Table Of ContentRecent Progress in Inequalities
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centrefor Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 430
Recent Progress
in Inequalities
edited by
G. V. Milovanovic
University of Nis,
Faculty of Electronic Engineering,
Nis, Yugoslavia
Springer Science+Business Media, LLC
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4945-2 ISBN 978-94-015-9086-0 (eBook)
DOI 10.1007/978-94-015-9086-0
Printed on acid-free paper
All Rights Reserved
© 1998 Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1998
Softcover reprint of the hardcover 1s t edition 1998
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,
inc\uding photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
This Volume is Dedicated to
s.
Professor Dragoslav Mitrinovic
(1908 - 1995)
Table of Contents
Preface ................................................................... xi
Life and Inequalities: D. S. Mitrinovic (1908-1995)
G. V. Milovanovic ........................................................ 1
Publications of D. S. Mitrinovic
R. Z. Djordjevic and R. R. Janic 11
Invited Papers
Complex Polynomials and Maximal Ranges: Background and Applications
V. V. Andrievskii and S. Ruscheweyh ..................................... 31
Exact Classical Polynomial Inequalities in Hp for 0 ~ p ~ 00
V. V. Arestov ............................................................ 55
Vietoris's Inequalities and Hypergeometric Series
R. Askey ................................................................. 63
Inequalities for Norms of Intermediate Derivatives and Some Their
Applications
V. F. Babenko .................................... . . . . . . . . . . . . . . . . . . . . . . . 77
Table of Inequalities in Elliptic Boundary Value Problems
C. Bandie and M. Flucher ................................................ 97
A Catalogue of Help and Help-type Integral and Series Inequalities
M. Benammar, C. Bennewitz, M. J. Beynon, B. M. Brown, N. G. J. Dias,
W. D. Evans, W. N. Everitt, V. G. Kirby, and L. L. Littlejohn ........... 127
Remarks of the Jackson and Whitney Constants
B. Bojanov ............................................................... 161
On the Application of the Peano Representation of Linear Functionals
in Nu merical Analysis
H. Brass and K.-J. Förster ............................................... 175
Inequalities Due to T. S. Nanjundiah
P. S. Bullen .............................................................. 203
Marcinkiewicz-Zygmund Inequalities: Methods and Results
D. S. Lubinsky ........................................................... 213
Shapiro's Inequality
A. M. Fink ............................................................... 241
Bernstein Type Inequalities for Rational Functions With Prescribed Poles
N. K. Govil and R. N. Mohapatra ........................................ 249
vii
viii TABLE OF CONTENTS
Some Generalisations and Refinements of the Hardy Inequality
H. Heining, A. Kufner, and L. E. Persson ................................ 271
Discrete Inequalities of Wirtinger's Type
G. V. Milovanovic and 1. Z. Milovanovic ................................. 289
Convexity Properties of Special Functions and Their Zeros
M. E. Muldoon ........................................................... 309
Inequalities in Circular Arithmetie: A Survey
Lj. D. Petkovic and M. S. Petkovic ....................................... 325
Properties of Isometries and Approximate Isometries
Th. M. Rassias ........................................................... 341
Inequalities for the Zeros of an Orthogonal Expansion of a Polynomial
G. Schmeisser ............................................................ 381
Error Inequalities for Discrete Hermite and Spline Interpolation
P. J. Y. Wong and R. P. Agarwal ........................................ 397
Contributed Papers
An Inequality Concerning Symmetrie Functions and Some Applications
D. Andrica and L. Mare .................................................. 425
A Note on the Second Largest Eigenvalue of Star-like Trees
F. K. Bell and S. K. Simic ............................................... 433
Refinements of Ostrowski's and Fan-Todd's Inequalities
M. Bjelica ................................................................ 445
On the Stability of the Quadratie Functional Equation and Related Topies
S. Czerwik ............................................................... 449
A Diriehlet-type Integral Inequality
W. N. Everitt ............................................................ 457
On the Hyers-Ulam-Rassias Stability of Mappings
P. Gavruta ............................................................... 465
Functions With Quasieonvex Derivatives
V. Govedarica and M. Jovanovic ......................................... 471
Local Approximation by Quasi-polynomials
Yu. Kryakin ............................................................. 475
Logarithmic Concavity of Distribution Functions
M. Merkle ................................................................ 481
Sharpening of Cauchy Inequality
Z. Mijalkovic and M. Mijalkovic .......................................... 485
TABLE OF CONTENTS ix
A Note on the Least Constant in Landau Inequality on a Finite Interval
A. Yu. Shadrin ........................................................... 489
Some Inequalities Involving Harmonie Numbers
M. S. Stankovic, B. M. Dankovic, and S. B. Trickovic .................... 493
Inequalities for Polynomials in Lo Norm
E. A. Storozenko ......................................................... 499
Some Inequalities for Altitudes and Other Elements of Triangle
M. R. Ziiovic and M. R. Stevanovic ...................................... 505
Author Index ............................................................. 511
Preface
This volume is dedieated to Professor Dragoslav S. Mitrinovic (1908-1995), one
of the most accomplished masters in the domain of inequalities. Inequalities are
everywhere and play an important and significant role in almost all subjects of
mathematies including other areas of sciences. Professor Mitrinovic often used to
say: "There are no equalities, even in the human life, the inequalities are always
met".
Inequalities present a very active and attractive field of research. As Richard
Bellman has so elegantly said at the Second International Conference on General
Inequalities (Oberwolfach, July 30 - August 5, 1978): "There are three reasons for
the study of inequalities: praetieal, theoretieal, and aesthetie." On the aesthetie
aspects he said: "As has been pointed out, beauty is in the eyes of the beholder.
However, it is generally agreed that eertain pieees of musie, art, or mathematies
are beautiful. There is an eleganee to inequalities that makes them very attraetive. "
A great progress in inequalities was made by seven Oberwolfach conferences on
inequalities with the corresponding seven volumes under the title General Inequal
ities 1 - 7, published by Birkhäuser (1978, 1980, 1983, 1984, 1987, 1992, and
1997), as weIl as by several other international conferences dedieated to inequali
ties. One of these conferences was held in 1987 at the University of Birmingham,
England, under the auspices of the London Mathematical Society, and dedieated
to the work of G. H. Hardy, J. E. Littlewood and G. P6lya in writing the book
Inequalities, whieh was first published by the Cambridge University Press in 1934.
This book has to be counted as one of the outstanding achievements in mathemat
ical scholarship in this century, as said Norrie Everitt in the Preface of the volume
Inequalities - Fifty years on /rom Hardy, Littlewood and P61ya (Marcel Dekker,
1991). Norrie said also: "Of great intrinsie interest, indeed, faseination, the book
has proved an invaluable referenee work for more than fifty years, and a souree
of lasting inspiration to workers in the vineyard of inequalities." Until the early
sixties only this classieal work intended to transform the field of inequalities from
a collection of isolated formulas into a systematie discipline. Since that time, other
books on inequalities have appeared, especially two Springer's: Inequalities (1971)
by E. F. Beckenbach and R. Bellman, and Analytic Inequalities (1970) by D. S.
Mitrinovic. After the classieal Inequalities by Hardy, Littlewood and P6lya, this
MitrinoviC's famous work is the most referred to books in the field of inequalities.
Mitrinovic was interested in all kinds of inequalities, from elementary inequali
ties, geometrie inequalities, inequalities with means, inequalities in analysis and
approximation theory, including inequalities in number theory. In collaboration
with fellow colleagues he produced several books in different subjects concerning
inequalities during the last ten years of his life. Five of them have been pub
lished by Kluwer: Means and Their Inequalities (1988) with P. S. Bullen and
P. M. Vasic, Reeent Advanees in Geometrie Inequalities (1989) with J. E. Pecaric
and V. Volenec, Inequalities Involving Functions and Their Integrals and Deriva
tives (1991) with J. E. Pecaric and A. M. Fink, Classieal and New Inequalities in
xi
xii PREFACE
Analysis (1993) with J. E. Pecaric and A. M. Fink, Handbook 0/ Number Theory
(1986) with J. Sandor and B. Crstici, and one book was published by World Scien
tifie: Topics in Polynomials: Extremal Problems, Inequalities, Zeros (1994) with
G. V. Milovanovic and Th. M. Rassias.
In order to provide a multi-disciplinary forum of discussion in mathematics and its
applications in which the essentiality of inequalities is highlighted, a new journal
with title Journal 0/ Inequalities and Applications is just started this year by
Gordon and Breach Science Publishers.
An International Memorial Conference dedicated to the late Professor D. S. Mitri
novic was held at the Faculty of Electronic Engineering, University of NiS, Yu
goslavia, from June 20-22, 1996. This conference was organised by the foHowing
institutions: The Serbian Scientific Society (Belgrade), The Mathematics Institute
of Serbian Academy of Sciences and Arts (Belgrade ), Faculty of Electrical Engi
neering (Belgrade), and Faculty of Electronic Engineering (Nis). There were 93
participants from 17 countries and the work on the conference was organised in
the foHowing three sections: Recent Progress in Inequalities, Advances in Mathe
matical Analysis, and Topics in Mathematics with Applications. More than 140
authors sent their survey and contributed papers to the Program Committee. Af
ter a refereeing process, a number of selected papers on inequalities are included
in this volume. Ten members of the Editorial Board of the Journal o/Inequalities
and Applications appear as authors in this volume.
This book is divided into three sections: An introduction to the life and scientific
work of Professor Mitrinovic, Invited Papers, and Contributed Papers. In each
section the papers appear in alphabeticalorder according to the initial of the last
name of the first-named author. An author index is also included at the end of
the book.
Lastly, I wish to express my warmest thanks to all of the scientists who contributed
to this volume, as weH as to all of my coHeagues from the Department of Math
ematics, University of Nis, who helped in the preparation of this volume. The
financial support for preparing this book is given by Nis Assembly. It is, also, a
pleasure to acknowledge the superb assistance that the staff of Kluwer Academic
Publishers provided.
Nis, June 1997 Gradimir V. Milovanovic
