ON A CERTAIN SUBCLASS OF STARLIKE
FUNCTIONS WITH NEGATIVE COEFFICIENTS
Subclass of Starlike
Functions
A. Y. LASHIN
Department of Mathematics
Faculty of Science
Mansoura University
Mansoura, 35516, EGYPT.
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
Title Page
EMail: aylashin@yahoo.com
Received:
11 December, 2007
Accepted:
29 May, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
30C45.
Key words:
Univalent functions, Starlike functions, Integral means, Neighborhoods, Partial
sums.
Abstract:
We introduce the class H(α, β) of analytic functions with negative coefficients.
In this paper we give some properties of functions in the class H(α, β) and we
obtain coefficient estimates, neighborhood and integral means inequalities for the
function f (z) belonging to the class H(α, β). We also establish some results
concerning the partial sums for the function f (z) belonging to the class H(α, β).
Acknowledgements:
The author would like to thank the referee of the paper for his helpful suggestions.
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Contents
1
Introduction
3
2
Coefficient Estimates
5
3
Some Properties of the Class H(α, β)
7
4
Neighborhood Results
8
5
Integral Means Inequalities
10
6
Partial Sums
12
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
Title Page
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1.
Introduction
Let A denote the class of functions of the form
∞
X
(1.1)
f (z) = z +
ak z k ,
k=2
which are analytic in the unit disc U = {z : |z| < 1}. And let S denote the subclass
of A consisting of univalent functions f (z) in U.
A function f (z) in S is said to be starlike of order α if and only if
0 zf (z)
Re
>α
(z ∈ U ),
f (z)
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
Title Page
for some α (0 ≤ α < 1). We denote by S ∗ (α) the class of all functions in S which
are starlike of order α. It is well-known that
S ∗ (α) ⊆ S ∗ (0) ≡ S ∗ .
Further, a function f (z) in S is said to be convex of order α in U if and only if
00
zf (z)
Re 1 + 0
>α
(z ∈ U ),
f (z)
for some α (0 ≤ α < 1). We denote by K(α) the class of all functions in S which
are convex of order α.
The classes S ∗ (α), and K(α) were first introduced by Robertson [8], and later
were studied by Schild [10], MacGregor [4], and Pinchuk [7].
Let T denote the subclass of S whose elements can be expressed in the form:
(1.2)
f (z) = z −
∞
X
k=2
ak z k
(ak ≥ 0).
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We denote by T ∗ (α) and C(α), respectively, the classes obtained by taking the intersections of S ∗ (α) and K(α) with T,
T ∗ (α) = S ∗ (α) ∩ T
and C(α) = K(α) ∩ T.
The classes T ∗ (α) and C(α) were introduced by Silverman [11].
Let H(α, β) denote the class of functions f (z) ∈ A which satisfy the condition
2 00
αz f (z) zf 0 (z)
Re
+
>β
f (z)
f (z)
for some α ≥ 0, 0 ≤ β < 1, f (z)
6= 0 and z ∈ U.
z
The classes H(α, β) and H(α, 0) were introduced and studied by Obraddovic and
Joshi [5], Padmanabhan [6], Li and Owa [2], Xu and Yang [14], Singh and Gupta
[13], and others.
Further, we denote by H(α, β) the class obtained by taking intersections of the
class H(α, β) with T , that is
H(α, β) = H(α, β) ∩ T.
We note that
H(0, β) = T ∗ (β)
(Silverman [11]).
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
Title Page
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2.
Coefficient Estimates
Theorem 2.1. A function f (z) ∈ T is in the class H(α, β) if and only if
(2.1)
∞
X
[(k − 1)(αk + 1) + (1 − β)]ak ≤ 1 − β.
k=2
Subclass of Starlike
Functions
The result is sharp.
Proof. Assume that the inequality (2.1) holds and let |z| < 1. Then we have
2 00
P∞
αz f (z) zf 0 (z)
− k=2 (k − 1)(αk + 1)ak z k−1 P
f (z) + f (z) − 1 = k−1
1− ∞
a
z
k
k=2
P∞
(k − 1)(αk + 1)ak
P
≤ 1 − β.
≤ k=2
1− ∞
k=2 ak
00
αz 2 f (z)
f (z)
zf 0 (z)
f (z)
This shows that the values of
+
lie in the circle centered at w = 1
whose radius is 1 − β. Hence f (z) is in the class H(α, β).
To prove the converse, assume that f (z) defined by (1.2) is in the class H(α, β).
Then
2 00
αz f (z) zf 0 (z)
(2.2)
Re
+
f (z)
f (z)
P
1− ∞
− 1) + k)]ak z k−1
k=2 [αk(k
P
= Re
>β
k−1
1− ∞
k=2 ak z
00
for z ∈ U. Choose values of z on the real axis so that
αz 2 f (z)
f (z)
+
zf 0 (z)
f (z)
is real. Upon
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
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clearing the denominator in (2.2) and letting z → 1− through real values, we have
!
∞
∞
X
X
[αk(k − 1) + k]ak ,
β 1−
ak ≤ 1 −
k=2
k=2
which obviously is the required result (2.1).
Finally, we note that the assertion (2.1) of Theorem 2.1 is sharp, with the extremal
function being
Subclass of Starlike
Functions
A. Y. Lashin
(2.3)
1−β
f (z) = z −
zk
[(k − 1)(αk + 1) + (1 − β)]
(k ≥ 2).
vol. 10, iss. 2, art. 40, 2009
Title Page
Corollary 2.2. Let f (z) ∈ T be in the class H(α, β). Then we have
(2.4)
ak ≤
1−β
[(k − 1)(αk + 1) + (1 − β)]
(k ≥ 2).
Equality in (2.4) holds true for the function f (z) given by (2.3).
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3.
Some Properties of the Class H(α, β)
Theorem 3.1. Let 0 ≤ α1 < α2 and 0 ≤ β < 1. Then H(α2 , β) ⊂ H(α1 , β).
Proof. It follows from Theorem 2.1. That
∞
X
[(k − 1)(α1 k + 1) + (1 − β)]ak
Subclass of Starlike
Functions
k=2
<
∞
X
A. Y. Lashin
[(k − 1)(α2 k + 1) + (1 − β)]ak ≤ 1 − β
vol. 10, iss. 2, art. 40, 2009
k=2
for f (z) ∈ H(α2 , β). Hence f (z) ∈ H(α1 , β).
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Corollary 3.2. H(α, β) ⊆ T ∗ (β).
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The proof is now immediate because α ≥ 0.
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4.
Neighborhood Results
Following the earlier investigations of Goodman [1] and Ruscheweyh [9], we define
the δ− neighborhood of function f (z) ∈ T by:
)
(
∞
∞
X
X
k |ak − bk | ≤ δ .
bk z k ,
Nδ (f ) = g ∈ T : g(z) = z −
k=2
k=2
Subclass of Starlike
Functions
In particular, for the identity function
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
e(z) = z,
we immediately have
Title Page
(
(4.1)
Nδ (e) =
g ∈ T : g(z) = z −
∞
X
bk z k ,
k=2
Theorem 4.1. H(α, β) ⊆ Nδ (e), where δ =
∞
X
)
k |bk | ≤ δ
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.
k=2
2(1−β)
.
(2α+2−β)
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Proof. Let f (z) ∈ H(α, β). Then, in view of Theorem 2.1, since [(k − 1)(αk + 1) +
(1 − β)] is an increasing function of k (k ≥ 2), we have
(2α + 2 − β)
∞
X
ak ≤
k=2
∞
X
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[(k − 1)(αk + 1) + (1 − β)]ak ≤ 1 − β,
k=2
which immediately yields
(4.2)
∞
X
k=2
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ak ≤
1−β
.
(2α + 2 − β)
Close
On the other hand, we also find from (2.1)
(α + 1)
∞
X
k=2
kak − β
∞
X
ak ≤
k=2
=
(4.3)
∞
X
k=2
∞
X
[(α(k − 1) + 1)k − β)]ak
[(k − 1)(αk + 1) + (1 − β)]ak ≤ 1 − β.
Subclass of Starlike
Functions
k=2
From (4.3) and (4.2), we have
(α + 1)
A. Y. Lashin
∞
X
kak ≤ (1 − β) + β
k=2
∞
X
vol. 10, iss. 2, art. 40, 2009
ak
k=2
1−β
≤ (1 − β) + β
(2α + 2 − β)
2(α + 1)(1 − β)
≤
,
(2α + 2 − β)
that is,
(4.4)
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∞
X
k=2
kak ≤
2 (1 − β)
= δ,
(2α + 2 − β)
which in view of the definition (4.1), prove Theorem 4.1.
Letting α = 0, in the above theorem, we have:
Corollary 4.2. T ∗ (β) ⊆ Nδ (e), where δ =
2(1−β)
.
(2−β)
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5.
Integral Means Inequalities
We need the following lemma.
Lemma 5.1 ([3]). If f and g are analytic in U with f ≺ g, then
Z 2π
Z 2π
g(reiθ )δ dθ ≤
f (reiθ )δ dθ,
0
0
Subclass of Starlike
Functions
where δ > 0, z = reiθ and 0 < r < 1.
A. Y. Lashin
Applying Lemma 5.1, and (2.1), we prove the following theorem.
Theorem 5.2. Let δ > 0. If f (z) ∈ H(α, β), then for z = reiθ , 0 < r < 1, we have
Z 2π
Z 2π
f (reiθ )δ dθ ≤
f2 (reiθ )δ dθ,
0
0
where
(1 − β)
z2.
f2 (z) = z −
(2α + 2 − β)
(5.1)
Proof. Let f (z) defined by (1.2) and f2 (z) be given by (5.1). We must show that
δ
δ
Z 2π Z 2π ∞
X
(1
−
β)
k−1 1 −
dθ.
ak z dθ ≤
z
1 −
(2α
+
2
−
β)
0
0
k=2
By Lemma 5.1, it suffices to show that
1−
∞
X
k=2
ak z k−1 ≺ 1 −
(1 − β)
z.
(2α + 2 − β)
vol. 10, iss. 2, art. 40, 2009
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Setting
(5.2)
1−
∞
X
ak z k−1 = 1 −
k=2
(1 − β)
w(z).
(2α + 2 − β)
From (5.2) and (2.1), we obtain
∞
X
(2α + 2 − β)
|w(z)| = ak z k−1 (1 − β)
k=2
≤ |z|
Subclass of Starlike
Functions
A. Y. Lashin
∞
X
[(k − 1)(αk + 1) + (1 − β)]
1−β
k=2
vol. 10, iss. 2, art. 40, 2009
ak ≤ |z| .
Title Page
This completes the proof of the theorem.
Contents
Letting α = 0 in the above theorem, we have:
∗
iθ
Corollary 5.3. Let δ > 0. If f (z) ∈ T (β), then for z = re , 0 < r < 1, we have
Z 2π
Z 2π
δ
iθ
f (re ) dθ ≤
f2 (reiθ )δ dθ,
0
0
where
f2 (z) = z −
(1 − β) 2
z .
(2 − β)
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6.
Partial Sums
In this section we will examine the ratio of a function of the
(1.2) to its sequence
Pform
n
of partial sums defined by f1 (z) = z and fn (z) = z − k=2 ak z k when the coefficients of f are sufficiently
the
determine
small
to satisfy
condition
0 (2.1). Wewill
f (z)
fn (z)
f (z)
fn0 (z)
sharp lower bounds for Re fn (z) , Re f (z) , Re f 0 (z) and Re f 0 (z) .
n
In what follows, we will use the well known result that
Re
1 − w(z)
> 0,
1 + w(z)
if and only if
w(z) =
∞
X
Subclass of Starlike
Functions
A. Y. Lashin
z ∈ U,
vol. 10, iss. 2, art. 40, 2009
Title Page
ck z
k
Contents
k=1
satisfies the inequality |w(z)| ≤ |z| .
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Theorem 6.1. If f (z) ∈ H(α, β), then
J
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(6.1)
f (z)
1
Re
≥1−
fn (z)
cn+1
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(z ∈ U, n ∈ N )
and
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(6.2)
where ck =:
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Re
fn (z)
f (z)
≥
[(k−1)(αk+1)+(1−β)]
1−β
cn+1
1 + cn+1
(z ∈ U, n ∈ N ) ,
. The estimates in (6.1) and (6.2) are sharp.
Close
Proof. We employ the same
P technique used by Silverman [12]. The function f (z) ∈
H(α, β), if and only if ∞
k=2 ck ak ≤ 1. It is easy to verify that ck+1 > ck > 1. Thus,
(6.3)
n
X
ak + cn+1
∞
X
ak ≤
k=n+1
k=2
∞
X
ck ak ≤ 1.
k=2
We may write
P
P
k−1
1 − nk=2 ak z k−1 − cn+1 ∞
f (z)
1
k=n+1 ak z
P
cn+1
− 1−
=
fn (z)
cn+1
1 − nk=2 ak z k−1
1 + D(z)
=
.
1 + E(z)
Then
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
Title Page
Set
so that
Subclass of Starlike
Functions
1 + D(z)
1 − w(z)
=
,
1 + E(z)
1 + w(z)
E(z) − D(z)
w(z) =
.
2 + D(z) + E(z)
P
k−1
cn+1 ∞
k=n+1 ak z
Pn
P
w(z) =
k−1
2 − 2 k=2 ak z k−1 − cn+1 ∞
k=n+1 ak z
and
P∞
|w(z)| ≤
cn+1 k=n+1 ak
Pn
P
.
2 − 2 k=2 ak − cn+1 ∞
k=n+1 ak
Now |w(z)| ≤ 1 if and only if
n
X
k=2
ak + cn+1
∞
X
k=n+1
ak ≤ 1,
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which is true by (6.3). This readily yields the assertion (6.1) of Theorem 6.1.
To see that
f (z) = z −
(6.4)
z n+1
cn+1
gives sharp results, we observe that
zn
f (z)
=1−
.
fn (z)
cn+1
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
Letting z → 1− , we have
f (z)
1
=1−
,
fn (z)
cn+1
which shows that the bounds in (6.1) are the best possible for each n ∈ N. Similarly,
we take
P
P
k−1
1 − nk=2 ak z k−1 + cn+1 ∞
fn (z)
cn+1
k=n+1 ak z
P
(1 + cn+1 )
−
=
k−1
f (z)
1 + cn+1
1− ∞
k=2 ak z
1 − w(z)
:=
,
1 + w(z)
where
P
(1 + cn+1 ) ∞
k=n+1 ak
Pn
P
|w(z)| ≤
.
2 − 2 k=2 ak + (1 − cn+1 ) ∞
k=n+1 ak
Now |w(z)| ≤ 1 if and only if
n
X
k=2
ak + cn+1
∞
X
k=n+1
ak ≤ 1,
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which is true by (6.3). This immediately leads to the assertion (6.2) of Theorem 6.1.
The estimate in (6.2) is sharp with the extremal function f (z) given by (6.4). This
completes the proof of Theorem 6.1.
Letting α = 0 in the above theorem, we have:
Corollary 6.2. If f (z) ∈ T ∗ (β), then
n
f (z)
Re
≥
,
fn (z)
(n + 1 − β)
(z ∈ U )
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
and
n+1−β
fn (z)
≥
,
(z ∈ U ) .
f (z)
(n + 2 − 2β)
The result is sharp for every n, with the extremal function
Re
f (z) = z −
(6.5)
1−β
z n+1 .
(n + 1 − β)
We now turn to the ratios involving derivatives. The proof of Theorem 6.3 below
follows the pattern of that in Theorem 6.1, and so the details may be omitted.
Theorem 6.3. If f (z) ∈ H(α, β), then
Re
(6.6)
f 0 (z)
n+1
≥1−
0
fn (z)
cn+1
(6.7)
Re
fn0 (z)
f 0 (z)
≥
cn+1
n + 1 + cn+1
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(z ∈ U ) ,
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and
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(z ∈ U, n ∈ N ) .
The estimates in (6.6) and (6.7) are sharp with the extremal function given by (6.4).
Letting α = 0 in the above theorem, we have:
Corollary 6.4. If f (z) ∈ T ∗ (β), then
Re
and
f 0 (z)
βn
≥
,
0
fn (z)
(n + 1 − β)
f 0 (z)
n+1−β
Re n0
≥
,
f (z)
n + (1 − β)(n + 2)
(z ∈ U ) ,
(z ∈ U ) .
The result is sharp for every n, with the extremal function given by (6.5).
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
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References
[1] A.W. GOODMAN, Univalent functions and analytic curves, Proc. Amer. Math.
Soc., 8(3) (1957), 598–601.
[2] J.L. LI AND S. OWA, Sufficient conditions for starlikeness, Indian J. Pure Appl.
Math., 33 (2002), 313-318.
[3] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London
Math. Soc., 23 (1925), 481–519.
[4] T.H. MacGREGOR, The radius for starlike functions of order 12 , Proc. Amer.
Math. Soc., 14 (1963), 71–76.
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
Title Page
[5] M. OBRADOVIC AND S.B. JOSHI, On certain classes of strongly starlike
functions, Taiwanese J. Math., 2(3) (1998), 297–302.
[6] K.S. PADMANABHAN, On sufficient conditions for starlikeness, Indian J.
Pure Appl. Math., 32(4) (2001), 543–550.
[7] B. PINCHUK, On starlike and convex functions of order a, Duke Math. J., 35
(1968), 89–103.
[8] M.S. ROBERTSON, On the theory of univalent functions, Ann. of Math., 37
(1936), 374–408.
[9] S. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer. Math.
Soc., 81(4) (1981), 521–527.
[10] A. SCHILD, On starlike functions of order a, Amer. J. Math., 87 (1965), 65–70.
[11] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., 51 (1975), 109–116.
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[12] H. SILVERMAN, Partial sums of starlike and convex functions, J. Math. Anal.
Appl., 209 (1997), 221–227.
[13] S. SINGH AND S. GUPTA, First order differential subordinations and starlikeness of analytic maps in the unit disc, Kyungpook Math. J., 45 (2005), 395–404.
[14] N. XU AND D. YANG, Some criteria for starlikeness and strongly starlikeness,
Bull. Korean Math. Soc., 42(3) (2005), 579–590.
Subclass of Starlike
Functions
A. Y. Lashin
vol. 10, iss. 2, art. 40, 2009
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