Analytic and Geometric Inequalities and Applications

376 Pages·1999·14.56 MB·English

by  Tsuyoshi Ando (auth.)

Analytic and Geometric Inequalities and Applications
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 478
Analytic and Geometric
Inequalities and Applications
Edited by
Themistocles M. Rassias
Department of Mathematics,
National Technical University of Athens,
Zografou Campus,
Athens, Greece
and
Hari M. Srivastava
Department of Mathematics and Statistics,
University of Victoria,
Victoria, British Columbia, Canada
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V .
A CLP. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-5938-1 ISBN 978-94-011-4577-0 (eBook)
DOI 10.1007/978-94-011-4577-0
Printed on acid-free paper
All Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Table of Contents
Preface vii
Problem of Infimum in the Positive Cone
Tsuyoshi Ando
Open Problems on Eigenvalues of the Laplacian
Mark S. Ashbaugh 13
On an Inequality of S. Bernstein and the Gauss-Lucas Theorem
A. Aziz and N .A. Rather 29
Second Order Opial Inequalities in LP Spaces and Applications
R. Brown, V. Burenkov, S. Clark and D. Hinton 37
On Multi-Dimensional Integral Inequalities and Applications
Wing-Sum Cheung and Themistocles M. Rassias 53
Some Inequalities for Triangle: Old and New Results
Radosav Z. Djordjevic 69
A Treatise on Gruss' Inequality
A.M. Fink 93
A 2:: B 2:: 0 Ensures (B¥ AP B¥) ~ 2:: (B¥ BP B¥) ~ for r 2:: 0, p 2:: 0,
q 2:: 1 with (1 +r)q 2:: p + r and its Recent Applications
Takayuki Furuta 115
Degree of Convergence for a Class of Linear Operators
N.K. Govil and R.N. Mohapatra 135
Some Comments and a Bibliography on the Laguerre-Samuelson
Inequality with Extensions and Applications in Statistics and Mat
rix Theory
Shane T. Jensen and George P.H. Styan 151
Fractional Order Inequalities of Hardy Type
Alois Kufner 183
Theory of Differential and Integral Inequalities with Initial Time Differ-
ence and Applications
V. Lakshmikantham and A.S. Vatsala 191
Numerical Radii of Some Companion Matrices and Bounds for the Zeros
of Polynomials
Hansjorg Linden 205
Bounds on Entropy Measures for Mixed Populations
M. Matic, C.E.M. Pearce and J. Pecaric 231
vi
Extremal Problems and Inequalities of Markov-Bernstein Type for Poly
nomials
Gradimir V. Milovanovic 245
On Alpha-Quasi-Convex Functions Defined by Convolution with Incom-
plete Beta Functions
Khalida Inayat Noor 265
Fourth Order Obstacle Boundary Value Problems
M. Aslam Noor and Eisa A. AI-Said 277
On Some Generalized Opial Type Inequalities
B.G. Pachpatte 301
Controlling the Velocity of Brownian Motion by its Terminal Value
Goran Peskir 323
Chebyshev Polynomials with Integer Coefficients
Igor E. Pritsker 335
Distortion Inequalities for Analytic and Univalent Functions Associated
with Certain Fractional Calculus and Other Linear Operators
H.M. Srivastava 349
Index 375
PREFACE
Analytic and Geometric Inequalities and Applications is devoted to
recent advances in a variety of inequalities of Mathematical Analysis and Geo
metry. Subjects dealt with in this volume include: Fractional order inequalities
of Hardy type, differential and integral inequalities with initial time differ
ence, multi-dimensional integral inequalities, Opial type inequalities, Gruss'
inequality, Furuta inequality, Laguerre-Samuelson inequality with extensions
and applications in statistics and matrix theory, distortion inequalities for ana
lytic and univalent functions associated with certain fractional calculus and
other linear operators, problem of infimum in the positive cone, alpha-quasi
convex functions defined by convolution with incomplete beta functions,
Chebyshev polynomials with integer coefficients, extremal problems for poly
nomials, Bernstein's inequality and Gauss-Lucas theorem, numerical radii of
some companion matrices and bounds for the zeros of polynomials, degree
of convergence for a class of linear operators, open problems on eigenvalues
of the Laplacian, fourth order obstacle boundary value problems, bounds on
entropy measures for mixed populations as well as controlling the velocity of
Brownian motion by its terminal value. A wealth of applications of the above
is also included.
We wish to express our appreciation to the distinguished mathematicians
who contributed to this volume. Finally, it is our pleasure to acknowledge the
fine cooperation and assistance provided by the staff of Kluwer Academic
Publishers.
June 1999 Themistocles M. Rassias
Hari M. Srivastava
vii
PROBLEM OF INFIMUM IN THE POSITIVE CONE
TSUYOSHI ANDO
Faculty of Economics, Hokusei Gakuen University, Sapporo 004-8631, Japan
Abstract. It is known that for (bounded) self-adjoint operators A, B on a Hilbert
space 11 the infimum A A B, with respect to the order induced by the cone of positive
(semi-definite) operators, exists only when A and B are comparable, that is, A 2:: B or
A S B. In this paper we present a necessary and sufficient condition for that, given
A, B 2:: 0, the infimum considered in the positive cone exists.
1. Problem
The space of (bounded) self-adjoint operators on a Hilbert space H is provided
with the order relation induced by positive semi-definiteness; for self-adjoint A, B
the order relation A ;::: B means A - B is positive (semi-definite). In particular,
A ;::: 0 means positive semi-definiteness of A. With respect to this order, the space
of self-adjoint operators does not become a lattice. More exactly, for self-adjoint
A, B the infimum A /\ B exists only when A and B is comparable, that is, A ;::: B
or A ::; B.
The situation is different if we take the positive cone P = P(H), the cone of posi
tive (semi-definite) operators, in place of the whole space of self-adjoint operators.
For instance, it is well known that for any two orthoprojections P, Q the infimum
P /\ Q in P always exists and is equal to the orthoprojection to the intersection
of the range of P and that of Q.
In this paper we present a necessary and sufficient condition for that, given A, B ;:::
0, the infimum A/\B in P exists. This problem has been studied by several authors
in mathematical physics (see [6], [7], [8] and [9]). Among others, Moreland and
Gudder [9] have solved this problem in the finite dimensional case. We explain
also how our result reduces to theirs in the finite dimensional case.
1991 Mathematics Subject Classification. Primary 47D20; Secondary 47 A63
Key words and phrases. Positive cone; Lattice, Operator inequalities
T.M. Rassias and H.M. Srivastava (eds.), Analytic and Geometric Inequalities and Applications, 1-12.
© 1999 Kluwer Academic Publishers.
2
2. Reduction
To discuss existence of the infimum A 1\ B in P, we may assume ker(A + B) =
{a}. (Here ker denotes the kernel.) In fact, let P be the orthoprojection to the
orthocomplement K of ker(A + B). Then the map
o :::; X cp(X) = P Xk
f------+
gives an affine, order-isomorphism from {X ; 0 :::; X :::; A, B} onto a subset Q
z :::;
of the positive cone P(K) satisfying that Y E Q,O :::; Y =? Z E Q, and
ker(cp(A) + cp(B)) = {O}.
When ker(A + B) = {O}, there is an affine, order-isomorphism r.p from the set
+
{X ; 0 :::; X :::; A B} onto the set of positive contractions, that is, {Y ; 0 :::;
Y:::; I}, given by
X = (A + B)1/2 . r.p(X) . (A + B)1/2. (1)
Here important is the fact that
r.p(A) + r.p(B) = 1,
in particular, r.p(A) and r.p(B) commute. Therefore we first discuss existence of
the infimum A 1\ B in P under the assumption A + B = 1.
3. Commuting Case
Suppose that
A, B ::::: 0, A + B = 1. (2)
Then according to the spectral theorem there is uniquely an increasing family of
orthoprojection {E(A) ; 0:::; A :::; I} with E(I) = 1 such that
l l
A = AdE(A) and B = (1 - A)dE(A). (3)
Lemma 1. Under the setting (3), if the infimum A 1\ B in P exists, then
11
A 1\ B = min()., 1 - A)dE(A).
Proof. Let
11
c ~ min(A, 1 - A)dE(A).
Then obviously 0 :::; C :::; A, B. We claim that C is a maximal element in the set
{X ; 0:::; X:::; A,B} (d. Ando [5]). To see this, take D such that
C:::;D:::;A, B.
3
Then
o ::; D - C ::; A - C, B - C
so that there exist contractions X, Y such that
(D - C)1/2 = (A - C)1/2 X and (D - C)1/2 = (B _ C)1/2y,
which implies
ran(D - C)1/2 C ran(A - C)1/2 n ran(B _ C)1/2.
Here ran denotes the range. Since by the spectral representation (3)
l
(A - C)1/2 Jmax(2). -l,O)dE(>') = h~ J2>' -ldE(A),
l 2
(B - C)1/2 Jmax(l - 2>', O)dE(>.) = 1~- Jl=2):dE(>'),
we can infer that
ran(A - C)1/2 n ran(B - C)1/2 = {O},
which implies
ran(D - C?/2 = {O},
or equivalently D - C = 0, that is, D = C. This proves our claim on maximality.
Now if the infimum A 1\ B in P exists, it must coincide with the maxirnial element
C. This completes the proof. 0
Theorem 2. Under the setting (3), the infimum A 1\ B in P exists if and only if
r r
1 >.dE(>'):2: 1 (1 - >.)dE(>.)
- -
Jo+ Jo+
or
r r
1 >.dE(>.)::; 1 (1 - >.)dE(>.).
- -
Jo+ Jo+
In each case, the infimum coincides with the smaller of the above two and is equal
to 11
rnin(>', 1 - >.)dE(>.).
Here remark
r
1- ).dE(>.) = A· {I - E( {O}) - E( {I})}, (4)
Jo+
t-
(1- >.)dE(>.) = B· {1- E({O}) - E({l})}. (5)
Jo+