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Inequalities and Applications

Conference on Inequalities and Applications, Noszvaj (Hungary), September 2007

herausgegeben von: Catherine Bandle, László Losonczi, Attila Gilányi, Zsolt Páles, Michael Plum

Verlag: Birkhäuser Basel

Buchreihe : ISNM International Series of Numerical Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Inequalities continue to play an essential role in mathematics. Perhaps, they form the last field comprehended and used by mathematicians in all areas of the discipline. Since the seminal work Inequalities (1934) by Hardy, Littlewood and Pólya, mathematicians have laboured to extend and sharpen their classical inequalities. New inequalities are discovered every year, some for their intrinsic interest whilst others flow from results obtained in various branches of mathematics. The study of inequalities reflects the many and various aspects of mathematics. On one hand, there is the systematic search for the basic principles and the study of inequalities for their own sake. On the other hand, the subject is the source of ingenious ideas and methods that give rise to seemingly elementary but nevertheless serious and challenging problems. There are numerous applications in a wide variety of fields, from mathematical physics to biology and economics.

This volume contains the contributions of the participants of the Conference on Inequalities and Applications held in Noszvaj (Hungary) in September 2007. It is conceived in the spirit of the preceding volumes of the General Inequalities meetings held in Oberwolfach from 1976 to 1995 in the sense that it not only contains the latest results presented by the participants, but it is also a useful reference book for both lecturers and research workers. The contributions reflect the ramification of general inequalities into many areas of mathematics and also present a synthesis of results in both theory and practice.

Inhaltsverzeichnis

Frontmatter

Inequalities Related to Ordinary and Partial Differential Equations

Frontmatter

A Rayleigh-Faber-Krahn Inequality and Some Monotonicity Properties for Eigenvalue Problems with Mixed Boundary Conditions

Abstract

An eigenvalue problem is considered whose eigenvalues appear in the interior and on the boundary. It has been shown in [1] that there exists an infinite sequence of positive and an infinite sequence of negative eigenvalues. The lowest positive and the largest negative eigenvalue λ ₁, resp. λ ₋₁ can be characterised by means of a Rayleigh principle. It turns out that among all domains of given volume the ball has the smallest λ ₁. A partial result in this direction is established for λ ₋₁. The proof uses the isoperimetric inequality of Krahn-Bossel-Daners. Some monotonicity properties similar to those for the elastically supported membrane are included.

Catherine Bandle

Lower and Upper Bounds for Sloshing Frequencies

Abstract

The calculation of the frequencies ω for small oscillations of an ideal liquid in a container results in a Steckloff eigenvalue problem. A procedure for calculating lower and upper bounds to the smallest eigenvalues is proposed. For the lower bound computation Goerisch’s generalization of Lehmann’s method is applied, trial functions are constructed with finite elements. Rounding errors are controlled with interval arithmetic.

Henning Behnke

On Spectral Bounds for Photonic Crystal Waveguides

Abstract

For a (d + 1)-dimensional photonic crystal with a linear defect strip (waveguide), we calculate real intervals containing spectrum of the associated spectral problem. If such an interval falls completely into a spectral gap of the unperturbed problem (without defect), this will prove the existence of additional spectrum induced by the waveguide.

B. Malcolm Brown, Vu Hoang, Michael Plum, Ian G. Wood

Real Integrability Conditions for the Nonuniform Exponential Stability of Evolution Families on Banach Spaces

Abstract

Let J be either ℝ or ℝ₊ := [0,∞). We prove that an evolution family U = {U(t, s)}_t≥s∈J which satisfies some natural assumptions is non-uniformly exponentially stable if there exist a positive real number α and a nondecreasing function φ : ℝ₊ → ℝ₊ with φ(t) positive for all positive t and such that for each s ∈ J, the following inequality

$$ \mathop {\sup }\limits_{t > s} \int_0^{t - s} {\varphi (e^{\alpha u} \left\| {U(s + u,s)x} \right\|)du = M_\varphi(s) < \infty } $$

holds true for all x ∈ X with ‖x‖ ≤ 1. We arrive at the same conclusion under the assumption that there exist three positive real numbers α, β and K such that for each t ∈ J the inequality

$$ \left( {\int_J {\chi ( - \infty ,t](\tau )e^{ - q\alpha \tau } \parallel U(t,\tau )^* x^* \parallel } )d\tau } \right)^{\tfrac{1} {q}}\leqslant Ke^{ - \beta t} $$

holds true, for all x* ∈ X* with ‖x*‖ ≤ 1 and for some q ≥ 1.

Constantin Buşe

Validated Computations for Fundamental Solutions of Linear Ordinary Differential Equations

Abstract

We present a method to enclose fundamental solutions of linear ordinary differential equations, especially for a one dimensional Schrödinger equation which has a periodic potential. Our method is based on Floquet theory and Nakao’s verification method for nonlinear equations. We show how to enclose fundamental solutions together with characteristic exponents and give a numerical example.

Kaori Nagatou

Integral Inequalities

Frontmatter

Equivalence of Modular Inequalities of Hardy Type on Non-negative Respective Non-increasing Functions

Abstract

Some weighted modular integrals inequalities with Volterra type operators are considered. The equivalence of such inequalities on the cones on non-negative respective non-increasing functions is established.

Sorina Barza, Lars-Erik Persson

Some One Variable Weighted Norm Inequalities and Their Applications to Sturm-Liouville and Other Differential Operators

Richard C. Brown, Don B. Hinton

Bounding the Gini Mean Difference

Abstract

Some recent results on bounding and approximating the Gini mean difference in which the author was involved for both general distributions and distributions supported on a finite interval are surveyed. The paper supplements the previous work utilising the Steffensen and Karamata type approaches in approximating and bounding the Gini mean difference.

Pietro Cerone

On Some Integral Inequalities

Abstract

An extension of inequalities (1.2) and (1.3) ([1]) is given and an open problem raised in [1] is solved.

Bogdan Gavrea

A New Characterization of the Hardy and Its Limit Pólya-Knopp Inequality for Decreasing Functions

Abstract

In this paper we present and prove a new alternative weight characterization for the Hardy inequality for decreasing functions. We also give an alternative approach to the characterization of the Hardy inequality using a fairly new equivalence theorem. In fact, this result shows that there are infinitely many possibilities to characterize the considered Hardy inequality for decreasing functions. We also state the corresponding weight characterization for the Pólya-Knopp inequality for decreasing functions.

Maria Johansson

Euler-Grüss Type Inequalities Involving Measures

Abstract

An inequality of Grüss type for a real Borel measure μ is proved. Some Euler-Grüss type inequalities are given, by using general Euler identities involving μ-harmonic sequences of functions with respect to a real Borel measure μ.

Ambroz Čivljak, Ljuban Dedić, Marko Matić

The ρ-quasiconcave Functions and Weighted Inequalities

Abstract

We present some facts from a general theory of ρ-quasiconcave functions defined on the interval I = (a, b) ⊆ ℝ and show how to use them to characterize the validity of weighted inequalities involving ρ-quasiconcave operators.

William Desmond Evans, Amiran Gogatishvili, Bohumír Opic

Inequalities for Operators

Frontmatter

Inequalities for the Norm and Numerical Radius of Composite Operators in Hilbert Spaces

Abstract

Some new inequalities for the norm and the numerical radius of composite operators generated by a pair of operators are given.

Silvestru S. Dragomir

Norm Inequalities for Commutators of Normal Operators

Abstract

Let S, T, and X be bounded linear operators on a Hilbert space. It is shown that if S and T are normal with the Cartesian decompositions S = A+iC and T = B+iD such that a ₁ ≤ A ≤ a ₂, b ₁ ≤ B ≤ b ₂, c ₁ ≤ C ≤ c ₂, and d ₁ ≤ D ≤ d ₂ for some real numbers a ₁, a ₂, b ₁, b ₂, c ₁, c ₂, d ₁, and d ₂, then for every unitarily invariant norm |||·|||,

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-7643-8773-0_14/978-3-7643-8773-0_14_Equa_HTML.gif

and

$$ \left\| {ST - TS} \right\| \leqslant \frac{1} {2}\sqrt {(a_2- a_1 )^2+ (c_2- c_1 )^2 } \sqrt {(b_2- b_1 )^2+ (d_2- d_1 )^2 } , $$

where ‖·‖ is the usual operator norm. Applications of these norm inequalities are given, and generalizations of these inequalities to a larger class of nonnormal operators are also obtained.

Fuad Kittaneh

Uniformly Continuous Superposition Operators in the Spaces of Differentiable Functions and Absolutely Continuous Functions

Abstract

Let I, J ⊂ ℝ be intervals. We prove that if a superposition operator H generated by a two place h : I × J → ℝ,

$$ H(\phi )(x): = h(x,\phi (x)), $$

maps the set C ^r(I, J) of all r-times continuously differentiable functions ϕ : I → J into the Banach space C ^r(I, ℝ) and is uniformly continuous with respect to C ^r-norm, then

$$ h(x,y) = a(x)y + b(x), x \in I, y \in J, $$

for some a, b ∈ C ^r(I, ℝ).

For the Banach space of absolutely continuous functions an analogous result is proved.

Janusz Matkowski

Tight Enclosures of Solutions of Linear Systems

Abstract

This paper is concerned with the problem of verifying the accuracy of an approximate solution of a linear system. A fast method of calculating both lower and upper error bounds of the approximate solution is proposed. By the proposed method, it is possible to obtain the error bounds which are as tight as needed. As a result, it can be verified that the obtained error bounds are of high quality. Numerical results are presented elucidating properties and efficiencies of the proposed verification method.

Takeshi Ogita, Shin’ichi Oishi

Inequalities in Approximation Theory

Frontmatter

Operators of Bernstein-Stancu Type and the Monotonicity of Some Sequences Involving Convex Functions

Abstract

By using the operators introduced by D.D. Stancu in 1969, we show that the results obtained in the papers [1], [2] and [5] follow from the properties of these operators. We also present some improvements and generalizations of the results obtained in the above mentioned papers.

Ioan Gavrea

Inequalities Involving the Superdense Unbounded Divergence of Some Approximation Processes

Abstract

Estimations concerning the norm of the approximating functionals associated to some approximation procedures are given, in order to deduce their superdense unbounded divergence.

Alexandru Ioan Mitrea, Paulina Mitrea

An Overview of Absolute Continuity and Its Applications

Abstract

The aim of this paper is to illustrate the usefulness of the notion of absolute continuity in a series of fields such as Functional Analysis, Approximation Theory and PDE.

Constantin P. Niculescu

Generalizations of Convexity and Inequalities for Means

Frontmatter

Normalized Jensen Functional, Superquadracity and Related Inequalities

Abstract

In this paper we generalize the inequality

$$ MJ_n (f,x,q) \geqslant J_n (f,x,p) \geqslant mJ_n (f,x,q) $$

where

$$ J_n (f,x,p) = \sum\limits_{i = 1}^n {p_i f(x_i ) - f\left( {\sum\limits_{i = 1}^n {p_i x_i } } \right)} , $$

obtained by S.S. Dragomir for convex functions. We show that for the class of functions that we call superquadratic, strictly positive lower bounds of J _n (f, x, p)—mJ _n (f, x, q) and strictly negative upper bounds of J _n (f, x, p)∔MJ _n (f, x, q) exist when the functions are also nonnegative. We also provide cases where we can improve the bounds m and M for convex functions and superquadratic functions. Finally, an inequality related to the Čebyšev functional and superquadracity is also given.

Shoshana Abramovich, Silvestru S. Dragomir

Comparability of Certain Homogeneous Means

Abstract

We present some inequalities between two variables homogeneous means. Namely, we give necessary as well as sufficient condition on the comparability of Daróczy means.

Pál Burai

On Some General Inequalities Related to Jensen’s Inequality

Abstract

We present several general inequalities related to Jensen’s inequality and the Jensen-Steffensen inequality. Some recently proved results are obtained as special cases of these general inequalities.

Milica Klaričić Bakula, Marko Matić, Josip Pečarić

Schur-Convexity, Gamma Functions, and Moments

Abstract

The gamma function is a central function that arises in many contexts. A wide class of inequalities is obtained by showing that certain gamma functions are Schur-convex coupled with majorization of two vectors.

Albert W. Marshall, Ingram Olkin

A Characterization of Nonconvexity and Its Applications in the Theory of Quasi-arithmetic Means

Abstract

In this paper, we give necessary and sufficient conditions for the comparison, equality and homogeneity problems of two-variable means of the form

$$ M(A_{\phi ,w_1 } (x,y), \ldots ,A_{\phi ,w_n } (x,y)) (x,y \in I) $$

where M is an n-variable mean on the open interval I and $ A_{\varphi ,w_i } $ denotes the weighted quasi-arithmetic mean generated by a strictly increasing continuous function ϕ : I → ℝ and by a weight function w _i : I ² →]0, 1[. The approach is based on a characterization of lower semicontinuous nonconvex function.

Zoltán Daróczy, Zsolt Páles

Approximately Midconvex Functions

Abstract

Let X be a vector space and let D ⊂ X be a nonempty convex set. We say that a function f is δ-midconvex if

$$ f\left( {\frac{{x + y}} {2}} \right) \leqslant \frac{{f(x) + f(y)}} {2} + \delta\;\;for x,y \in D. $$

We find the smallest function C : [0, 1] ∩ ℚ → ℝ such that for every δ-midconvex function f : D → ℝ the following estimate holds

$$ f(qx + (1 - q)y) \leqslant qf(x) + (1 - q)f(y) + C(q)\delta $$

for x, y ∈ D, q ∈ [0, 1] ⋂ ℚ.

Jacek Mrowiec, Jacek Tabor, Józef Tabor

Inequalities, Stability, and Functional Equations

Frontmatter

Sandwich Theorems for Orthogonally Additive Functions

Abstract

Let p be an orthogonally subadditive mapping, q an orthogonally superadditive mapping such that p ≤ q or q ≤ p. We prove that under some additional assumptions there exists a unique orthogonally additive mapping f such that p ≤ f ≤ q or q ≤ f ≤ p, respectively.

Włodzimierz Fechner, Justyna Sikorska

On Vector Pexider Differences Controlled by Scalar Ones

Abstract

We deal with a functional inequality

$$ \left\| {F(x + y) - G(x) - H(y)} \right\| \leqslant g(x) + h(y) - f(x + y) $$

where F, G, H map a given commutative (semi)group (S, +) into a Banach space and f, g, h : S → ℝ are given scalar functions. This is a pexiderized version of the stability problem

$$ \left\| {F(x + y) - F(x) - F(y)} \right\| \leqslant f(x) + f(y) - f(x + y) $$

examined in connection with the singular case (p = 1) in

$$ \left\| {F(x + y) - F(x) - F(y)} \right\| \leqslant \varepsilon \left( {\left\| x \right\|^p+ \left\| y \right\|^p } \right) $$

We show, among others, that the maps F, G and H have to be, in a sense, close to an additive map provided that the function g + h − 2f is upper bounded.

Roman Ger

A Characterization of the Exponential Distribution through Functional Equations

Abstract

In this paper we give a characterization for the exponential distribution by using functional equations.

Gyula Maksa, Fruzsina Mészáros

Approximate Solutions of the Linear Equation

Abstract

In this paper we obtain a stability result for the general linear equation in Hyers-Ulam sense.

Dorian Popa

On a Functional Equation Containing Weighted Arithmetic Means

Abstract

In this paper we solve the functional equation

$$ \sum\limits_{i = 1}^n {a_i f(\alpha _i x + (1 - \alpha _i )y) = 0} $$

which holds for all x, y ∈ I, where I ⊂ ℝ is a non-void open interval, f : I → ℝ is an unknown function and the weights α _i ∈ (0, 1) are arbitrarily fixed (i = 1, . . ., n). It will be proved that all solutions are generalized polynomials of degree at most n − 2. Furthermore we give a sufficient condition for the existence of nontrivial solutions.

Adrienn Varga, Csaba Vincze

Titel: Inequalities and Applications
herausgegeben von: Catherine Bandle
László Losonczi
Attila Gilányi
Zsolt Páles
Michael Plum
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8773-0
Print ISBN: 978-3-7643-8772-3
DOI: https://doi.org/10.1007/978-3-7643-8773-0