Hardy, Littlewood and P6lya's famous monograph on inequalities [17J has served as an introduction to hard analysis for many mathema ticians. Some of its most interesting results center around Hilbert's inequality and generalizations. This family of inequalities determines the best bound of a family of operators on /p. When such inequalities are restricted only to finitely many variables, we can then ask for the rate at which the bounds of the restrictions approach the uniform bound. In the context of Toeplitz forms, such research was initiated over fifty years ago by Szego [37J, and the chain of ideas continues to grow strongly today, with fundamental contributions having been made by Kac, Widom, de Bruijn, and many others. In this monograph I attempt to draw together these lines of research from the point of view of sharpenings of the classical inequalities of [17]. This viewpoint leads to the exclusion of some material which might belong to a broader-based discussion, such as the elegant work of Baxter, Hirschman and others on the strong Szego limit theorem, and the inclusion of other work, such as that of de Bruijn and his students, which is basically nonlinear, and is therefore in some sense disjoint from the earlier investigations. I am grateful to Professor Halmos for inviting me to prepare this volume, and to Professors John and Olga Todd for several helpful comments. Philadelphia, Pa. H.S.W.

Survey on Classical Inequalities provides a study of some of the well known inequalities in classical mathematical analysis. Subjects dealt with include: Hardy-Littlewood-type inequalities, Hardy's and Carleman's inequalities, Lyapunov inequalities, Shannon's and related inequalities, generalized Shannon functional inequality, operator inequalities associated with Jensen's inequality, weighted Lp -norm inequalities in convolutions, inequalities for polynomial zeros as well as applications in a number of problems of pure and applied mathematics. It is my pleasure to express my appreciation to the distinguished mathematicians who contributed to this volume. Finally, we wish to acknowledge the superb assistance provided by the staff of Kluwer Academic Publishers. June 2000 Themistocles M. Rassias Vll LYAPUNOV INEQUALITIES AND THEIR APPLICATIONS RICHARD C. BROWN Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA. email address:[email protected] DON B. HINTON Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA. email address: [email protected] Abstract. For nearly 50 years Lyapunov inequalities have been an important tool in the study of differential equations. In this survey, building on an excellent 1991 historical survey by Cheng, we sketch some new developments in the theory of Lyapunov inequalities and present some recent disconjugacy results relating to second and higher order differential equations as well as Hamiltonian systems. 1. Introduction Lyapunov's inequality has proved useful in the study of spectral properties of ordinary differential equations. Typical applications include bounds for eigenvalues, stability criteria for periodic differential equations, and estimates for intervals of disconjugacy.

This memoir describes a new way of looking at the classical inequalities. The most famous such results, (those of Hilbert, Hardy, and Copson) may be interpreted as inclusion relationships, l[superscript italic]p [subset equality symbol] [italic capital]Y, between certain (Banach) sequence spaces, the norm of the injection being the best constant of the particular inequality. The inequalities of Hilbert, Hardy, and Copson all share the same space [italic capital]Y. That space -- alias [italic]ces([italic]p) -- is central to many celebrated inequalities, and thus is studied here in considerable detail.

Nullane de tantis gregibus tibi digna videtur? rara avis in terra nigroque simillima cygno. Juvenal Sat. VI 161, 165. 1966-JNC visits AN at CornelI. An idea emerges. 1968-JNC is at V. c. L.A. for the Logic Year. The Los Angeles ma- script appears. 1970-AN visits JNC at Monash. 1971-The Australian manuscript appears. 1972-JNC visits AN at Cornell. Here is the result. We gratefully acknowledge support from Cornell Vniversity, Vni versity of California at Los Angeles, Monash Vniversity and National Science Foundation Grants GP 14363, 22719 and 28169. We are deeply indebted to the many people who have helped uso Amongst the mathe maticians, we are particularly grateful to J.C.E. Dekker, John Myhill, Erik Ellentuck, Peter AczeI, Chris Ash, Charlotte ehell, Ed Eisenberg, Dave Gillam, Bill Gross, Alan Hamilton, Louise Hay, Georg Kreisel, Phil Lavori, Ray Liggett, Al Manaster, Michael D. Morley, Joe Rosen stein, Graham Sainsbury, Bob Soare and Michael Venning. Last, but by no means least, we thank Anne-Marie Vandenberg, Esther Monroe, Arletta Havlik, Dolores Pendell, and Cathy Stevens and the girls of the Mathematics Department of VCLA in 1968 for hours and hours of excellent typing. Thanksgiving November 1972 J.N. Crossley Ithaca, New Y ork Anil Nerode Contents O. Introduction ... 1 Part 1. Categories and Functors 3 1. Categories ... 3 2. Morphism Combinatorial Functors 3 3. Combinatorial Functors ... 18 Part H. Model Theory . . 18 4. Countable Atomic Models 18 5. Copying . 22 6. Dimension ... 26 Part III.

In the last few decades the theory of ordinary differential equations has grown rapidly under the action of forces which have been working both from within and without: from within, as a development and deepen ing of the concepts and of the topological and analytical methods brought about by LYAPUNOV, POINCARE, BENDIXSON, and a few others at the turn of the century; from without, in the wake of the technological development, particularly in communications, servomechanisms, auto matic controls, and electronics. The early research of the authors just mentioned lay in challenging problems of astronomy, but the line of thought thus produced found the most impressive applications in the new fields. The body of research now accumulated is overwhelming, and many books and reports have appeared on one or another of the multiple aspects of the new line of research which some authors call" qualitative theory of differential equations". The purpose of the present volume is to present many of the view points and questions in a readable short report for which completeness is not claimed. The bibliographical notes in each section are intended to be a guide to more detailed expositions and to the original papers. Some traditional topics such as the Sturm comparison theory have been omitted. Also excluded were all those papers, dealing with special differential equations motivated by and intended for the applications.