Refinements of some classical inequalities via superquadraticity

The values of p and q for Lp(Lq) that satisfy the extension of Paley and Hardy inequalities for vector-valued HI functions are characterized. In particular, it is shown that L2(L1) is a Paley space that fails Hardy inequality. INTRODUCTION In [BP] the vector-valued analogue of two classical inequalities in the theory of Hardy spaces were investigated. A complex Banach space X is said to be a Paley space if /00 1/2 (P) (E IIf(2k)II2 ? CIfflI for all f E H1(X). k=0 A complex Banach space X is said to verify vector-valued Hardy inequality (for short X is a (HI)-space) if (H) IIf(n)I' < Cllfl II for all f e H1 (X), n=O where H1(X) = {f E L1(T, X): f(n) = O for n < O}. Both inequalities can be regarded in the framework of vector-valued extensions of multipliers from H1 to II . Recall that a sequence (mn) is a (H1 -1')multiplier, to be denoted by mn E (H1 /1), if Tmn (f) = (f(n)mO) defines a bounded operator from H1 into II . The (H1 I1)-multipliers were characterized by C. Fefferman in the following way (see [SW] for a proof):

Improving interpolated Hardy and trace Hardy inequalities on bounded domains

We use the properties of superquadratic functions to produce various improvements and popularizations on time scales of the Hardy form inequalities and their converses. Also, we include various examples and interpretations of the disparities in the literature that exist. In particular, our findings can be seen as refinements of some recent results closely linked to the time-scale inequalities of the classical Hardy, Pólya-Knopp, and Hardy-Hilbert. Some continuous inequalities are derived from the main results as special cases. The essential results will be proved by making use of some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality on time scales.

The problem of estimating functions on the basis of indirect measurements is a universal problem of Statistical Learning Theory. In this paper, some important inequalities on Sugeno measure space are presented and proven, such as Hölder's inequality, Minkowski inequality, and Jensen's inequality. Furthermore, a useful theorem about the problem of estimating functions on the basis of indirect measurements on Sugeno measure space is given and proven by using Minkowski inequality and Jensen's inequality.

Preface. Organization of the Book. Notations. I. Convex Functions and Jensen's Inequality. II. Some Recent Results Involving Means. III. Bernoulli's Inequality. IV. Cauchy's and Related Inequalities. V. Hoelder and Minkowski Inequalities. VI. Generalized Hoelder and Minkowski Inequalities. VII. Connections Between General Inequalities. VIII. Some Determinantal and Matrix Inequalities. IX. Cebysev's Inequality. X. Gruss' Inequality. XI. Steffensen's Inequality. XII. Abel's and Related Inequalities. XIII. Some Inequalities for Monotone Functions. XIV. Young's Inequality. XV. Bessel's Inequality. XVI. Cyclic Inequations. XVII. The Centroid Method in Inequalities. XVII. Triangle Inequalities. XVIII. Norm Inequalities. XIX. More on Norm Inequalities. XX. Gram's Inequality. XXI. Frejer-Jackson's Inequalities and Related Results. XXII. Mathieu's Inequality. XXIII. Shannon's Inequality. XXIV. Turan's 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.

We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with best constants. Besides, we improve two Rellich type inequalities by using the improved Hardy type inequality.

The present paper is devoted to the study of Jensen type inequalities for fractional integration on finite subintervals of the real axis. The complete form of Jensen's inequality and the generalized Jensen's inequality are investigated by using the Chebyshev inequality. As applications, some new integral inequalities, including Holder's and Minkowski's inequalities, are obtained by using Jensen's inequality via Riemann-Liouville fractional integrals.

This survey deals with inequalities satisfied by γ-quasiconvex functions which are one of the many variants of convex functions. The γ-quasiconvex functions have already been dealt with by S. Abramovich, L.-E. Persson and N. Samko. Among the applications we demonstrate here are Jensen, Hardy, Holder, Minkowski, Jensen-Steffensen and Slater-Pecaric inequalities. These inequalities can be seen as extensions and refinements of inequalities satisfied by convex functions.

Classical and new inequalities in analysis : D.S. Mitrinovic,J.E. Pecaric and A.M. Fink,Kluwer, Dordrecht, 1992. 725 pp., Dfl.425, US$285, UK£170.50, ISBN 0-792-32064-6

First we present and discuss an important proof of Hardy's inequality via Jensen's inequality which Hardy and his collaborators did not discover during the 10 years of research until Hardy finally proved his famous inequality in 1925. If Hardy had discovered this proof, it obviously would have changed this prehistory, and in this article the authors argue that this discovery would probably also have changed the dramatic development of Hardy type inequalities in an essential way. In particular, in this article some results concerning power-weight cases in the finite interval case are proved and discussed in this historical perspective. Moreover, a new Hardy type inequality for piecewise constant p = p(x) is proved with this technique, limiting cases are pointed out and put into this frame. Mathematics Subject Classification: 26D15.

We obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.

Preface. Lyapunov Inequalities and their Applications R.C. Brown, D.B. Hinton. Classical Hardy's and Carleman's Inequalities and Mixed Means A. Cizmesija, J. Pecaric. Operator Inequalities Associated with Jensen's Inequality F. Hansen. Hardy-Littlewood-type Inequalities and their Factorized Enhancement L. Leindler. Shannon's and Related Inequalities in Information Theory M. Matic, C.E.M. Pearce, J. Pecaric. Inequalities for Polynomial Zeros G.V. Milovanovic, T.M. Rassias. On Generalized Shannon Functional Inequality and its Applications P.K. Sahoo. Weighted Lp-norm Inequalities in Convolutions S. Saitoh. Index. List of Tables: Table 1. Sample Table. Table 2. Sample table caption. Table 3. Sample Table. Table 4. Sample table caption.

The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond- derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen's inequality on time scales via the diamond- integral and present some corollaries, including Holder's and Minkowski's diamond- integral inequalities.

Generalized pseudo-integral Jensen's inequality for ((⊕1,⊗1),(⊕2,⊗2))-pseudo-convex functions

We present a unified approach to improved $L^p$ Hardy inequalities in $\R^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where distance is taken from a surface of codimension $1<k<N$. In our main result we add to the right hand side of the classical Hardy inequality, a weighted $L^p$ norm with optimal weight and best constant. We also prove non-homogeneous improved Hardy inequalities, where the right hand side involves weighted L^q norms, q \neq p.