Open Access

Some Vector Inequalities for Continuous Functions of Self-Adjoint Operators in Hilbert Spaces

Journal of Inequalities and Applications20112011:564836

DOI: 10.1155/2011/564836

Received: 24 November 2010

Accepted: 21 February 2011

Published: 15 March 2011

Abstract

On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq1_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq2_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq3_HTML.gif and for various classes of continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq4_HTML.gif are given. Applications for the power function and the logarithmic function are also provided.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq5_HTML.gif be a self-adjoint operator on the complex Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq6_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq7_HTML.gif included in the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq8_HTML.gif for some real numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq9_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq10_HTML.gif be its spectral family. Then, for any continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq11_HTML.gif , it is well known that we have the following spectral representation in terms of the Riemann-Stieltjes integral:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ1_HTML.gif
(1.1)
which in terms of vectors can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ2_HTML.gif
(1.2)
for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq12_HTML.gif . The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq13_HTML.gif is of bounded variation on the interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq14_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ3_HTML.gif
(1.3)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq15_HTML.gif . It is also well known that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq16_HTML.gif is monotonic nondecreasing and right continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq17_HTML.gif .

Utilising the spectral representation from (1.2), we have established the following Ostrowski-type vector inequality [1].

Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq18_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq19_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq20_HTML.gif for some real numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq21_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq22_HTML.gif be its spectral family. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq23_HTML.gif is a continuous function of bounded variation on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq24_HTML.gif , then one has the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ4_HTML.gif
(1.4)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq25_HTML.gif and for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq26_HTML.gif .

Another result that compares the function of a self-adjoint operator with the integral mean is embodied in the following theorem [2].

Theorem 1.2.

With the assumptions in Theorem 1.1 one has the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ5_HTML.gif
(1.5)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq27_HTML.gif .

The trapezoid version of the above result has been obtained in [3] and is as follows.

Theorem 1.3.

With the assumptions in Theorem 1.1 one has the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ6_HTML.gif
(1.6)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq28_HTML.gif .

For various inequalities for functions of self-adjoint operators, see [48]. For recent results see [1, 912].

In this paper, we investigate the quantity
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq29_HTML.gif are vectors in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq31_HTML.gif is a self-adjoint operator with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq32_HTML.gif , and provide different bounds for some classes of continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq33_HTML.gif . Applications for some particular cases including the power and logarithmic functions are provided as well.

2. Some Vector Inequalities

The following representation in terms of the spectral family is of interest in itself.

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq34_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq35_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq36_HTML.gif for some real numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq37_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq38_HTML.gif be its spectral family. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq39_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq40_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq41_HTML.gif , then one has the representation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ8_HTML.gif
(2.1)

Proof.

We observe
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ9_HTML.gif
(2.2)

which is an equality of interest in itself.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq42_HTML.gif are projections, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq43_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq44_HTML.gif and then we can write
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ10_HTML.gif
(2.3)
Integrating by parts in the Riemann-Stieltjes integral and utilizing the spectral representation (1.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ11_HTML.gif
(2.4)

which together with (2.3) and (2.2) produce the desired result (2.1).

The following vector version may be stated as well.

Corollary 2.2.

With the assumptions of Lemma 2.1 one has the equality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ12_HTML.gif
(2.5)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq45_HTML.gif .

The following result that provides some bounds for continuous functions of bounded variation may be stated as well.

Theorem 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq46_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq47_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq48_HTML.gif for some real numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq49_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq50_HTML.gif be its spectral family. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq51_HTML.gif is a continuous function of bounded variation on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq52_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq53_HTML.gif , then we have the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ13_HTML.gif
(2.6)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq54_HTML.gif .

Proof.

It is well known that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq55_HTML.gif is a bounded function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq56_HTML.gif is of bounded variation, and the Riemann-Stieltjes integral https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq57_HTML.gif exists, then the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ14_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq58_HTML.gif denotes the total variation of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq59_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq60_HTML.gif .

Utilising this property and the representation (2.5), we have by the Schwarz inequality in Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq61_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ15_HTML.gif
(2.8)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq62_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq63_HTML.gif are projections, in this case we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ16_HTML.gif
(2.9)

then from (2.8), we deduce the first part of (2.6).

Now, by the same property (2.7) for vector-valued functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq64_HTML.gif with values in Hilbert spaces, we also have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ17_HTML.gif
(2.10)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq66_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq67_HTML.gif in the operator order, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq68_HTML.gif which gives that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq69_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq70_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq71_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq72_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq73_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq74_HTML.gif which together with (2.10) prove the last part of (2.6).

The case of Lipschitzian functions is as follows.

Theorem 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq75_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq76_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq77_HTML.gif for some real numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq78_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq79_HTML.gif be its spectral family. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq80_HTML.gif is a Lipschitzian function with the constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq81_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq82_HTML.gif and with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq83_HTML.gif , then one has the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ18_HTML.gif
(2.11)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq84_HTML.gif .

Proof.

Recall that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq85_HTML.gif is a Riemann integrable function and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq86_HTML.gif is Lipschitzian with the constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq87_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ19_HTML.gif
(2.12)
then the Riemann-Stieltjes integral https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq88_HTML.gif exists and the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ20_HTML.gif
(2.13)
Now, on applying this property of the Riemann-Stieltjes integral, then we have from the representation (2.5) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ21_HTML.gif
(2.14)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq89_HTML.gif and the first inequality in (2.11) is proved.

Further, observe that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ22_HTML.gif
(2.15)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq90_HTML.gif .

If we use the vector-valued version of the property (2.13), then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ23_HTML.gif
(2.16)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq91_HTML.gif and the second part of (2.11) is proved.

Further on, by applying the double-integral version of the Cauchy-Buniakowski-Schwarz inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ24_HTML.gif
(2.17)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq92_HTML.gif .

Now, by utilizing the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq93_HTML.gif   are projections for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq94_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ25_HTML.gif
(2.18)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq95_HTML.gif .

If we integrate by parts and use the spectral representation (1.2), then we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ26_HTML.gif
(2.19)
and by (2.18), we then obtain the following equality of interest:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ27_HTML.gif
(2.20)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq96_HTML.gif .

On making use of (2.20) and (2.17), we then deduce the third part of (2.11).

Finally, by utilizing the elementary inequality in inner product spaces
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ28_HTML.gif
(2.21)
we also have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ29_HTML.gif
(2.22)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq97_HTML.gif , which proves the last inequality in (2.11).

The case of nondecreasing monotonic functions is as follows.

Theorem 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq98_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq99_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq100_HTML.gif for some real numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq101_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq102_HTML.gif be its spectral family. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq103_HTML.gif is a monotonic nondecreasing function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq104_HTML.gif , then one has the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ30_HTML.gif
(2.23)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq105_HTML.gif .

Proof.

From the theory of Riemann-Stieltjes integral, it is also well known that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq106_HTML.gif is of bounded variation and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq107_HTML.gif is continuous and monotonic nondecreasing, then the Riemann-Stieltjes integrals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq109_HTML.gif exist and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ31_HTML.gif
(2.24)
Now, on applying this property of the Riemann-Stieltjes integral, we have from the representation (2.5) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ32_HTML.gif
(2.25)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq110_HTML.gif , which proves the first inequality in (2.23).

On utilizing the Cauchy-Buniakowski-Schwarz-type inequality for the Riemann-Stieltjes integral of monotonic nondecreasing integrators, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ33_HTML.gif
(2.26)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq111_HTML.gif .

Observe that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ34_HTML.gif
(2.27)
and, integrating by parts in the Riemann-Stieltjes integral, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ35_HTML.gif
(2.28)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq112_HTML.gif .

On making use of the equalities (2.28), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ36_HTML.gif
(2.29)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq113_HTML.gif .

Therefore, we obtain the following equality of interest in itself as well:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ37_HTML.gif
(2.30)

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq114_HTML.gif

On making use of the inequality (2.26), we deduce the second inequality in (2.23).

The last part follows by (2.21), and the details are omitted.

3. Applications

We consider the power function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq115_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq117_HTML.gif . The following power inequalities hold.

Proposition 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq118_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq119_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq120_HTML.gif for some real numbers with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq121_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq122_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq123_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ38_HTML.gif
(3.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ39_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ40_HTML.gif
(3.3)

The proof follows from Theorem 2.4 applied for the power function.

Proposition 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq124_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq125_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq126_HTML.gif for some real numbers with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq127_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq128_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq129_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ41_HTML.gif
(3.4)

The proof follows from Theorem 2.5.

Now, consider the logarithmic function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq130_HTML.gif . We have the following

Proposition 3.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq131_HTML.gif be a self-adjoint operator in the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq132_HTML.gif with the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq133_HTML.gif for some real numbers with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq134_HTML.gif . Then one has the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ42_HTML.gif
(3.5)

The proof follows from Theorems 2.4 and 2.5 applied for the logarithmic function.

Authors’ Affiliations

(1)
Mathematics, School of Engineering & Science, Victoria University
(2)
School of Computational & Applied Mathematics, University of the Witwatersrand

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Copyright

© S. S. Dragomir. 2011

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