INEQUALITIES IN THE COMPLEX PLANE
RÓBERT SZÁSZ
Romania
EMail: szasz_robert2001@yahoo.com
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
Received:
28 July, 2006
Title Page
Accepted:
04 January, 2007
Contents
Communicated by:
H.M. Srivastava
2000 AMS Sub. Class.:
30C99.
Key words:
Differential subordination, Extreme point, Locally convex linear topological
space, Convex functional.
Abstract:
A differential inequality is generalised and improved. Several other differential
inequalities are considered.
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Contents
1
Introduction
3
2
Preliminaries
4
3
The Main Result
6
4
Particular Cases
9
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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1.
Introduction
Let H(U ) be the set of holomorfic functions defined on the unit disc U = {z ∈ C :
|z| < 1}.
In [2, pp. 38 Example 2.4. d] and [3, pp. 192 Example 9.3.4] the authors have
proved, as an application of the developed theory, the implication:
If f ∈ H(U ), f (0) = 1 and
Re(f (z) + zf 0 (z) + z 2 f 00 (z)) > 0, z ∈ U then Re f (z) > 0, z ∈ U.
The aim of this paper is to generalise this inequality and to determine the biggest
α ∈ R for which the implication
f (0) = 1, Re(f (z)+zf 0 (z)+z 2 f 00 (z)) > 0, (∀) z ∈ U ⇒ Re f (z) > α, (∀) z ∈ U
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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holds true.
In this paper each many-valued function is taken with the principal value.
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2.
Preliminaries
In our study we need the following definitions and lemmas:
Let X be a locally convex linear topological space. For a subset U ⊂ X the closed
convex hull of U is defined as the intersection of all closed convex sets containing U
and will be denoted by co(U ). If U ⊂ V ⊂ X then U is called an extremal subset of
V provided that whenever u = tx + (1 − t)y where u ∈ U, x, y ∈ V and t ∈ (0, 1)
then x, y ∈ U.
An extremal subset of U consisting of just one point is called an extreme point of
U.
The set of the extreme points of U will be denoted by EU.
Lemma 2.1 ([1, pp. 45]). If J : H(U ) → R is a real-valued, continuous convex
functional and F is a compact subset of H(U ), then
max{J(f ) : f ∈ co(F)} = max{J(f ) : f ∈ F} = max{J(f ) : f ∈ E(co(F))}.
In the particular case if J is a linear map then we also have:
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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min{J(f ) : f ∈ co(F)} = min{J(f ) : f ∈ F} = min{J(f ) : f ∈ E(co(F))}.
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Suppose that f, g ∈ H(U ). The function f is subordinate to g if there exists a
function θ ∈ H(U ) such that θ(0) = 0, |θ(z)| < 1, z ∈ U and f (z) = g(θ(z)),
z ∈ U.
The subordination will be denoted by f ≺ g.
Observation 1. Suppose that f, g ∈ H(U ) and g is univalent. If f (0) = g(0) and
f (U ) ⊂ g(U ) then f ≺ g.
When F ∈ H(U ) we use the notation
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s(F ) = {f ∈ H(U ) : f ≺ F }.
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Lemma 2.2 ([1, pp. 51]). Suppose that Fα is defined by the equality
α
1 + cz
, |c| ≤ 1, c 6= −1.
Fα (z) =
1−z
If α ≥ 1 then co(s(Fα )) consists of all functions in H(U ) represented by
α
Z 2π 1 + cze−it
f (z) =
dµ(t)
1 − ze−it
0
where µ is a positive measure on [0, 2π] having the property µ([0, 2π]) = 1 and
1 + cze−it E(co(s(Fα ))) =
t ∈ [0, 2π] .
1 − ze−it Observation 2. If L : H(U ) → H(U ) is an invertible linear map and F ⊂ H(U ) is
a compact subset, then L(co(F)) = co(L(F)) and the set E(co(F)) is in one-to-one
correspondence with EL(co(F)).
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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3.
The Main Result
Theorem 3.1. If f ∈ H(U ), f (0) = 1; m, p ∈ N∗ ; ak ∈ R, k = 1, p and
q
(3.1)
Re m f (z) + a1 zf 0 (z) + · · · + ap z p f (p) (z) > 0, z ∈ U
then
∞
X
Inequalities in the
Complex Plane
!
Pm
m−1
k
Cm
Cm+n−k−1
(3.2) 1 + inf Re
zn
z∈U
P (n)
n=1
!
∞ Pm
m−1
k
X
C
C
k=0 m m+n−k−1 n
< Re f (z) < 1 + sup Re
z ,
P (n)
z∈U
n=1
k=0
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
z∈U
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where P (x) = 1 + a1 x + a2 x(x − 1) + · · · + ap x(x − 1) · · · (x − p + 1).
Proof. The condition of the theorem can be rewritten in the form
q
1+z
m
f (z) + a1 zf 0 (z) + · · · + ap z p f (p) (z) ≺
,
1−z
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f (z) + a1 zf 0 (z) + · · · + ap z p f (p) (z) ≺
1+z
1−z
m
.
p (p)
f (z) + a1 zf (z) + · · · + ap z f
Z
(z) =
0
2π
1 + ze−it
1 − ze−it
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According to the results of Lemma 2.2,
where µ([0, 2π]) = 1.
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which is equivalent to
0
JJ
m
dµ(t) = h(z),
If
f (z) = 1 +
∞
X
bn z n ,
z∈U
n=1
then
0
p (p)
f (z) + a1 zf (z) + · · · + ap z f
(z) = 1 +
∞
X
bn P (n)z n .
n=1
On the other hand
! Z
m
Z 2π ∞
n
2π
X
X
1 + ze−it
m−1
n
k
C
C
z
e−int dµ(t),
dµ(t)
=
1
+
m m+n−k−1
−it
1
−
ze
0
0
n=1
k=0
with Cpq = 0 if q > p. The equalities Cpq = 0 if q > p imply also that:
n
X
m−1
k
Cm
Cm+n−k−1
=
m
X
The above two developments in power series imply that:
! Z
∞
∞
m
X
X
X
m−1
k
zn
1+
bn P (n)z n = 1 +
Cm
Cm+n−k−1
n=0
n=1
and
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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m−1
k
Cm+n−k−1
.
Cm
k=0
k=0
Inequalities in the
Complex Plane
2π
e−int dµ(t)
0
k=0
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bn =
1
P (n)
m
X
m−1
k
Cm
Cm+n−k−1
!Z
2π
e−int dµ(t).
0
k=0
Consequently,
f (z) = 1 +
∞
X
n=1
1
P (n)
m
X
k=0
!
m−1
k
Cm
Cm+n−k−1
z
n
Z
0
2π
e−int dµ(t).
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If
m
Z 2π 1 + ze−it
dµ(t), z ∈ U, µ([0, 2π]) = 1 ,
B = h ∈ H(U ) h(z) =
1 − ze−it
0
q
m
0
p
(p)
f (z) + a1 zf (z) + · · · + ap z f (z) > 0, z ∈ U
C = f ∈ H(U ) Re
then the correspondence L : B → C, L(h) = f defines an invertible linear map and
according to Observation 2 the extreme points of the class C are
!
m
∞
X
X
1
m−1
k
z n e−int , z ∈ U, t ∈ [0, 2π).
ft (z) = 1 +
Cm
Cm+n−k−1
P
(n)
n=1
k=0
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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This result and Lemma 2.1 implies the assertion of Theorem 3.1.
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4.
Particular Cases
If we put p = 2, a1 = a2 = m = 1 in Theorem 3.1 then we get:
Corollary 4.1. If f ∈ H(U ), f (0) = 1 and
(4.1)
Re(f (z) + zf 0 (z) + z 2 f 00 (z)) > 0,
z ∈ U,
then
π(e2π + 1)
2πeπ
>
Re
f
(z)
>
,
e2π − 1
e2π − 1
and these results are sharp in the sense that
(4.2)
sup Re f (z) =
z∈U
f ∈C
π(e2π + 1)
e2π − 1
Inequalities in the
Complex Plane
z∈U
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
and
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π
inf Re f (z) =
z∈U
f ∈C
2πe
.
−1
e2π
Proof. Theorem 3.1 implies the following inequalities:
!
!
∞
∞
X
X
2
2
z n < Re f (z) < 1 + sup Re
zn .
1 + inf Re
2
2
z∈U
n +1
n +1
z∈U
n=1
n=1
The minimum and maximum principle for harmonic functions imply that
!
!
∞
∞
X
X
2
2
n
int
sup Re
z
= sup Re
e
n2 + 1
n2 + 1
z∈U
t∈[0,2π]
n=1
n=1
!
!
∞
∞
X
X
2
2
n
int
z
= inf Re
e
.
inf Re
z∈U
t∈[0,2π]
n2 + 1
n2 + 1
n=1
n=1
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By considering the integral
Z
In =
|z|=n+ 12
eizt
dz,
(z 2 + 1)(e2πiz − 1)
t ∈ [0, 2π),
using the equality limn→∞ In = 0 and residue theory we deduce that
!
∞
X
2
π(et + e2π−t )
ikt
e
=
, t ∈ [0, 2π)
1 + Re
k2 + 1
e2π − 1
k=1
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
and so we get
2π
π
π(e + 1)
2πe
> Re(f (z)) > 2π
,
2π
e −1
e −1
z ∈ U.
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If we put m = 2, a1 = 0, a2 = 4, Theorem 3.1 implies
Corollary 4.2. If f ∈ H(U ), f (0) = 1 and
p
(4.3)
Re f (z) + 4z 2 f 00 (z) > 0,
z ∈ U,
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then
(4.4)
∞
X
(−1)k · k
Re f (z) > 1 + 4
,
2
(2k
−
1)
k=1
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z∈U
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and this result is sharp.
Proof. Theorem 3.1 and the minimum principle imply that
∞
X
eikt · k
Re f (z) > 1 + 4 inf Re
t∈[0,2π]
(2k − 1)2
k=1
!
.
It is easy to observe that
∞
∞
∞
X
X
X
keikt
eikt
eikt
2
=
+
(2k − 1)2
2k − 1 n=1 (2k − 1)2
n=1
k=1
Z 1X
Z 1Z
∞
2 k−1 ikt
=
(x ) e dx +
0
Z
=
(4.5)
0
Since
Re
0
k=1
1
it
e
dx +
1 − x2 eit
Z
eit
−1
≥
,
2
it
1−x e
1 + x2
0
1Z
0
1
0
∞
1X
(x2 y 2 )k−1 eikt dxdy
k=1
it
e
dxdy,
1 − x2 y 2 eit
Inequalities in the
Complex Plane
t ∈ [0, 2π).
x ∈ [0, 1], t ∈ [0, 2π)
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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and
eit
−1
≥
, x, y ∈ [0, 1], t ∈ [0, 2π),
2
2
it
1−x y e
1 + x2 · y 2
by integrating we get that
Z 1
Z 1
1
eit
dx ≥ −
dx
Re
2
it
2
0 1+x
0 1−x e
and
Z 1
Z 1
1
eit
Re
dxdy ≥ −
dxdy.
2
2
it
2 2
0 1−x y e
0 1+x y
In the derived inequalities, equality occurs if t = π, this means that
∞
∞
X
X
(−1)k · k
keikt
inf Re
=
t∈[0,2π)
(2k − 1)2
(2k − 1)2
k=1
k=1
Re
and the inequality (4.4) holds true.
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References
[1] D.J. HALLENBECK AND T.H. MAC GREGOR, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman Advanced Publishing
Program, 1984.
[2] S.S. MILLER AND P.T. MOCANU, Differential Subordinations, Marcel Decker
Inc. New York. Basel(2000).
[3] P.T. MOCANU, TEODOR BULBOACĂ AND G. Şt. SĂLĂGEAN, Toeria
Geoemtrică a Funcţiilor Univalente, Casa Cărţii de Ştiinţă1, Cluj-Napoca, 1991.
Inequalities in the
Complex Plane
Róbert Szász
vol. 8, iss. 1, art. 27, 2007
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