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Recent 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
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