12_743_darus.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 1086-1092

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Coefficient Inequalities for concave Cesáro Operator of

Non-concave Analytic Functions

Maslina Darus∗, Rabha W. Ibrahim

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,

Bangi 43600, Selangor Darul Ehsan, Malaysia

Abstract. In this article, we determined the coefficient inequalities for concave Cesáro operator which

applied on non-concave analytic functions f (z) =
∑∞

n=0
an( f )z

n, a0 = 0, a1 = 2 in an open unit disk

U := {z : |z|< 1}. Also we discussed the univalence of this operator by using per-Schwarzian derivative.

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Meromorphic univalent functions, Concave functions, Convex set, Per-

Schwarzian derivative; Cesáro operator

1. Introduction and Preliminaries

The Cesáro operator C acts formally on the power series f (z) =
∑∞

n=0 an( f )z
n as

C f (z) =

∞
∑

n=0

� 1

n+ 1

n
∑

k=0

ak( f )
�

zn.

In the past few years, many authors focused on the boundedness and compactness of extended

Cesáro operator between several spaces of holomorphic functions. The history of the Cesáro

operator goes back to Hardy, who was amongst the first to show that C is bounded on H2.

The boundedness of this operator on various spaces has attracted a lot of attention. In fact

that the Cesáro operator is bounded follows from the work of Siskakis [13]. The boundedness

∗Corresponding author.

Email addresses: maslina�ukm.my (M. Darus), rabhaibrahim�yahoo.
om (R. Ibrahim)

http://www.ejpam.com 1086 c© 2010 EJPAM All rights reserved.



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 3 (2010), 1086-1092 1087

of C on H1 was proved with a different method based on a result of Hardy and Littlewood

[14]. With similar techniques, Miao [11] proved that C is bounded even on Hp, p ∈ (0,1).

The Cesáro operator is unbounded on H∞ (see [6]), so that it is reasonable to work in larger

spaces of analytic functions.

In the theory of univalent functions the most important question is to find the coefficient

estimates for functions

g(z) = z +

∞
∑

n=2

bn(g)z
n (1)

that are analytic and univalent in the unit disk U = {z : |z| < 1}. Let Co(p) be the family of

functions g : U → C where p ∈ (0,1) that satisfy the following assumption

Assumption (A):

(i) g is meromorphic in U and has a simple pole at the point p.

(ii) g(0) = g′(0)− 1= 0.

(iii) g maps U conformally onto a set whose complement with respect to C is convex.

The family Co(p) has been investigated recently in [1-4,7,15]. In [10], Livingston introduced

a necessary and sufficient condition for a function f to be in Co(p)

ℜ{−(1+ p2) + 2pz −
(z − p)(1− pz)g′′(z)

g′(z)
} > 0, ∀z ∈ U .

Later Avkhadiev and Wirths (see [4]) proved that for each g ∈ Co(p) with the expansion in

(1) the inequality

|bn(g)−
1− p2n+2

pn−1(1− p4)
| ≤ |

p2(1− p2n−2)

pn−1(1− p4)
|

is valid. Equality is attained if and only if

g(z) =
z − p

1+p2 (1+ eiθ )z2

(1− z

p
)(1− zp)

. (2)

Recently, Bhowmik and Pommerenke (see [5]) obtained certain coefficient estimates for func-

tions have the Laurent expansion

g(z) =

∞
∑

n=−1

Bn(g)(z− p)n, z ∈ t r iangle

where △ := {z ∈ C : |z − p| < 1− p} and p ∈ (0,1),

|Bn−2 −
(1− p2Bn−1)

p
| ≤

p

(1− p4)(1− p)n−1
[1− (

1− p4

p4
)|B−1 +

p2

1− p4
|2, n≥ 3

and of the form (2).



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 3 (2010), 1086-1092 1088

In our investigation, we shall use the analytic functions f (z) in the open disk U take the form

f (z) =

∞
∑

n=0

an( f )z
n, (z ∈ U)

such that a0 = 0 and a1 = 2. Applied the Cesáro operator C on f we obtain the operator

C f (z) =

∞
∑

n=0

� 1

n+ 1

n
∑

k=0

ak( f )
�

zn, (z ∈ U). (3)

It is clear that C f (z) is normalized as followsC f (0) = 0 andC f (0)′ = 1. Assume thatC f (z)

satisfies the assumption (A). Moreover, it satisfies the expansion

C f (z) =

∞
∑

n=0

An(z − p)n, (z ∈△). (4)

Our aim is to determine some estimates bound of An and an( f ) for n≥ 2.

We need to the following result in the sequel.

Theorem 1 ([14]). For each f ∈ Co(p), there exists a function ω holomorphic in U such that

ω(U)⊂ U and

f (z) =
z − p

1+p2 (1+ω(z))z
2

(1− z

p
)(1− zp)

, (z ∈ U). (5)

Next we discuss some other properties of the operator (3) such as univalence of this oper-

ator by using per-Schwarzian derivative.

Let h be analytic and locally univalent in U . The pre-Schwarzian derivative Th of h is defined

by

Th(z) =
h′′(z)

h′(z)
, (z ∈ U) (6)

with the norm

‖Th‖= supz∈U |Th|(1− |z|
2).

It is known that ‖Th‖<∞ if and only if h is uniformly locally univalent. It is also known that

‖Th‖ ≤ 6 for h ∈ S the class of starlike functions and that ‖Th‖ ≤ 4 for h ∈ K the class of

convex functions (see [9]).

2. Coefficient Estimates

In this section, we introduce some coefficient estimates for operator (3) and have the

expansion (4). Now, we state our first results

Theorem 2. Let p ∈ (0,1) and C f (z) ∈ Co(p) have the expansion (4). Then

|A0| ≤
p

1+ p2
. (7)

The inequality is sharp.



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 3 (2010), 1086-1092 1089

Proof. Let C f (z) ∈ Co(p). Then by Theorem 1, there exists a function ω(z) holomorphic

in U and ω(U)⊂ U satisfying (5). Assume that

ω(z) =

∞
∑

n=0

cn(z − p)n, z ∈ △. (8)

Using these two expansions (4) and (8), the power series formulation of (5) takes the form

∞
∑

n=0

An(z − p)n(1−
z

p
)(1− zp) = z −

p

1+ p2
[1+

∞
∑

n=0

cn(z− p)n]z2. (9)

Comparing the coefficient of z on both sides of (9), yields the assertion (7).

Corollary 1. Let p ∈ (0,1) and C f ∈ Co(p) have the expansion (4). Then

|A1| ≤
p2(3+ p2)

(1− p4)(1+ 2p3)
. (10)

The result is sharp.

Proof. Comparing the coefficient of z2 on both sides of (9), we obtain

A1 =

p2

1−p2 (1+ c0) + pA0

1+ 2p3
. (11)

Thus in virtue of Theorem 2 and let |c0| ≤ 1 we obtain the assertion (10).

In general we have the following result for n≥ 2.

Theorem 3. Let p ∈ (0,1) and C f (z) ∈ Co(p) have the expansion (4). Then

|An| ≤
p

(1− p)n(1+ p2)2
, n≥ 2. (12)

The inequality is sharp.

Proof. Let p ∈ (0,1) and C f (z) ∈ Co(p). Then by compering the coefficient of (z− p)n on

both sides of (9), we obtain

An =
p

1+ p2
cn. (13)

But since

|cn| ≤
1− |c0|

2

(1− p)n(1+ p)

(see [14]) then yields the assertion (12).

Consequently, the next result present sharp coefficient estimates for all n≥ 2 if

C f ∈ Co(p) of the form (3) and has the expansion (4).



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 3 (2010), 1086-1092 1090

Theorem 4. Let p ∈ (0,1) and C f ∈ Co(p) of the form (3) and have the expansion (4). Then

|
n
∑

k=0

an( f )| ≤
p(n+ 1)

(1− p)n(1+ p2)2
, n≥ 2. (14)

The inequality is sharp.

Proof. Equating the right sides of (3) and (4) and applying Theorem 3.

Corollary 2. Let p ∈ (0,1) and C f ∈ Co(p) have the expansion (4). Then

|an( f )| ≤
n
∑

k=2

(k+ 1)p

(1− p)k(1+ p2)2
+ (n− 1), n≥ 2. (15)

The result is sharp.

Proof. By applying Theorem 4.

3. Norm Estimates of the per-Schwarzian Derivative

In this section we determined the norm estimates of the per-Schwarzian derivative for the

operator (3).

Theorem 5. Let p ∈ (0,1) and C f ∈ Co(p) of the form (3). Then for z→ 0 the per-Schwarzian

derivative of C f satisfies the inequality

‖TC f ‖ ≤
(2p+ 1)2

p
. (16)

The result is sharp.

Proof. Let p ∈ (0,1) and C f ∈ Co(p) then in view of Theorem 1, C f takes the form (5).

Differentiating both sides of (5) we obtain

C f ′(z) =
H(z)(1−W ′(z))− (z −W (z))H ′(z)

H2(z)

where H(z) := (1− z

p
)(1− zp) and W (z) :=

p

1+p2 (1+ω(z))z
2, or equivalent to

lnC f ′(z) = ln[H(z)(1−W ′(z))− (z −W (z))H ′(z)]− 2 ln H(z).

Take the derivative for the above equality we receive

C f ′′(z)

C f ′(z)
=

Q′(z)

Q(z)
−

2H ′(z)

H(z)

where Q(z) := [H(z)(1−W ′(z))− (z −W (z))H ′(z)]. Now for z→ 0 we obtain the assertion

(16).



M. Darus, R. Ibrahim / Eur. J. Pure Appl. Math, 3 (2010), 1086-1092 1091

Corollary 3. Let p ∈ (0,1) and C f ∈ Co(p). Then C f is uniformly locally univalent when

z→ 0.

Proof. By applying Theorem 5, we get ‖TC f ‖ <∞ henceC f is uniformly locally univalent.

Consider the class Σ of all analytic functions F satisfy

‖F‖Σ = supz∈U (1− |z|
2)|

F ′(z)

F(z)
| <∞.

Also denoted by Rg(z) := zg′(z). Define the extended Cesáro operator in term of integral

operator as follows (see [15])

Cg[ f ](z) =

∫ 1

0

f (ξz)Rg(ξz)
dξ

ξ
. (17)

Then we have the following result

Theorem 6. Let f and Rg in the class Σ. Then TCg[ f ]
is bounded and uniformly locally univalent.

Proof. Differentiating both sides of (17) we obtain

Cg[ f ]
′(z) = f (z)Rg(z).

Or equivalent to

lnCg[ f ]
′(z) = ln f (z) + lnRg(z).

Take the derivative for both sides of the above equality

Cg[ f ]
′′(z)

Cg[ f ]
′(z)

=
f ′(z)

f (z)
+

Rg′(z)

Rg(z)
.

Hence we obtain

|TCg[ f ]
| ≤ |

f ′(z)

f (z)
|+ |

Rg′(z)

Rg(z)
|

≤ supz∈U (1− |z|
2)|

f ′(z)

f (z)
|+ supz∈U (1− |z|

2)|
Rg′(z)

Rg(z)
|

= ‖ f ‖Σ + ‖Rg‖Σ <∞

yields that TCg[ f ]
is bounded and uniformly locally univalent.

ACKNOWLEDGEMENTS The work presented here was supported by UKM-ST-06-FRGS0107-

2009.



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