥m (n − pα) (α − 1) hn
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F UNCTIONS WITH P OSITIVE C OEFFICIENTS
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R EFERENCES
[1] R.M. ALI, M. HUSSAIN KHAN, V. RAVICHANDRAN AND K.G. SUBRAMANIAN, A class of
multivalent functions with negative coefficients defined by convolution, preprint.
[2] O.P. AHUJA, Hadamard products of analytic functions defined by Ruscheweyh derivatives, in Current Topics in Analytic Function Theory, 13–28, World Sci. Publishing, River Edge, NJ.
[3] S. OWA AND H.M. SRIVASTAVA, Some generalized convolution properties associated with certain subclasses of analytic functions, J. Inequal. Pure Appl. Math., 3(3) (2002), Article 42, 13 pp.
[ONLINE: http://jipam.vu.edu.au/article.php?sid=194]
[4] V. RAVICHANDRAN, On starlike functions with negative coefficients, Far East J. Math. Sci., 8(3)
(2003), 359–364.
[5] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975),
109–116.
[6] B.A. URALEGADDI, M.D. GANIGI AND S.M. SARANGI, Univalent functions with positive coefficients, Tamkang J. Math., 25(3) (1994), 225–230.
J. Inequal. Pure and Appl. Math., 6(1) Art. 22, 2005
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