Journal of Inequalities in Pure and
Applied Mathematics
PARTIAL SUMS OF SOME MEROMORPHIC FUNCTIONS
S. LATHA AND L. SHIVARUDRAPPA
Department of Mathematics
Maharaja’s College
University of Mysore
Mysore - 570005, INDIA.
EMail: drlatha@gmail.com
Department of Mathematics, S.J.C.E
Visweshwarayah Technological University
Belgam, INDIA.
EMail: shivarudrappa@lycos.com
volume 7, issue 4, article 140,
2006.
Received 05 July, 2005;
accepted 18 October, 2006.
Communicated by: G. Kohr
Abstract
Contents
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2000
Victoria University
ISSN (electronic): 1443-5756
203-05
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Abstract
In the present paper we give some results concerning partial sums of certain
meromorphic functions.We also consider the partial sums of certain integral
operator.
2000 Mathematics Subject Classification: 30C45.
Key words: Partial sums, Meromorphic functions, Integral operators, Meromorphic
starlike functions, Meromorphic convex functions, Meromorphic close to
convex functions.
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Some Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
7
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1.
Introduction
Let Σ be the class consisting of functions of the form
∞
f (z) =
(1.1)
1 X
+
ak z k
z k=1
which are regular in the punctured disc E = {z : 0 < |z| < 1} with a simple
pole at the origin and residue 1 there.
P
Let fn (z) = z1 + nk=1 ak z k be the nth partial sum of the series expansion
for f (z) ∈ Σ. Let Σ∗ (A, B), ΣK (A, B), Σc (A, B), −1 ≤ A < B ≤ 1 be the
subclasses of functions in Σ satisfying
0 zf (z)
1 + Az
(1.2)
−
≺
,
z ∈ U = E ∪ {0}
f (z)
1 + Bz
(1.3)
−
zf 00 (z)
1 + Az
+1 ≺
,
0
f (z)
1 + Bz
z ∈ U.
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
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(1.4)
−z 2 f 0 (z) ≺
∗
1 + Az
,
1 + Bz
z∈U
respectively [5]. The classes Σ (2α − 1, 1) and ΣK (2α − 1, 1) are respectively
the well known subclasses of Σ consisting of functions meromorphic starlike
of order α and meromorphic convex of order α and meromorphically close to
convex of order α denoted by Σ∗ (α), ΣK (α) and Σc (α) respectively.
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P
P∞
1
k
k
If f (z) = z1 + ∞
k=1 ak z and g(z) = z +
k=1 bk z , then their Hadamard
product (or convolution), denoted by f (z) ∗ g(z) is the function defined by the
power series
∞
1 X
f (z) ∗ g(z) = +
ak b k z k .
z k=1
In the present paper, we give sufficient conditions for f (z) to be in Σ∗ (A, B),
ΣK (A, B) and further investigate the ratio of a function of the form (1.1) to its
sequence of partial sums when the coefficients are sufficiently small to satisfy
conditions
∞
X
k{k(1 + B) + (1 + A)}|ak | ≤ B − A,
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
k=1
∞
X
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{k(1 + B) + (1 + A)}|ak | ≤ B − A.
Contents
k=1
n
o
n
o
(z)
More precisely, we will determine sharp lower bounds for < ffn(z)
, < ffn(z)
,
(z)
n 0 o
n 0 o
(z)
< ff0 (z)
and < ffn0 (z)
. Further, we give a property for the partial sums of
n (z)
certain integral operators in connection with functions belonging to the class
Σc (A, B).
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2.
Some Preliminary Results
Theorem 2.1. Let f (z) =
(2.1)
∞
X
1
z
+
P∞
k=1
ak z k ,
z ∈ E. If
k{k(1 + B) + (1 + A)}|ak | ≤ B − A,
then
f (z) ∈ ΣK (A, B).
k=1
Proof. It suffices to show that
1 + zf 00 (z) + 1 f (z)
< 1,
zf 00 (z)
B 1 + f 0 (z) + A that is,
zf 00 (z) + 2f 0 (z)
Bzf 00 (z) + (A + B)f 0 (z) < 1.
Consider
00
0
zf
(z)
+
2f
(z)
(2.2) Bzf 00 (z) + (A + B)f 0 (z) P∞
k(k + 1)ak z k+1
k=1
P∞
P
=
∞
(B − A) + B k=1 k(k + 1)ak z k+1 − (2B − A) k=1 kak z k+1 Σk(k + 1)|ak |
P
≤
.
(B − A) − ∞
k=1 k(kB + A)|ak |
(2.2) is bounded by 1 if
∞
X
k=1
k(k + 1)|ak | ≤ (B − A)
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
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∞
X
k=1
k(kB + A)|ak |
J. Ineq. Pure and Appl. Math. 7(4) Art. 140, 2006
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which reduces to (2.1)
Similarly we can prove the following theorem.
P
k
Theorem 2.2. Let f (z) = z1 + ∞
z ∈ E. If
k=1 ak z ,
(2.3)
∞
X
{k(1 + B) + (1 + A)}|ak | ≤ B − A,
then f (z) ∈ Σ∗ (A, B).
k=1
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
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3.
Main Results
Theorem 3.1. If f (z) of the form (1.1) satisfies (2.3), then
f (z)
2(n + 1 + A)
≥
, z ∈ U.
<
fn (z)
2n + 2 + A + B
The result is sharp for every n, with extremal function
f (z) =
(3.1)
1
B−A
+
z n+1 ,
z 2n + 2 + A + B
n ≥ 0.
Partial Sums of Some
Meromorphic Functions
Proof. Consider
S. Latha and L. Shivarudrappa
f (z)
2n + A + B
−
fn (z) 2n + 2 + A + B
P∞
P
k+1
1 + nk=1 ak z k+1 + 2n+2+A+B
k=n+1 ak z
B−A
P
=
1 + nk=1 ak z k+1
1 + w(z)
=
,
1 − w(z)
2n + 2 + A + B
B−A
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where
w(z) =
2+
P∞
2n+2+A+B
k+1
k=n+1 ak z
B−A
P
P∞
2 nk=1 ak z k+1 − 2n+2+A+B
k=n+1
B−A
and
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ak z k+1
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P∞
2n+2+A+B
|w(z)| ≤
B−A
2−2
Pn
k=1 |ak | −
k=1 |ak |
P∞
2n+2+A+B
k=n+1
B−A
|ak |
.
J. Ineq. Pure and Appl. Math. 7(4) Art. 140, 2006
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Now
|w(z)| ≤ 1
if and only if
2
2n + 2 + A + B
B−A
X
∞
|ak | ≤ 2 − 2
k=n+1
n
X
|ak |,
k=1
which is equivalent to
n
X
(3.2)
|ak | +
k=1
2n + 2 + A + B
B−A
∞
X
Partial Sums of Some
Meromorphic Functions
|ak | ≤ 1.
It suffices to show that the left hand side of (3.2) bounded above by
∞
X
2k + A + B
k=1
(B − A)
k=1
B−A
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|ak |,
which is equivalent to
n X
2(k + A)
S. Latha and L. Shivarudrappa
k=n+1
∞ X
2(k − n − 1)
|ak | +
|ak | ≥ 0.
B−A
k=n+1
To see that the function f (z) given by (3.1) gives the sharp result, we observe
for
πi
z = re n+2
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that
f (z)
B−A
B−A
2(n + 1 + A)
= 1+
z n+2 → 1−
=
fn (z)
2n + 2 + A + B
2n + 2 + A + B
2(n + 1) + A + B
when r → 1− .
Therefore we complete the proof of Theorem 3.1.
Corollary 3.2. For A = 2α − 1, B = 1, we get Theorem 2.1 in [3] which states
as follows:
If f (z) of the form (1.1) satisfies condition
∞
X
(k + α)|ak | ≤ 1 − α,
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
1
then
f (z)
n + 2α
<
≥
, z ∈ U.
fn (z)
n+1+α
The result is sharp for every n, with extremal function
1
1 − α n+1
f (z) = +
z , n ≥ 0.
z n+1+α
Theorem 3.3. If f (z) of the form (1.1) satisfies (2.1), then
f (z)
(n + 2)(2n + A + B)
<
≥
, z ∈ U.
fn (z)
(n + 1)(2n + 2 + A + B)
f (z) =
1
B−A
+
z n+1 ,
z (n + 1)(2n + 2 + A + B)
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The result is sharp for every n, with extremal function
(3.3)
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n ≥ 0.
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Proof. Consider
(n + 1)(2n + 2 + B + A) f (z)
(n + 2)(2n + A + B)
−
B−A
fn (z) (n + 1)(2n + 2 + A + B)
P
P∞
k+1
1 + nk=1 ak z k+1 + (n+1)(2n+2+A+B)
1 + w(z)
k=n+1 ak z
Pn B−A k+1
=
:=
,
1 + k=1 ak z
1 − w(z)
where
w(z) =
2+
(n+1)(2n+2+A+B) P∞
k+1
k=n+1 ak z
B−A
P
P∞
2 nk=1 ak z k+1 + (n+1)(2n+2+B+A)
k=n+1
B−A
Partial Sums of Some
Meromorphic Functions
ak z k+1
.
Now
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(n+1)(2n+2+A+B) P∞
|w(z)| ≤
2−2
|ak |
P∞
k=n+1
B−A
(n+1)(2n+2+A+B)
k=1 |ak | −
B−A
Pn
k=n+1
|ak |
if
(3.4)
S. Latha and L. Shivarudrappa
∞
(n + 1)(2n + 2 + A + B) X
|ak | +
|ak | ≤ 1.
B−A
k=1
k=n+1
n
X
≤1
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The left hand side of (3.4) is bounded above by
∞
X
k=1
k(2k + A + B)
|ak |
B−A
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if
1
B−A
( n
X
+
(k(2k + A + B) − (B − A)) |ak |
k=1
∞
X
)
(k(2k + A + B) − (n + 1)(2n + 2 + A + B)) |ak |
≥ 0,
k=n+1
and the proof is completed.
Partial Sums of Some
Meromorphic Functions
Corollary 3.4. For A = 2α − 1, B = 1, we get Theorem 2.2 in [3] which reads:
If f (z) of the form (1.1) satisfies condition
∞
X
Title Page
k(k + α)|ak | ≤ 1 − α,
Contents
1
then
<
f (z)
fn (z)
S. Latha and L. Shivarudrappa
(n + 2)(n + α)
≥
,
(n + 1)(n + 1 + α)
z ∈ U.
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The result is sharp for every n, with extremal function
1
1−α
f (z) = +
z n+1 ,
z (n + 1)(n + 1 + α)
o
n
fn (z)
We next determine bounds for < f (z) .
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Theorem 3.5.
(a) If f (z) of the form (1.1) satisfies the condition (2.3), then
fn (z)
2n + 2 + A + B
≥
, z ∈ U.
<
f (z)
n+2
(b) If f (z) of the form (1.1) satisfies condition (2.1), then
fn (z)
2(n + 1)(2n + 2 + A + B)
≥
,
<
f (z)
2(n + 1)(n + 2) − n(B − A)
z ∈ U.
Equalities hold in (a) and (b) for the functions given by (3.1) and (3.3)
respectively.
Proof. We prove (a). The proof of (b) is similar to (a) and will be omitted.
Consider
2(n + 2) fn (z) 2n + 2 + A + B
−
B−A
f (z)
2(n + 2)
Pn
P∞
k+1
1 + k=1 ak z k+1 + 2n+2+A+B
1 + w(z)
k=n+1 ak z
B−A
Pn
:=
,
=
1 + k=1 ak z k+1
1 − w(z)
where
|w(z)| ≤
n+2
B−A
1−
Pn
k=1
|ak | −
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P∞
k=n+1 |ak |
P∞
n+A+B
k=n+1
B−A
Partial Sums of Some
Meromorphic Functions
|ak |
≤ 1.
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This last inequality is equivalent to
∞
2n + 2 + A + B X
|ak | +
|ak | ≤ 1.
B−A
k=1
k=n+1
n
X
(3.5)
Since the left hand side of (3.5) is bounded above by
∞
X
2k + A + B
B−A
k=1
|ak |,
Partial Sums of Some
Meromorphic Functions
the proof is completed.
S. Latha and L. Shivarudrappa
Corollary 3.6. For A = 2α − 1, B = 1, we get Theorem 2.3 in [3] which reads:
Title Page
(a) If f (z) of the form (1.1) satisfies condition
∞
X
Contents
(k + α)|ak | ≤ 1 − α,
1
then
<
fn (z)
f (z)
(n + 1 + α)
≥
,
(n + 2)
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z ∈ U.
(b) If f (z) of the form (1.1) satisfies condition
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∞
X
1
k(k + α)|ak | ≤ 1 − α,
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then
<
fn (z)
f (z)
≥
(n + 1)(n + 1 + α)
,
(n + 1)(n + 2) − n(1 − α)
z ∈ U.
Equalities hold in (a) and (b) for the functions given by
f (z) =
f (z) =
1
1−α
+
z n+1 ,
z (n + 1 + α)
n ≥ 0,
1
1−α
+
z n+1 ,
z (n + 1)(n + 1 + α)
n≥0
respectively.
We turn to ratios involving derivatives. The proof of Theorem 3.7 is similar
to that in Theorem 3.1 and (a) of Theorem 3.5 and so the details may be
omitted.
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
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Theorem 3.7. If f (z) of form (1.1) satisfies condition (2.3) with A = −B,
then
0 f (z)
<
(a)
≥ 0, z ∈ U,
fn0 (z)
0
1
fn (z)
(b)
≥
, z ∈ U.
<
f 0 (z)
2
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In both the cases, the extremal function is given by (3.1) with A = −B.
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Theorem 3.8. If f (z) of form (1.1) satisfies condition (2.1) then,
0 f (z)
2(n + A + B)
(a)
<
≥
, z ∈ U,
fn0 (z)
2n + 2 + A + B
0
fn (z)
2n + 2 + A + B
(b)
<
≥
, z ∈ U.
0
f (z)
2(n + 2)
In both the cases, the extremal function is given by (3.3)
Proof. It is well known that f (z) ∈ ΣK (A, B) if and only if −zf 0 (z) ∈
Σ∗ (A, B). In particular, f (z) satisfies condition (2.1) if and only if −zf 0 (z)
satisfies condition (2.3). Thus (a) is an immediate consequence of Theorem
3.1 and (b) follows directly from (a) of Theorem 3.5.
For a function f (z) ∈ Σ, we define the integral operator F (z) as follows
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
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F (z) =
1
z2
Z
z
tf (t)dt =
0
∞
X
1
1
+
ak z k ,
z k=1 k + 2
z ∈ E.
The nth partial sum Fn (z) of the integral operator F (z) is given by
n
Fn (z) =
1 X 1
+
ak z k ,
z k=1 k + 2
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z ∈ E.
The following lemmas will be required for the proof of Theorem 3.11 below.
P
cos(kθ)
Lemma 3.9. For 0 ≤ θ ≤ π, 12 + m
k=1 k+1 ≥ 0
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Lemma 3.10. Let P be analytic in U with P (0) = 1 and <{P (z)} > 12 in U.
For any function Q analytic in U the function P ∗ Q takes values in the convex
hull of the image on U under Q.
Lemma 3.9 is due to Rogosinski and Szego [4] and Lemma 3.10 is a well
known result ([2] and [6]) that can be derived from the Herglotz representation
for P. Finally we derive
Theorem 3.11. If f (z) ∈ Σc (A, B), then Fn (z) ∈ Σc (A, B).
Partial Sums of Some
Meromorphic Functions
Proof. Let f (z) be the form (1.1) and belong to the class Σc (A, B).
We have,
(
)
∞
X
1
1
(3.6)
< 1−
kak z k+1 > , z ∈ U.
B − A k=1
2
Applying the convolution properties of power series to
(3.7)
−z
2
Fn0 (z)
n
X
=1−
k=1
=
1−
Putting z = reiθ ,
Fn0 (z)
S. Latha and L. Shivarudrappa
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we may write
k
ak z k+1
k+2
1
B−A
∞
X
Close
kak z k+1
0 ≤ r < 1,
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!
k=1
JJ
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∗
1 + (B − A)
∞
X
1 k
z
k+1
k=n+1
!
.
0 ≤ |θ| ≤ π, and making use of the minimum
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principle for harmonic functions along with Lemma 3.9, we obtain
(
)
n+1
n+1 k
X
X
1 k
r cos(kθ)
(3.8)
< 1 + (B − A)
z
= 1 + (B − A)
k+1
k+1
k=1
k=1
> 1 + (B − A)
n+1
X
cos kθ
k+1
B−A
.
2
k=1
≥
1−
In view of (3.6), (3.7), (3.8) and Lemma 3.10 we deduce that
B−A
2 0
−<{z Fn (z)} > 1 −
, 0 ≤ A + B < 2,
2
Partial Sums of Some
Meromorphic Functions
S. Latha and L. Shivarudrappa
z ∈ U,
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which completes the proof of Theorem 3.11
Corollary 3.12. For A = 2α − 1, B = 1, we obtain Theorem 2.8 in [3] which
reads:
If f (z) ∈ Σc (α), then Fn (z) ∈ Σc (α).
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References
[1] S. RUSCHWEYH, Convolutions in Geometric Function Theory, Les
Presses de l’Universite de Montreal, 1982.
[2] A.W. GOODMAN, Univalent Functions, Vol. I, Mariner Publ. Co., Tampa,
Fl., (1983).
[3] NAK EUN CHO AND S. OWA, On partial sums of certain meromorphic
functions, J. Inequal. in Pure and Appl. Math., 5(2) (2004), Art. 30. [ONLINE: http://jipam.vu.edu.au/article.php?sid=377].
[4] W. ROGOSINSKI AND G. SZEGO, Uber die abschimlte von potenzreihen
die in ernein kreise be schranket bleiben, Math. Z., 28 (1928), 73–94.
[5] S. LATHA, Some studies in the theory of univalent and multivalent functions, PhD Thesis, (1994).
[6] R. SINGH AND S. SINGH, Convolution properties of a class of starlike
fuctions, Proc. Amer. Math. Soc., 106 (1989), 145–152.
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Meromorphic Functions
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