General Mathematics Vol. 15, Nr. 2–3(2007), 108-117
Certain inequalities concerning some
complex and positive functionals
Emil C. Popa
Abstract
In this paper we study an inequality for the complex and positive
functionals. Some applications for the Carlson0 s inequality and for
complex matrices on given.
2000 Mathematical Subject Classification: 26D07, 26D15
Key words and phrases: complex functionals, positive functionals, Carlson0 s
inequality, complex matrices.
1
Introduction
Let X be a complex algebra and F : X × X → C a complex functional with
the following properties
i) F (αx1 + βx2 , y) = αF (x1 , y) + βF (x2 , y)for all x1 , x2 , y ∈ X and
α, β ∈ C
108
Certain inequalities concerning some complex ...
109
ii) F (x, y) = F (y, x) for all x, y ∈ X
iii) F (x, x) ≥ 0 for all x ∈ X.
Appling the Cauchy - Schwartz - Buniakowski inequality we have
|F (yx, z)|2 ≤ F (yx, yx) · F (z, z)
(1)
for all x, y, z ∈ X.
1
Let y0 = λw1 + w2 be an element of X where w1 , w2 ∈ X, λ ∈ C,
λ
Re(λ) 6= 0, Im(λ) 6= 0, |y0 | 6= 0.
We have
µ
(2)
¶
1
1
F (y0 x, y0 x)= F λw1 x + w2 x, λw1 x + w2 x =
λ
λ
µ
¶
λ
1
2
= |λ| F (w1 x, w1 x) + 2 F (w2 x, w2 x) + 2Re
F (w1 x, w2 x) .
|λ|
λ
We can formulate the next lemma
Lemma 1. If |F (w1 x, w1 x)|6=0, |F (w2 x, w2 x)| =
6 0 and Re (F (w1 x, w2 x))6=0,
there exist a complex number λ = p + qi, such that
s
F (w2 x, w2 x)
(3)
|λ|2 =
F (w1 x, w1 x)
µ
(4)
(5)
Re
¶
λ
F (w1 x, w2 x) = 0
λ
p2 ≥ q 2
110
Emil C. Popa
Proof. We denote F (w1 x, w2 x) = a + bi, a 6= 0 and from (4) we obtain
µ ¶2
p
p
a
− 2b − a = 0
q
q
with b2 + a2 > 0 and x1 x2 = 1 (x1 , x2 roots). Hence |x¯1 | ¯≥ 1 or |x2 | ≥ 1.
¯p¯
Then exists p, q ∈ R, p 6= 0, q 6= 0 such that ¯¯ ¯¯ ≥ 1 or |p| ≥ |q|
q
satisfying (3), (4), (5).
With this λ, from (1) and (2) we obtain
p
|F (y0 x, z)|2 ≤ 2 F (w1 x, w1 x) · F (w2 x, w2 x) · F (z, z)
so
|F (y0 x, z)|4 ≤ 4F 2 (z, z) · F (w1 x, w1 x) · F (w2 x, w2 x).
(6)
We name (6) the Carlson - type inequality for complex and positive functionals, because of (6) we obtain for example the classical Carlson integral
inequality.
2
Applications
1. Let X be the complex algebra of the complex and integrable functions
defined on [a, ∞), a > 0. We consider
Z∞
F (f, g) =
f (t)g(t)dt
a
where f, g ∈ X.
F verify the conditions i), ii), iii) of introduction, evidently.
Certain inequalities concerning some complex ...
111
Now we consider x(t), y0 (t), z(t) ∈ X, non-nulls, and w1 , w2 : [a, ∞) →
(0, ∞) two continuously differentiable functions such that
w20 (t)w1 (t) − w2 (t)w10 (t) ≥ m > 0.
It is clear that
Z∞
w12 (t)|x(t)|2 dt ≥ 0,
F (w1 x, w1 x) =
a
Z∞
w22 (t)|x(t)|2 dt ≥ 0,
F (w2 x, w2 x) =
a
Z∞
w1 (t)w2 (t)|x(t)|2 dt ≥ 0,
F (w1 x, w2 x) =
a
2
and hence, using (4) we have p = q 2 .
Of (6) we obtain
¯4
¯∞
¯
¯Z
¯
¯
¯ y0 (t)x(t)z(t)dt¯ ≤
¯
¯
¯
¯
a
2 ∞
∞
Z
Z∞
Z
2
2
4  z(t)z(t)dt · w1 (t)|x(t)| dt · w22 (t)|x(t)|2 dt.
(7)
a
a
Since |y0 (t)| 6= 0 we choose z(t) =
(8)
a
1
y0 (t)
in (7) and we get
¯∞
¯4
2 ∞
∞
¯Z
¯
Z
Z∞
Z
¯
¯
dt
¯ x(t)dt¯ ≤ 4 
 · w12 (t)|x(t)|2 dt · w22 (t)|x(t)|2 dt.
¯
¯
2
|y
(t)|
0
¯
¯
a
a
a
a
Clearly
Z
dt
=
|y0 (t)|2
Z
dt
¯2 =
¯
¯
¯
1
¯λw1 (t) + w2 (t)¯
¯
¯
λ
Z
|λ|2
w12 (t)
¯
¯ dt.
¯ 2 w2 (t) ¯2
¯λ +
¯
¯
w1 (t) ¯
112
Emil C. Popa
Since λ = p + qi, p2 = q 2 , we have
¯
¯
2
¯ 2 w2 (t) ¯2
¯λ +
¯ = |λ|4 + w2 (t) .
¯
w1 (t) ¯
w12 (t)
Hence
Z
dt
=
|y0 (t)|2
Z
µ
(9)
≤
1
m
Z
1
|λ|2 w12 (t)
µ
¶2 dt ≤
w2 (t)
1+
|λ|2 w1 (t)
¶0
w2 (t)
1
w2 (t)
|λ|2 w1 (t)
arctg
µ
¶2 dt =
m
|λ|2 w1 (t)
w2 (t)
1+
|λ|2 w1 (t)
and we have the following result
Theorem 1.Let x(t) : [a, ∞) → C, a > 0, an integrable function and
w1 (t), w2 (t) : [a, ∞) → (0, ∞) two continuously differentiable functions
w2 (t)
= ∞.
with w20 (t)w1 (t) − w2 (t)w10 (t) ≥ m > 0, lim
t→∞ w1 (t)
Then
¯∞
¯4
¯Z
¯
¶2
µ
¯
¯
¯ x(t)dx¯ ≤ 4 π − 1 arctg w2 (a)
·
¯
¯
2m m
c · w1 (a)
¯
¯
a
(10)
Z∞
Z∞
2
2
· w1 (t)|x(t)| dt · w22 (t)|x(t)|2 dt
a
where
a
v
u Z∞
u
u w2 (t)|x(t)|2 dt
u
2
u
ua
c = c(w1 , w2 ) = u Z∞
u
u
t w12 (t)|x(t)|2 dt
a
Certain inequalities concerning some complex ...
113
Z∞
w12 (t)|x(t)|2 dt > 0.
and
a
Proof. Of (8) and (9) we obtain
¯4
¯∞
¯
¯Z
µ
¶2
¯
¯
π
1
w
(a)
2
¯ x(t)dt¯ ≤ 4
·
−
arctg
¯
¯
2m m
|λ|2 w1 (a)
¯
¯
a
Z∞
Z∞
w12 (t)|x(t)|2 dt ·
·
a
where
w22 (t)|x(t)|2 dt
a
v
u Z∞
u
u w2 (t)|x(t)|2 dt
u
2
u
u
|λ|2 = u Za∞
u
u
t w12 (t)|x(t)|2 dt
a
in conformity with (3).
Remark 1. When w1 (t) = 1, w2 (t) = t then the inequality (10) reduces to
(11)
¯∞
¯4
¯Z
¯
¶2 Z∞
µ
Z∞
¯
¯
a
π
2
¯ x(t)dt¯ ≤ 4
− arctg
· |x(t)| dt · t2 |x(t)|2 dt.
¯
¯
2
c(1,
t)
¯
¯
a
a
a
When a → 0, inequality (11) reduces to the well known Carlson0 s integral
inequality
(12)
¯4
¯∞
¯
¯Z
Z∞
Z∞
¯
¯
2
2
¯ x(t)dt¯ ≤ π
|x(t)| dt · t2 |x(t)|2 dt
¯
¯
¯
¯
0
(see [7]).
0
0
114
Emil C. Popa
Hence (10) and (11) are an improvement of (12).
2. We consider now the complex algebra of square matrices with complex elements X = Mn (C). If A is a n × n matrix, we write tr A to denote
the trace of A.
If

a11 a12 ... a1n


 a21 a22 ... a2n
A=

 ... ... ... ...

an1 an2 ... ann
we denote





 ,



a11 a21 ... an1


 a12 a22 ... an2
∗
A =

 ... ... ... ...

a1n a2n ... ann
aij ∈ C




.



Let F be a complex functional defined by
F (x, y) = tr(y ∗ x) ,
F :X ×X →C
which verify i), ii), iii), evidently.
Using (6) we get
(13)
|tr(z ∗ y0 x)|4 ≤ 4tr2 (z ∗ z) · tr(x∗ w1∗ w1 x) · tr(x∗ w2∗ w2 x)
1
where x, z, w1 , w2 ∈ X and y0 = λw1 + w2 , y0 ∈ X, with λ of (3), (4), (5).
λ
If y0∗ y0 = In then choosing in (13) z = y0 we obtain
(14)
|trx|4 ≤ 4tr2 (y0∗ y0 ) · tr(x∗ w1∗ w1 x) · tr(x∗ w2∗ w2 x)
and we have the next
Certain inequalities concerning some complex ...
115
Theorem 2. Let x, w1 , w2 be some matrices of Mn (C). If
µ
¶∗ µ
¶
1
1
λw1 + w2
λw1 + w2 = In
λ
λ
with λ complex number which verify
s
tr(x∗ w2∗ w2 x)
(15)
|λ|2 =
tr(x∗ w1∗ w1 x)
µ
(16)
Re
¶
λ
∗ ∗
· tr(x w2 w1 x) = 0
λ
(Re(λ))2 ≥ (Im(λ))2 ,
(17)
then we have the following inequality of Carlson0 s type
|tr x|4 ≤ 4n2 · tr(x∗ w1∗ w1 x) · tr(x∗ w2∗ w2 x).
(18)
Proof. Using the inequality (14) and (3), (4), (5) we get (15), evidently.
Remark 1. For w1 = w2 =
µ
1
λw1 + w2
λ
¶∗ µ
1
w, where p = Re(λ) and w∗ w = In , we obtain
2p
1
λw1 + w2
λ
¶
2
1 (λ2 + 1)(λ + 1) ∗
= 2
w w=
4p
λλ
2
1 (λ2 + 1)(λ + 1)
= 2
In .
4p
λλ
Since (15), (16), (17) we have |λ|2 = 1 and (Re(λ))2 = (In(λ))2 . Hence
|λ|2 = 2p2 = 1 and
µ
1
λw1 + w2
λ
¶∗ µ
1
λw1 + w2
λ
¶
= In .
116
Emil C. Popa
Therefore from (18) it follows that
|trx|4 ≤
n2 2 ∗
tr (x x).
4p4
This implies
|trx|2 ≤ n · tr(x∗ x).
(19)
Remark 2.We observe the fact that from (19) we get the well known inequality
|a1 + a2 + ... + an |2 ≤ n(|a1 |2 + |a2 |2 + ... + |an |2 )
for ai ∈ C, i = 1, 2.
References
[1] Barza S., Pečarić J., Persson L. E., Carlson type inequalities, J. Inequal.
Appl. 2, 2 (1998), 121-135.
[2] Barza S., Popa E. C., Inequalities related with Carlson0 s inequality,
Tamkang J. Math., 29, 1 (1998), 59-64.
[3] Barza C., Popa E. C., Weighted multiplicative integral inequalities,
Y.I.P.A.M., Vol 7(5), Art. 169, 2006.
[4] Carlson F., Une inégalité, Ark. Mat. Astr. Fysik 26B, 1 (1934).
[5] Hardy G. H., A note on two inequalities, J. London Math. Soc. 11
(1936), 167-170.
Certain inequalities concerning some complex ...
117
[6] Hardy G. H., Littlewood J. E., Pólya G., Inequalities, Cambridge, University Press, 1952, 2d ed..
[7] Larsson L., Maligranda L., Peěcarić J., Persson L. E., Multiplicative
Inequalities of Carlson Type and Interpolation, World Scientific, 2006.
Emil C. Popa
Department of Mathematics
Faculty of Science
University ”Lucian Blaga” of Sibiu
Str. I. Ratiu, no. 5-7,
550012 - Sibiu, Romania
Email address: emil.popa@ulbsibiu.ro
ecpopa2002@yahoo.com