Improved Young and Heinz operator inequalities with Kantorovich constant
UDC 517.9 We present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert –  Schmidt norm of matrices. &nbs...
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| author | Beiranvand, A. Ghazanfari, A. G. Beiranvand, A. Ghazanfari, A. G. |
| author_facet | Beiranvand, A. Ghazanfari, A. G. Beiranvand, A. Ghazanfari, A. G. |
| author_sort | Beiranvand, A. |
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| datestamp_date | 2025-03-31T08:49:21Z |
| description | UDC 517.9
We present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert –  Schmidt norm of matrices.
  |
| doi_str_mv | 10.37863/umzh.v73i1.901 |
| first_indexed | 2026-03-24T02:06:01Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i1.901
UDC 517.9
A. Beiranvand, A. G. Ghazanfari (Lorestan Univ., Khoramabad, Iran)
IMPROVED YOUNG AND HEINZ OPERATOR INEQUALITIES
WITH KANTOROVICH CONSTANT
ВДОСКОНАЛЕНI ОПЕРАТОРНI НЕРIВНОСТI ЯНГА I ХАЙНЦА
З КОНСТАНТОЮ КАНТОРОВИЧА
We present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities
to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert – Schmidt
norm of matrices.
Отримано ряд покращень нерiвностi Янга за допомогою константи Канторовича. Цi покращенi нерiвностi ви-
користовуються для встановлення вiдповiдних операторних нерiвностей у просторi Гiльберта та деяких нових
нерiвностей, що включають норми Гiльберта – Шмiдта для матриць.
1. Introduction and preliminaries. Let Mm,n(\BbbC ) be the space of m \times n complex matrices and
Mn(\BbbC ) = Mn,n(\BbbC ). Let \| .\| denote any unitarily invariant norm on Mn(\BbbC ). So, \| UAV \| = \| A\| for
all A \in Mn(\BbbC ) and for all unitary matrices U, V \in Mn(\BbbC ). The Hilbert – Schmidt and trace class
norm of A = [aij ] \in Mn(\BbbC ) are denoted by
\| A\| 2 =
\left( n\sum
j=1
s2j (A)
\right) 1
2
, \| A\| 1 =
n\sum
j=1
sj(A),
where s1(A) \geq s2(A) \geq . . . \geq sn(A) are the singular values of A, which are the eigenvalues of the
positive semidefinite matrix | A | = (A\ast A)
1
2 , arranged in decreasing order and repeated according
to multiplicity. For Hermitian matrices A,B \in Mn(\BbbC ), we write that A \geq 0 if A is positive
semidefinite, A > 0 if A is positive definite, and A \geq B if A - B \geq 0.
Let a, b \geq 0 and 0 \leq \nu \leq 1. Young’s inequality for real numbers states that
a\nu b1 - \nu \leq \nu a+ (1 - \nu )b (1.1)
with equality if and only if a = b. This inequality has numerous applications in various fields.
Young’s inequality and its reverse have received renewed attention in recent years and a remarkable
variety of refinements and generalizations have been found (see, for example, [1, 2, 8, 9, 15, 17]).
Zhao and Wu in [15] obtained refinements of the Young inequality and its reverses in the follo-
wing forms:
if 0 < \nu \leq 1/2, then
\nu a+ (1 - \nu )b \geq a\nu b1 - \nu + \nu
\bigl( \surd
a -
\surd
b
\bigr) 2
+ r1
\bigl( 4
\surd
ab -
\surd
b
\bigr) 2
,
\nu a+ (1 - \nu )b \leq a\nu b1 - \nu + (1 - \nu )
\bigl( \surd
a -
\surd
b
\bigr) 2 - r1
\bigl( 4
\surd
ab -
\surd
a
\bigr) 2
;
(1.2)
if 1/2 < \nu < 1, then
c\bigcirc A. BEIRANVAND, A. G. GHAZANFARI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1 23
24 A. BEIRANVAND, A. G. GHAZANFARI
\nu a+ (1 - \nu )b \geq a\nu b1 - \nu + (1 - \nu )
\bigl( \surd
a -
\surd
b
\bigr) 2
+ r1
\bigl( 4
\surd
ab -
\surd
b
\bigr) 2
,
\nu a+ (1 - \nu )b \leq a\nu b1 - \nu + \nu
\bigl( \surd
a -
\surd
b
\bigr) 2 - r1
\bigl( 4
\surd
ab -
\surd
a
\bigr) 2
,
(1.3)
where r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu , 1 - \nu \} and r1 = \mathrm{m}\mathrm{i}\mathrm{n}\{ 2r, 1 - 2r\} .
A multiple-term refinement of Young’s inequality presented in [12] as follows:
a\nu b1 - \nu + SN (\nu ; a, b) \leq \nu a+ (1 - \nu )b, (1.4)
where SN (N ; a, b) is the following nonnegative function:
N\sum
j=1
sj(\nu )
\Bigl(
2j
\sqrt{}
b2
j - 1 - kj(\nu )akj(\nu ) - 2j
\sqrt{}
akj(\nu )+1b2
j - 1 - kj(\nu ) - 1
\Bigr) 2
.
The Kantorovich constant is defined as
K(t, 2) =
(t+ 1)2
4t
for t > 0.
Zuo et al. in [17] improved the classical Young’s inequality (1.1) via the Kantorovich constant as
follows:
a\nabla \nu b = \nu a+ (1 - \nu )b \geq K(h, 2)ra\nu b1 - \nu (1.5)
for all \nu \in [0, 1], where r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu , 1 - \nu \} and h =
b
a
.
Liao and Wu in [9] gave refinements of inequalities (1.2) and (1.3) with the Kantorovich constant:
if 0 < \nu \leq 1/2, then
\nu a+ (1 - \nu )b \geq \nu
\bigl( \surd
a -
\surd
b
\bigr) 2
+ r1
\bigl( 4
\surd
ab -
\surd
b
\bigr) 2
+K
\bigl( 4
\surd
h, 2
\bigr) \^r1a\nu b1 - \nu ; (1.6)
if 1/2 < v < 1, then
\nu a+ (1 - \nu )b \geq (1 - \nu )
\bigl( \surd
a -
\surd
b
\bigr) 2
+ r1
\bigl( 4
\surd
ab -
\surd
b
\bigr) 2
+K
\bigl( 4
\surd
h, 2
\bigr) \^r1a\nu b1 - \nu , (1.7)
where h =
b
a
, r = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu , 1 - \nu \} , r1 = \mathrm{m}\mathrm{i}\mathrm{n}\{ 2r, 1 - 2r\} and \^r1 = \mathrm{m}\mathrm{i}\mathrm{n}\{ 2r1, 1 - 2r1\} . Using the
Kantorovich constant a refinement of (1.4) given in [13] as follows:
K
\Biggl(
2N
\sqrt{}
b
a
, 2
\Biggr) \beta N (\nu )
a\nu b1 - \nu + SN (\nu ; a, b) \leq \nu a+ (1 - \nu )b, (1.8)
where \beta N (\nu ) is a special function which defined therein.
The Heinz means are defined as follows:
H\nu (a, b) =
a\nu b1 - \nu + a1 - \nu b\nu
2
.
It is easy to show that the Heinz means interpolate between the geometric mean and the arithmetic
mean:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
IMPROVED YOUNG AND HEINZ OPERATOR INEQUALITIES WITH KANTOROVICH CONSTANT 25
\surd
ab \leq H\nu (a, b) \leq
a+ b
2
. (1.9)
The second inequality of (1.9) is know as the Heinz inequality for nonnegative real numbers.
Let A,B \in B(H) be two positive operators, \nu \in [0, 1]. The \nu -weighted arithmetic mean of A
and B, denoted by A\nabla \nu B, is defined as A\nabla \nu B = (1 - \nu )A+\nu B. If A is invertible, the \nu -geometric
mean of A and B, denoted by A\sharp \nu B, is defined as
A\sharp \nu B = A
1
2 (A
- 1
2 BA
- 1
2 )\nu A
1
2 .
The Heinz operator mean is defined by
H\nu (A,B) =
A\sharp \nu B +A\sharp 1 - \nu B
2
,
where A and B are two invertible positive operators in B(H).
Zuo et al. in [17] show that the inequality (1.5) admits an operator extension
A\nabla \nu B \geq K(h, 2)rA\sharp \nu B
for positive operators A,B on a Hilbert space.
Bakherad et al. in [1] proved that if \nu \geq 0 or \nu \leq - 1, then
A\nabla - \nu B \leq A\sharp - \nu B.
In [9], the authors have presented operator versions of inequalities (1.6) and (1.7) on a Hilbert
space and corresponding inequalities with the Hilbert – Schmidt norm.
Let A be positive operator acting on a Hilbert space H and \lambda \in [0, 1]. Hölder – McCarthy
inequality states that
\langle Ax, x\rangle \lambda \geq \langle A\lambda x, x\rangle for all unit vectors x \in H.
It is known that the Hölder – McCarthy inequality and the Young inequality are equivalent, e.g.,
[5] (\S 3.1.3).
Fujii and Nakamoto in [4] proved that for A \geq 0 and 0 \leq \mu , \nu \leq 1 the following refinement of
the Young inequality:
\mu A+ 1 - \mu - A\mu \geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
1 - \mu
1 - \nu
,
\mu
\nu
\biggr\}
(\nu A+ 1 - \nu - A\nu )
is also equivalent to the following refinement of the Hölder – McCarthy inequality
1 - \langle A\mu x, x\rangle
\langle Ax, x\rangle \mu
\geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
1 - \mu
1 - \nu
,
\mu
\nu
\biggr\} \biggl(
1 - \langle A\nu x, x\rangle
\langle Ax, x\rangle \nu
\biggr)
for unit vector x.
For more information on the equivalent between Hölder – McCarthy inequality and Young in-
equality, the reader is referred to [4] and the references therein.
Sababheh and Moslehian in [13] obtained several multiterm refinements of Young type inequali-
ties for both real numbers and operators. They also proved the following operator inequality:
K
\Biggl(
2N
\sqrt{}
M
m
, 2
\Biggr) \beta N (\nu )
A\sharp \nu B+
+
N\sum
j=1
sj(\nu )
\bigl(
A\sharp \alpha j(\nu )B +A\sharp 21 - j\alpha j(\nu )B - 2A\sharp 2 - j\alpha j(\nu )B
\bigr)
\leq A\nabla \nu B
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
26 A. BEIRANVAND, A. G. GHAZANFARI
for the positive operators mI \leq A, B \leq MI, where sj(\nu ) and \alpha j(\nu ) are special functions which
have been defined therein.
In this paper, we present numerous refinements of the Young inequality by the Kantorovich con-
stant that improve several known results. We use these improved inequalities to obtain corresponding
operator inequalities on a Hilbert space. Moreover, some new Young type inequalities involving the
Hilbert – Schmidt norm are established.
2. Main results. 2.1. Several refinements of the Young inequality. First of all, we state a
refinement of the weighted arithmetic-geometric mean inequality for n positive numbers, which was
shown by Pečarić et al., see [10, p. 717] (Theorem 1) and [3].
Lemma 1. Let x1, . . . , xn belong to a closed interval I = [a, b], a < b, p1, . . . , pn \geq 0 with\sum n
i=1
pi = 1 and \lambda = \mathrm{m}\mathrm{i}\mathrm{n}\{ p1, . . . , pn\} . If f is a convex function on I, then
n\sum
i=1
pif(xi) - f
\Biggl(
n\sum
i=1
pixi
\Biggr)
\geq n\lambda
n\sum
i=1
1
n
f(xi) - f
\Biggl(
1
n
n\sum
i=1
xi
\Biggr)
.
Before exposing the main results, we state the following corollary from Lemma 1, which we will
use in what follows.
Corollary 1. If xi \in [a, b], 0 < a < b, p1, . . . , pn \geq 0 with
\sum n
i=1
pi = 1 and \lambda =
= \mathrm{m}\mathrm{i}\mathrm{n}\{ p1, . . . , pn\} , then \sum n
i=1
pixi\prod n
i=1
xpii
\geq
\left( 1
n
\sum n
i=1
xi\prod n
i=1
x
1
n
i
\right)
n\lambda
.
Suppose that f is a real convex (concave) function on [0, 1] and n \in \BbbN \cup \{ 0\} .
Let A0,0 = [0, 1], An,i = [2 - ni, 2 - n(i+ 1)) for n = 1, 2, 3, . . . , i = 0, 1, . . . , 2n - 1, and
fn(\nu ) =
2n - 1\sum
i=0
\bigl[
(i+ 1 - 2n\nu )f(2 - ni) + (2n\nu - i)f(2 - n(i+ 1))
\bigr]
\chi An,i(\nu ). (2.1)
It can be easily shown that fn is continuous on [0, 1] for every n \in \BbbN , and \{ fn\} is a decreasing
(increasing) sequence that converges pointwise to f. An example of such functions fn for n = 2
given in [16] (Theorem 2.1). Recently the authors in [14] extended this result for any integer n \geq 2.
Theorem 1. Suppose that 0 \leq \nu \leq 1, a, b \geq 0, with the assumption (2.1) for convex function
f(\nu ) = a\nu b1 - \nu . Then
a\nu b1 - \nu \leq K
\Biggl(
2n
\sqrt{}
b
a
, 2
\Biggr) \lambda n
a\nu b1 - \nu \leq fn(\nu ) \leq \nu a+ (1 - \nu )b, (2.2)
where \lambda n =
\sum 2n - 1
i=0
\mathrm{m}\mathrm{i}\mathrm{n}\{ i+ 1 - 2n\nu , 2n\nu - i\} \chi An,i(\nu ).
Proof. If \nu \in [2 - ni, 2 - n(i + 1)), by substitution x1 = f(2 - ni), x2 = f(2 - n(i + 1)), p1 =
= i+ 1 - 2n\nu and p2 = 2n\nu - i in Corollary 1, we get
f(2 - ni)i+1 - 2n\nu f(2 - n(i+ 1))2
n\nu - i
\left( 1
2
(f(2 - ni) + f(2 - n(i+ 1)))
f(2 - ni)
1
2 f(2 - n(i+ 1))
1
2
\right)
2\lambda n
\leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
IMPROVED YOUNG AND HEINZ OPERATOR INEQUALITIES WITH KANTOROVICH CONSTANT 27
\leq (i+ 1 - 2n\nu )f(2 - ni) + (2n\nu - i)f(2 - n(i+ 1)).
This implies that
a\nu b1 - \nu
\left(
\Biggl( \biggl(
b
a
\biggr) 2 - n
+ 1
\Biggr) 2
4
\biggl(
b
a
\biggr) 2 - n
\right)
\lambda n
\leq (i+ 1 - 2n\nu )f(2 - ni) + (2n\nu - i)f(2 - n(i+ 1))
or
K(h, 2)\lambda na\nu b1 - \nu \leq (i+ 1 - 2n\nu )f(2 - ni) + (2n\nu - i)f(2 - n(i+ 1)). (2.3)
Using Young’s inequality (1.1), we obtain
(i+ 1 - 2n\nu )f(2 - ni) + (2n\nu - i)f(2 - n(i+ 1)) \leq
\leq (i+ 1 - 2n\nu )
\bigl(
2 - nia+ (1 - 2 - ni)b
\bigr)
+
+(2n\nu - i)
\bigl(
2 - n(i+ 1)a+ (1 - 2 - n(i+ 1))b
\bigr)
= a\nabla \nu b. (2.4)
From inequalities (2.3) and (2.4), we deduce the desired inequalities (2.2).
Theorem 1 is proved.
Remark 1. Let 0 \leq \nu \leq 1, n \in \BbbN and r0 = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu , 1 - \nu \} . For n = 1, 2, . . . and i =
= 0, 1, . . . , 2n - 1 - 1 let
En,i =
\bigl[
2 - ni, 2 - n(i+ 1)) \cup (1 - 2 - n(i+ 1), 1 - 2 - ni
\bigr]
.
If a, b \geq 0, since H\nu (a, b) is symmetric with respect to \nu = 1/2, that is, H\nu (a, b) = H1 - \nu (a, b),
\nu \in [0, 1], then the following inequalities hold:
K
\Biggl(
2n
\sqrt{}
b
a
, 2
\Biggr) \lambda n
H\nu (a, b) \leq
2n - 1 - 1\sum
i=0
\bigl[
(i+ 1 - 2nr0)H2 - ni(a, b)+
+(2nr0 - i)H2 - n(i+1)(a, b)
\bigr]
\chi En,i(\nu ) \leq
\leq a+ b
2
, (2.5)
where \lambda n =
\sum 2n - 1 - 1
i=0
\mathrm{m}\mathrm{i}\mathrm{n}\{ i + 1 - 2nr0, 2
nr0 - i\} \chi En,i(\nu ). Clearly, the inequalities (2.5) are
refinements of inequalities (2.2) in [14].
By the same argument used in the proof of Theorem 1, we give new inequalities as to the Young
inequality in the following theorem.
Theorem 2. Suppose that 0 \leq \nu \leq 1, a, b \geq 0 with the assumption (2.1) for concave function
g(\nu ) =
\sqrt{}
\nu a+ (1 - \nu )b. Then the following inequalities hold:
a\nu b1 - \nu \leq g2n(\nu ) =
2n - 1\sum
i=0
\Bigl[
(i+ 1 - 2n\nu )
\sqrt{}
a\nabla 2 - nib +
+(2n\nu - i)
\sqrt{}
a\nabla 2 - n(i+1)b
\Bigr] 2
\chi An,i(\nu ) \leq \nu a+ (1 - \nu )b. (2.6)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
28 A. BEIRANVAND, A. G. GHAZANFARI
Proof. Similarly as in Theorem 1, for concave function g(\nu ) =
\sqrt{}
\nu a+ (1 - \nu )b there exists a
sequence
gn(\nu ) =
2n - 1\sum
i=0
\Bigl[
(i+ 1 - 2n\nu )
\sqrt{}
a\nabla 2 - nib+ (2n\nu - i)
\sqrt{}
a\nabla 2 - n(i+1)b
\Bigr]
\chi An,i(\nu ),
such that gn(\nu ) \leq
\sqrt{}
\nu a+ (1 - \nu )b. Using Young’s inequality (1.1), we obtain
(i+ 1 - 2n\nu )g(2 - ni) + (2n\nu - i)g(2 - n(i+ 1)) \geq
\geq (i+ 1 - 2n\nu )
\bigl(
a2
- n - 1ib
1
2
- 2 - n - 1i
\bigr)
+ (2n\nu - i)
\bigl(
a2
- n - 1(i+1)b
1
2
- 2 - n - 1(i+1)
\bigr)
\geq
\geq
\surd
a\nu b1 - \nu .
Theorem 2 is proved.
Furthermore, converging gn(\nu ) to
\sqrt{}
\nu a+ (1 - \nu )b and inequality (1.8), imply that, for any
N \in \BbbN , there exists a positive integer n1 such that, for every n > n1,
K
\Biggl(
2N
\sqrt{}
b
a
, 2
\Biggr) \beta N (\nu )
a\nu b1 - \nu + SN (\nu ; a, b) \leq g2n(\nu ) \leq \nu a+ (1 - \nu )b. (2.7)
Hence (2.7) is a refinement of (1.8). Moreover, a benefit of (2.6) is that it has an explicit formula
which doesn’t depend on certain functions.
2.2. Some matrix versions of Young and Heinz inequalities. Let A,B,X \in Mn(\BbbC ) be such
that A and B are positive semidefinite and 0 \leq \nu \leq 1. Hirzallah and Kittaneh in [6] proved that\bigm\| \bigm\| A\nu XB1 - \nu
\bigm\| \bigm\| 2
2
+ r20
\bigm\| \bigm\| AX - XB
\bigm\| \bigm\| 2
2
\leq
\bigm\| \bigm\| \nu AX + (1 - \nu )
\bigm\| \bigm\| 2
2
,
where r0 = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu , 1 - \nu \} .
Zou and Jiang [16] obtained refinements of the Heinz inequality for matrices in the following
forms:
Theorem 3. Let A,B,X \in Mn(\BbbC ) such that A and B are positive semidefinite and suppose
that
\phi (\nu ) =
\bigm\| \bigm\| A\nu XB1 - \nu +A1 - \nu XB\nu
\bigm\| \bigm\|
2
for \nu \in [0, 1]. Then
\phi (\nu ) \leq
\left\{
(1 - 4r0)\phi (0) + 4r0\phi
\biggl(
1
4
\biggr)
, \nu \in
\biggl[
0,
1
4
\biggr]
\cup
\biggl[
3
4
, 1
\biggr]
,
(4r0 - 1)\phi
\biggl(
1
2
\biggr)
+ 2(1 - 2r0)\phi
\biggl(
1
4
\biggr)
, \nu \in
\biggl[
1
4
,
3
4
\biggr]
,
where r0 = \mathrm{m}\mathrm{i}\mathrm{n}\{ \nu , 1 - \nu \} .
Krnić in [8] proved that\bigm\| \bigm\| A\nu XB1 - \nu +A1 - \nu XB\nu
\bigm\| \bigm\| 2
2
+ 4\nu (1 - \nu )\| AX - XB\| 22 \leq \| AX +XB\| 22. (2.8)
In the following theorem we give some refinements of the Young inequality for the Hilbert –
Schmidt norm based on the inequality (2.2).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
IMPROVED YOUNG AND HEINZ OPERATOR INEQUALITIES WITH KANTOROVICH CONSTANT 29
Theorem 4. Let n \in \BbbN \cup \{ 0\} and A,B,X \in Mm(\BbbC ) such that A and B are positive definite.
Let \mathrm{S}\mathrm{p}(A) = \{ \lambda 1, . . . , \lambda m\} be the spectrum of A, \mathrm{S}\mathrm{p}(B) = \{ \mu 1, . . . , \mu m\} and let
Kn = \mathrm{m}\mathrm{i}\mathrm{n}
\Biggl\{
K
\Biggl( \biggl(
\lambda j
\mu k
\biggr) 2 - n
, 2
\Biggr)
: k, j = 1, . . . ,m
\Biggr\}
.
Then the following inequalities hold:
K\gamma n
n \| A\nu XB1 - \nu \| 2 \leq
2n - 1\sum
i=0
\bigm\| \bigm\| \bigm\| ((i+ 1 - 2n\nu )A2 - niXB1 - 2 - ni+
+( - i+ 2n\nu )A2 - n(i+1)XB1 - 2 - n(i+1)
\bigm\| \bigm\| \bigm\|
2
\chi An,i \leq
\bigm\| \bigm\| \nu AX + (1 - \nu )XB
\bigm\| \bigm\|
2
,
where \gamma n =
\sum 2n - 1
i=0
\mathrm{m}\mathrm{i}\mathrm{n}\{ i+ 1 - 2n\nu , 2n\nu - i\} \chi An,i .
Proof. Since A and B are positive definite, it follows by the spectral theorem that there exist
unitary matrices U, V \in Mm(\BbbC ) such that
A = U\Lambda 1U
\ast and B = V \Lambda 2V
\ast ,
where
\Lambda 1 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\lambda 1, . . . , \lambda m), \Lambda 2 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mu 1, . . . , \mu m), \lambda k, \mu k > 0, k, j = 1, . . . .
Let
Y = U\ast XV = [ykj ].
We have
A\nu XB1 - \nu = (U\Lambda 1U
\ast )\nu X(V \Lambda 2V
\ast )1 - \nu = U\Lambda \nu
1Y \Lambda 1 - \nu
2 V \ast .
Therefore,
K2\gamma n
n \| A\nu XB1 - \nu \| 22 = K2\gamma n
n \| \Lambda \nu
1Y \Lambda 1 - \nu
2 \| 22 = K2\gamma
0
m\sum
k,j=1
\bigl(
\lambda \nu
j\mu
1 - \nu
k
\bigr) 2 | ykj | 2,
\nu AX + (1 - \nu )XB = U(\nu \Lambda 1Y + (1 - \nu )Y \Lambda 2)V
\ast ,
and
2n - 1\sum
i=0
\bigm\| \bigm\| \bigm\| ((i+ 1 - 2n\nu )A2 - niXB1 - 2 - ni + ( - i+ 2n\nu )A2 - n(i+1)XB1 - 2 - n(i+1)
\bigm\| \bigm\| \bigm\| 2
2
\chi An,i =
=
2n - 1\sum
i=0
m\sum
k,j=1
\Bigl(
(i+ 1 - 2n\nu )\lambda 2 - ni
j \mu 1 - 2 - ni
k + ( - i+ 2n\nu )\lambda
2 - n(i+1)
j \mu
1 - 2 - n(i+1)
k
\Bigr) 2
| ykj | 2\chi An,i .
Now, from inequalities (2.2), we deduce
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
30 A. BEIRANVAND, A. G. GHAZANFARI
K2\gamma n
n \| A\nu XB1 - \nu \| 22 = K2\gamma n
n
m\sum
k,j=1
\bigl(
\lambda \nu
j\mu
1 - \nu
k
\bigr) 2 | ykj | 2 =
=
2n - 1\sum
i=0
m\sum
k,j=1
\Bigl(
(i+ 1 - 2n\nu )\lambda 2 - ni
j \mu 1 - 2 - ni
k + ( - i+ 2n\nu )\lambda
2 - n(i+1)
j \mu
1 - 2 - n(i+1)
k
\Bigr) 2
| ykj | 2\chi An,i \leq
\leq
m\sum
k,j=1
\bigl(
\nu \lambda j + (1 - \nu )\mu k
\bigr) 2| ykj | 2 = \| \nu AX + (1 - \nu )XB\| 22.
Theorem 4 is proved.
Since for every unitarily invariant norm | | | \cdot | | | , the function f(\nu ) =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| A\nu XB1 - \nu +A1 - \nu XB\nu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
is convex, using the same strategy as in the proof of Theorem 4, we can present new refinements of
matrix versions of the Heinz inequality.
Corollary 2. Let A,B,X \in Mm(\BbbC ) such that A and B are positive definite. Let \mathrm{S}\mathrm{p}(A) =
= \{ \lambda 1, . . . , \lambda m\} be the spectrum of A, \mathrm{S}\mathrm{p}(B) = \{ \mu 1, . . . , \mu m\} be the spectrum of B and let
Kn = \mathrm{m}\mathrm{i}\mathrm{n}
\Biggl\{
K
\Biggl( \biggl(
\lambda j
\mu k
\biggr) 2 - n
, 2
\Biggr)
: k, j = 1, . . . ,m
\Biggr\}
.
Then the following inequalities hold:
K\gamma n
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| A\nu XB1 - \nu +A1 - \nu XB\nu
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
2n - 1 - 1\sum
i=0
\Bigl(
(i+ 1 - 2nr0)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| A2 - niXB1 - 2 - ni +A1 - 2 - niXB2 - ni
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| +
+(2nr0 - i)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| A2 - n(i+1)XB1 - 2 - n(i+1) +A1 - 2 - n(i+1)XB2 - n(i+1)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \Bigr) \chi En,i \leq
\leq | | | AX +XB| | | , (2.9)
where \gamma n =
\sum 2n - 1 - 1
i=0
\mathrm{m}\mathrm{i}\mathrm{n}\{ i+ 1 - 2nr0, 2
nr0 - i\} \chi En,i .
Clearly, inequalities (2.9) are refinements of inequality (2.10) in [7].
2.3. Some operator versions of Young and Heinz inequalities. In this section, we give an
operator version of the inequalities (2.2). To reach inequalities for bounded self-adjoint operators on
Hilbert space, we shall use the following monotonicity property for operator functions:
if X \in Bh(H) with a spectrum \mathrm{S}\mathrm{p}(X) and f, g are continuous real-valued functions on \mathrm{S}\mathrm{p}(X),
then
f(t) \geq g(t), t \in \mathrm{S}\mathrm{p}(X) \Rightarrow f(X) \geq g(X). (2.10)
For more details about this property, the reader is referred to [11].
Theorem 5. Let n \in \BbbN \cup \{ 0\} and 0 \leq \nu \leq 1. If A, B are two invertible positive operators in
B(H) and h a positive real number such that either A < hA \leq B or A > hA \geq B, then
\bigl(
K(h2
- n
, 2)
\bigr) \lambda nA\sharp \nu B \leq
2n - 1\sum
i=0
\Bigl[
(i+ 1 - 2n\nu )A\sharp 2 - niB + (2n\nu - i)A\sharp 2 - n(i+1)B
\Bigr]
\chi An,i \leq A\nabla \nu B,
(2.11)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
IMPROVED YOUNG AND HEINZ OPERATOR INEQUALITIES WITH KANTOROVICH CONSTANT 31
where \lambda n =
\sum 2n - 1
i=0
\mathrm{m}\mathrm{i}\mathrm{n}\{ i+ 1 - 2n\nu , 2n\nu - i\} \chi An,i.
Proof. For \nu \in [2 - ni, 2 - n(i+1)) and A < hA \leq B, then it is clear that I < hI \leq A
- 1
2 BA
- 1
2
and 1 < h. Since the function K(t, 2) is continuous and monotone increasing on, therefore, for all
real numbers t such that 1 < h \leq t, we have\bigl(
K(h2
- n
, 2)
\bigr) \lambda n \leq
\bigl(
K(t2
- n
, 2)
\bigr) \lambda n . (2.12)
Inequalities (2.2), for b = 1, become\bigl(
K(a2
- n
, 2)
\bigr) \lambda na\nu \leq (i+ 1 - 2n\nu )a2
- ni + (2n\nu - i)a2
- n(i+1) \leq
\leq \nu a+ (1 - \nu ). (2.13)
According to (2.10), we can insert X = A
- 1
2 BA
- 1
2 in (2.12) to get\bigl(
K(h2
- n
, 2)
\bigr) \lambda nI \leq
\bigl(
K(X2 - n
, 2)
\bigr) \lambda n . (2.14)
Multiplying both sides of inequality (2.14) by X
\nu
2 on the left and right, we have\bigl(
K(h2
- n
, 2)
\bigr) \lambda nX\nu \leq X\nu /2
\bigl(
K(X2 - n
, 2)
\bigr) \lambda nX\nu /2 =
\bigl(
K(X2 - n
, 2)
\bigr) \lambda nX\nu . (2.15)
We also can insert X = A
- 1
2 BA
- 1
2 in (2.13) to deduce\bigl(
K(X2 - n
, 2)
\bigr) \lambda nX\nu \leq (i+ 1 - 2n\nu )X2 - ni + (2n\nu - i)X2 - n(i+1) \leq
\leq \nu X + (1 - \nu )I. (2.16)
From inequalities (2.15), (2.16), we obtain\bigl(
K(h2
- n
, 2)
\bigr) \lambda nX\nu \leq (i+ 1 - 2n\nu )X2 - ni + (2n\nu - i)X2 - n(i+1) \leq
\leq \nu X + (1 - \nu )I. (2.17)
Finally, if we multiply inequalities (2.17) by A
1
2 on the left- and right-sides, we get the desired
inequalities (2.11).
Theorem 5 is proved.
The assumptions of Theorem 5 are weaker than the assumptions of Theorem 7 in [17]. Because
if m, m\prime , M, M \prime are positive real numbers such that 0 < m\prime I \leq A \leq mI < MI \leq B \leq M \prime I, then
A <
M
m
A \leq B.
By the same method used in the proof of Theorem 5, we give new refinements of operator
versions of the Heinz inequality.
Corollary 3. Let A, B are two invertible positive operators in B(H) and h a positive real
number such that either A < hA \leq B or A > hA \geq B. Then\bigl(
K(h2
- n
, 2)
\bigr) \lambda n(A\sharp \nu B +A\sharp 1 - \nu B) \leq
\leq
2n - 1 - 1\sum
i=0
\Bigl[
(i+ 1 - 2nr0)(A\sharp 2 - niB +A\sharp 1 - 2 - niB)+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
32 A. BEIRANVAND, A. G. GHAZANFARI
+(2nr0 - i)(A\sharp 2 - n(i+1)B +A\sharp 1 - 2 - n(i+1)B)
\Bigr]
\chi En,i \leq
\leq A+B
2
,
where \lambda n =
\sum 2n - 1 - 1
i=0
\mathrm{m}\mathrm{i}\mathrm{n}\{ i+ 1 - 2nr0, 2
nr0 - i\} \chi En,i.
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Received 12.03.18,
after revision — 25.09.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
|
| id | umjimathkievua-article-901 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:01Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ff/9d403c7da36d036e6ac5b90f0cdb68ff.pdf |
| spelling | umjimathkievua-article-9012025-03-31T08:49:21Z Improved Young and Heinz operator inequalities with Kantorovich constant Improved Young and Heinz operator inequalities with Kantorovich constant Beiranvand, A. Ghazanfari, A. G. Beiranvand, A. Ghazanfari, A. G. Operator inequalities Young inequality Heinz mean Kantorovich constant Operator inequalities Young inequality Heinz mean Kantorovich constant UDC 517.9 We present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert –&nbsp; Schmidt norm of matrices. &nbsp; УДК 517.9 Вдосконаленi операторнi нерiвностi Янга та Хайнца з константою Канторовича Отримано ряд покращень нерівності Янга за допомогою константи Канторовича.Ці покращені нерівності використовуються для встановлення відповідних операторних нерівностей у просторі Гільберта та деяких нових нерівностей, що включають норми Гільберта&nbsp; –&nbsp; Шмідта для матриць. Institute of Mathematics, NAS of Ukraine 2021-01-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/901 10.37863/umzh.v73i1.901 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 1 (2021); 23 - 32 Український математичний журнал; Том 73 № 1 (2021); 23 - 32 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/901/8894 |
| spellingShingle | Beiranvand, A. Ghazanfari, A. G. Beiranvand, A. Ghazanfari, A. G. Improved Young and Heinz operator inequalities with Kantorovich constant |
| title | Improved Young and Heinz operator inequalities with Kantorovich constant |
| title_alt | Improved Young and Heinz operator inequalities with Kantorovich constant |
| title_full | Improved Young and Heinz operator inequalities with Kantorovich constant |
| title_fullStr | Improved Young and Heinz operator inequalities with Kantorovich constant |
| title_full_unstemmed | Improved Young and Heinz operator inequalities with Kantorovich constant |
| title_short | Improved Young and Heinz operator inequalities with Kantorovich constant |
| title_sort | improved young and heinz operator inequalities with kantorovich constant |
| topic_facet | Operator inequalities Young inequality Heinz mean Kantorovich constant Operator inequalities Young inequality Heinz mean Kantorovich constant |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/901 |
| work_keys_str_mv | AT beiranvanda improvedyoungandheinzoperatorinequalitieswithkantorovichconstant AT ghazanfariag improvedyoungandheinzoperatorinequalitieswithkantorovichconstant AT beiranvanda improvedyoungandheinzoperatorinequalitieswithkantorovichconstant AT ghazanfariag improvedyoungandheinzoperatorinequalitieswithkantorovichconstant |