A. M. Fink

Classical and New Inequalities in Analysis

Preface. Organization of the Book. Notations. I. Convex Functions and Jensens Inequality. II. Some Recent Results Involving Means. III. Bernoullis Inequality. IV. Cauchys and Related Inequalities. V. Holder and Minkowski Inequalities. VI. Generalized Holder and Minkowski Inequalities. VII. Connections Between General Inequalities. VIII. Some Determinantal and Matrix Inequalities. IX. Cebysevs Inequality. X. Gruss Inequality. XI. Steffensens Inequality. XII. Abels and Related Inequalities. XIII. Some Inequalities for Monotone Functions. XIV. Youngs Inequality. XV. Bessels Inequality. XVI. Cyclic Inequations. XVII. The Centroid Method in Inequalities. XVII. Triangle Inequalities. XVIII. Norm Inequalities. XIX. More on Norm Inequalities. XX. Grams Inequality. XXI. Frejer-Jacksons Inequalities and Related Results. XXII. Mathieus Inequality. XXIII. Shannons Inequality. XXIV. Turans Inequality from the Power Sum Theory. XXV. Continued Fractions and Pade Approximation Method. XXVI. Quasilinearization Methods for Proving Inequalities. XXVIII. Dynamic Programming and Functional Equation Approaches to Inequalities. XXIX. Interpolation Inequalities. XXX. Minimax Inequalities. Name Index.

Almost periodic differential equations

Almost periodic functions.- Uniformly almost periodic families.- The fourier series theory.- Modules and exponents.- Linear constant coefficient equations.- Linear almost periodic equations.- Exponential dichotomy and kinematic similarity.- Fixed point methods.- Asymptotic almost periodic functions and other weaker conditions.- Separated solutions.- Stable solutions.- First order equations.- Second order equations.- Averaging.

Inequalities involving functions and their integrals and derivatives

I. Landau-Kolmogorov and related inequalities.- II. An inequality ascribed to Wirtinger and related results.- III. Opials inequality.- IV. Hardys, Carlemans and related inequalities.- V. Hilberts and related inequalities.- VI. Inequalities of Lyapunov and of De la Vallee Poussin.- VII. Zmorovi?s and related inequalities.- VIII. Carlsons and related inequalities.- IX. Inequalities involving kernels.- X. Convolution, rearrangement and related inequalities.- XI. Inequalities of Caplygin type.- XII. Inequalities of Gronwall type of a single variable.- XIII. Gronwall inequalities in higher dimension.- XIV. Gronwall inequalities on other spaces: discrete, functional and abstract.- XV. Integral inequalities involving functions with bounded derivatives.- XVI. Inequalities of Bernstein-Mordell type.- XVII. Methods of proofs for integral inequalities.- XVIII. Particular inequalities.- Name Index.

Kolmogorov-Landau inequalities for monotone functions

where llfll = ess su~-~~~~~ If(t Kolmogorov [2] suceeded in finding the best possible constants in (1) where the norm is as in (2). Except for isolated results, no new progress was made until Schoenberg and Cavaretta [3] found the best possible constants for llfll = su~~>~ If(f We propose to prove inequalities about various clases of monotone functions which yield, as special cases, pointwise inequalities of the form (1). For example,

Liapunov functions and almost periodic solutions for almost periodic systems

Most of the well-known conditions for the existence of almost periodic solutions of systems of ordinary differential equations with almost periodic time dependence involve stability conditions of some sort on a bounded solution; cf. [I], [2], for examples. Although a very general condition developed by L. Amerio [3] does not explicitly involve stability concepts, its use usually involves imposing stability conditions, usually of global type, on bounded solutions. While recently an approach to the existence problem via dynamical systems has yielded conditions in terms of local stability [2], it has also been known that global stability of a conditional type is sufficient for the use of America’s theorem [4], [5]; such conditional stability does not seem to be easily applicable to the dynamical system approach. It is the purpose of this paper to develop conditions, in terms of Liapunov functions, sufficient for the uniqueness of bounded solutions of systems. These conditions essentially impose a type of conditional stability on such bounded solutions, and by use of Amerio’s theorem these again yield sufficient conditions that such bounded solutions be almost periodic. Hence, the Liapunov functions we use are not assumed positive definite near the origin. In fact, our hypotheses in these Liapunov functions are along the lines of those given by LaSalle for certain stability results for autonomous systems [7].

Positive almost periodic solutions of some delay integral equations

The delay integral equation x(t) = ∝tt − τ f(s, x(s)) ds which arises in models for the spread of epidemics, is studied with the aim of establishing the existence of positive almost periodic solutions for large values of τ when f(t, x) is uniformly almost periodic in t for x in compact subsets of R+. Under reasonable assumptions on f it is shown that there exist two positive numbers τ∗ τ0 they do exist. A priori bounds on the set of positive solutions and uniqueness results are also obtained.

Discrete Inequalities of Generalized Wirtinger Type

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Jensen inequalities for functions with higher monotonicities

SummaryWe investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)ϕ(∫f(x) dμ) ⩽ A ∫ ϕ(f(x) dμ, d∫ dμ = 1. Iff is an arbitrary nonnegativeLx function, this holds ifμ ⩾ 0,ϕ is convex andA = 1. Iff is monotone the measure μ need not be positive for (J) to hold for all convex ϕ withA = 1. If ϕ has higher monotonicity, e.g., ϕ′ is also convex, then we get a version of (J) withA < 1 and measures μ that need not be positive.

Conjugate Inequalities for Functions and Their Derivatives

We consider finding the best possible constants

Convergence and Almost Periodicity of Solutions of Forced Lienard Equations

C = C(n,alpha ,beta ,p,q)