Classical and New Inequalities in Analysis

Frontmatter

Chapter I. Convex Functions and Jensen’s Inequality

Abstract

In this Chapter we give some introductory material and basic inequalities which are repeatedly used. We will begin with convex functions and Jensen’s inequality.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter II. Some Recent Results Involving Means

Abstract

W. Sierpiński [1] proved in 1909, the following inequalities

$${A_n}{(\alpha )^{n - 1}}{H_n}(\alpha )a{G_n}{(a)^n}{A_n}(a){H_n}{(a)^{n - 1}}$$

(1)

. where A _n , G _n and H _n are respectively the arithmetic, geometric, and harmonic means of a sequence α = (α ₁,…, α _n ).

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter III. Bernoulli’s Inequality

Abstract

If x > −1 and if n is a positive integer, then

$$\left( {1.1} \right){\left( {1 + x} \right)^n}1 + nx.$$

(1.1)

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter IV. Cauchy’s and Related Inequalities

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter V. Hölder’s and Minkowski’s Inequalities

Abstract

One of the most important inequalities of analysis is Hölder’s inequality.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter VI. Generalized Hölder and Minkowski Inequalities

Abstract

Let E a nonempty set and L be a linear class of real valued functions g on E having the properties

$$ L1:f,g \in L \Rightarrow \left( {af + bg} \right) \in L\,for\,all\,a,b \in R;$$

$$ L2:1 \in L,that\,is\,if\,\left( t \right) = 1\left( {t \in E} \right),\,then\,f \in L.$$

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter VII. Connections between General Inequalities

Abstract

It is well-known that there exist many connections between general inequalities. Some of these connections were noted in previous chapters. So in Remark 1 of 4. of Chapter V we gave the equivalence

$$H \Leftrightarrow C,$$

(1)

where H is Hölder’s inequality, and C is Cauchy’s inequality.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter VIII. Some Determinantal and Matrix Inequalities

Abstract

In this Chapter we shall give some determinantal and matrix inequalities connected to the inequalities given in previous chapters.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter IX. Čebyšev’s Inequality

Abstract

In the case of Čebyšev’s inequality there are two recent reviews by D. S. Mitrinović and P. M. Vasić [1], respectively, D. S. Mitrinović and J. E. Pečarić [2] which trace completely the historical and chronological development of Čebyšev’s and related inequalities. These studies are important because in many instances incorrect quotations of results — sometimes by very distinguished mathematicians — have been uncritically transferred from book to book and paper to paper. Thus, there are many “attributions fauses” regarding Čebyšev’s inequality in the literature. Some results have been rediscovered several times and there is a substantial number of papers which offer apparent generalizations of the inequality as new results.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter X. Grüss’ Inequality

Abstract

A whole sequence of inequalities exist which give complements of the Čebyšev’s inequality (see Chapter IX). These are the results which give estimates for the Čebyšev’s quotient or difference on both sides.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XI. Steffensen’s Inequality

Abstract

An inequality first proved by J. F. Steffensen [1] in 1918 is the subject of this Chapter. It is curious that this inequality is not included in the monograph of G. H. Hardy, J. Littlewood, and G. Pólya [2] Steffensen’s paper was not reviewed in Jahrbuch über die Fortschritte der Matematik. It is however, mentioned by G. Szegö in his review of the papers [3] and [4] by T. Hayashi.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XII. Abel’s and Related Inequalities

Abstract

General linear inequalities are old inequalities. We are not sure who is the author of such inequalities. So we shall given here only some basic facts about such inequalities but only for monotonic functions and some related results.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XIII. Some Inequalities for Monotone Functions

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XIV. Young’s Inequality

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XV. Bessel’s Inequality

Abstract

Let us consider the problem of the best approximation of a vector x by vectors of an orthonormal system from a Hilbert space X. For every system of numbers λ₁,...,λ₂ we have

$$||x - \sum\limits_{k = 1}^n {{\lambda _k}{x_k}|{|^2}} \geqslant 0.$$

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XVI. Cyclic Inequalities

Abstract

The type of inequality studied in this Chapter are inequalities for forms which are symmetric in several variables. The simplest is Schur’s inequality. We end with some related inequalities.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XVII. Triangle Inequalities

Abstract

The triangle inequality for real and complex numbers are basic and appear in any analysis book.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XVIII. Norm Inequalities

Abstract

In this Chapter we look at inequalities for norms which are related to the triangle inequality. Several of these are attached to the names, e.g. Clarkson’s, Dunkl-Williams’ and Hlawka’s.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XIX. More on Norm Inequalities

Abstract

H. Xu and Z. Xu [1] claimed the following inequality in L ^p (1 < p < 2):

$${\left( {\left. {\parallel \lambda x + \mu y{\parallel ^2} + g\left( \mu \right)\parallel x - y{\parallel ^2}} \right)} \right.^{1/2}}{\left( {\lambda \parallel x{\parallel ^p} + \mu \parallel y{\parallel ^p}} \right)^{1/p}}$$

(1.1)

where x,y ∈ L ^p , 0 ≤ µ ≤ 1, λ = 1 − µ, in the case when the function g is given by g(µ) = µ λ(p − 1).

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XX. Gram’s Inequality

Abstract

Let x ₁,...,x _n be vectors of a unitary space X. Then

$$G\,\left( {{x_1}\,,\,...\,,{x_n}} \right)\, = \,\left[{\begin{array}{*{20}{c}}{\left( {{x_1},\,{x_n}} \right)\,...\,\left( {{x_1}\,,\,{x_n}} \right)} \\ \vdots \\ {\left( {{x_1}\,,\,{x_n}} \right)\,...\,\left( {{x_n}\,,{x_n}} \right)} \end{array}} \right]$$

is called the Gram matrix of the vectors x ₁,...,x _n .

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXI. Fejér-Jackson’s Inequalities and Related Results

Abstract

This Chapter is devoted to trigonometric polynomials and series of the form

$$\sum {{C_v}} {e^{vxi}},\;or\;\sum {{C_v}\sin vx} ,\;or\;\sum {{C_v}\cos vx} ,$$

with the assumption that their coefficients C _v , are positive and monotonic, e.g. for some p ≥ 0, they are p-monotone (C _n ≥ 0, ∆C _n ≤ 0, ... , (−1) ^p ∆ ^p C _n ≥ 0).

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXII. Mathieu’s Inequality

Abstract

Let c be a positive number, and

$$ S = \sum\limits_{n = 1}^\infty {\frac{{2n}} {{n^2 + \left. {c^2 } \right)^2 }}.} $$

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXIII. Shannon’s Inequality

Abstract

The notion of the Shannon entropy appears frequently and is important in many works. In this Chapter we will review some of the characterizations of it and of the concept of the gain of information with functional inequalities. Similarly, we shall present a characterization of Rényi’s generalized concept of information measure and gain of information with the aid of functional inequalities. These inequalities, to be discussed, have also other interpretations.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXIV. Turán’s Inequalities from the Power Sum Theory

Abstract

We will quote some of Turán’s inequalities and applications in this Chapter. His work and others is exposed in the monograph [1]. Here we will give a short introduction to the types of inequalities and applications one finds in that book and elsewhere. We will not give extensive references in this Chapter as we do in others, since [1] contains a rather complete bibliography up to the date of the book, 1984. The monograph [1] is an enlargement of the monograph: P. Turán, “Eine neue Methode in der Analysis and deren Anwendungen”, Budapest, 1953.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXV. Continued Fractions and Padé Approximation Method

Abstract

The modern theory of (infinite) continued fractions probably begins with Bombelli (1526–1672) [1] in which he computes the square roots of numbers by the following device.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXVI. Quasilinearization Methods for Proving Inequalities

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXVII. The Centroid Method in Inequalities

Abstract

The concept of the centroid, introduced most likely by Archimedes, can be applied in solving various Mathematical problems. We mention, for example, the papers of K. F. Gauss [1] and L. Fejér [2]. Here we shall give a chronological account of the use of the centroid in developing inequalities, pointing to some priorities which are neglected in the literature.

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXVIII. Dynamic Programming and Functional Equation Approaches to Inequalities

Abstract

The development of dynamic programming was closely related to the theory of inequalities. E. F. Beckenbach and R. Bellman [1] proved the arthmetic-geometric mean inequality through the functional equation approach in dynamic programming (see also [2]).

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXIX. Interpolation Inequalities

Abstract

The purpose of this Chapter is to give some results which extend known inequalities in a systematic way by interpolating the extremes. As a simple example we cite the AG inequality

$$\sum\limits_1^n {{p_i}} {a_i}\prod\limits_1^n {a_i^{{p_i}}} ,$$

(1.1)

with \(\sum\limits_1^n {{p_i}} = 1,{p_i}0,{a_i}0.\)

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

Chapter XXX. Convex Minimax Inequalities-Equalities

Abstract

The possibility of the interchange of the operations of sup and inf in a function of several variables is an important issue. For example, the main theorem of matrix game theory is the existence of a point (x ₀, y ₀) such that

$${x^T}A{y_0}x_0^TA{y_0}x_0^T{A_y}$$

for all x, y such that \({x_0}0,{y_i}0,\sum\limits_{i = 1}^n {{x_i} = 1} \), and \(\sum\limits_{i = 1}^n {{y_i} = 1} \).

D. S. Mitrinović, J. E. Pečarić, A. M. Fink

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