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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq1_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq2_HTML.gif , where http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq6_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq7_HTML.gif included in the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq8_HTML.gif for some real numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq9_HTML.gif and let http://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 http://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:
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ2_HTML.gif
(1.2)
for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq12_HTML.gif . The function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq14_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ3_HTML.gif
(1.3)

for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq15_HTML.gif . It is also well known that http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq19_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq20_HTML.gif for some real numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq21_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq22_HTML.gif be its spectral family. If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq24_HTML.gif , then one has the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ4_HTML.gif
(1.4)

for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq25_HTML.gif and for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ5_HTML.gif
(1.5)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ6_HTML.gif
(1.6)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ7_HTML.gif
(1.7)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq29_HTML.gif are vectors in the Hilbert space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq31_HTML.gif is a self-adjoint operator with http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq35_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq36_HTML.gif for some real numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq37_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq38_HTML.gif be its spectral family. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq39_HTML.gif is a continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq40_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq41_HTML.gif , then one has the representation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ8_HTML.gif
(2.1)

Proof.

We observe
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq42_HTML.gif are projections, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq43_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq44_HTML.gif and then we can write
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ12_HTML.gif
(2.5)

for any http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq47_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq48_HTML.gif for some real numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq49_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq50_HTML.gif be its spectral family. If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq52_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq53_HTML.gif , then we have the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ13_HTML.gif
(2.6)

for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq55_HTML.gif is a bounded function, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq57_HTML.gif exists, then the following inequality holds:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ14_HTML.gif
(2.7)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq58_HTML.gif denotes the total variation of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq59_HTML.gif on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq61_HTML.gif that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ15_HTML.gif
(2.8)

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

Since http://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
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ17_HTML.gif
(2.10)

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

Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq67_HTML.gif in the operator order, then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq68_HTML.gif which gives that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq69_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq70_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq71_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq72_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq73_HTML.gif . Therefore, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq76_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq77_HTML.gif for some real numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq78_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq79_HTML.gif be its spectral family. If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq81_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq82_HTML.gif and with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq83_HTML.gif , then one has the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ18_HTML.gif
(2.11)

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

Proof.

Recall that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq85_HTML.gif is a Riemann integrable function and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq86_HTML.gif is Lipschitzian with the constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq87_HTML.gif , that is,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq88_HTML.gif exists and the following inequality holds:
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ21_HTML.gif
(2.14)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ22_HTML.gif
(2.15)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ23_HTML.gif
(2.16)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ24_HTML.gif
(2.17)

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

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

for any http://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
http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ27_HTML.gif
(2.20)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ28_HTML.gif
(2.21)
we also have that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ29_HTML.gif
(2.22)

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

for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq106_HTML.gif is of bounded variation and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq109_HTML.gif exist and
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ32_HTML.gif
(2.25)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ33_HTML.gif
(2.26)

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

Observe that
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ35_HTML.gif
(2.28)

for any http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ36_HTML.gif
(2.29)

for any http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ37_HTML.gif
(2.30)

for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq115_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq116_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq119_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq120_HTML.gif for some real numbers with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq121_HTML.gif .

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq122_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq123_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ38_HTML.gif
(3.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_Equ39_HTML.gif
(3.2)
where
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq125_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq126_HTML.gif for some real numbers with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq127_HTML.gif .

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq128_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq129_HTML.gif
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq132_HTML.gif with the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq133_HTML.gif for some real numbers with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F564836/MediaObjects/13660_2010_Article_2350_IEq134_HTML.gif . Then one has the inequalities
http://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

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.