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Inequalities and Applications 2010

Dedicated to the Memory of Wolfgang Walter

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

Verlag: Springer 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 arise as an essential component in various mathematical areas. Besides forming a highly important collection of tools, e.g. for proving analytic or stochastic theorems or for deriving error estimates in numerical mathematics, they constitute a challenging research field of their own. Inequalities also appear directly in mathematical models for applications in science, engineering, and economics.

This edited volume covers divers aspects of this fascinating field. It addresses classical inequalities related to means or to convexity as well as inequalities arising in the field of ordinary and partial differential equations, like Sobolev or Hardy-type inequalities, and inequalities occurring in geometrical contexts. Within the last five decades, the late Wolfgang Walter has made great contributions to the field of inequalities. His book on differential and integral inequalities was a real breakthrough in the 1970’s and has generated a vast variety of further research in this field. He also organized six of the seven “General Inequalities” Conferences held at Oberwolfach between 1976 and 1995, and co-edited their proceedings. He participated as an honorary member of the Scientific Committee in the “General Inequalities 8” conference in Hungary. As a recognition of his great achievements, this volume is dedicated to Wolfgang Walter’s memory. The “General Inequalities” meetings found their continuation in the “Conferences on Inequalities and Applications” which, so far, have been held twice in Hungary. This volume contains selected contributions of participants of the second conference which took place in Hajdúszoboszló in September 2010, as well as additional articles written upon invitation. These contributions reflect many theoretical and practical aspects in the field of inequalities, and will be useful for researchers and lecturers, as well as for students who want to familiarize themselves with the area.

Inhaltsverzeichnis

Frontmatter

Boundary Value Problems

Frontmatter

Domain Derivatives for Energy Functionals with Boundary Integrals

Abstract

This paper deals with domain derivatives of energy functionals related to elliptic boundary value problems. Emphasis is put on boundary conditions of mixed type which give rise to a boundary integral in the energy. A formal computation for rather general functionals is given. It turns out that in the radial case the first derivative vanishes provided the perturbations are volume preserving. In the simplest case of a torsion problem with Robin boundary conditions, the sign of the first variation shows that the energy is monotone with respect to domain inclusion for nearly circular domains. In this case also the second variation is derived.

Catherine Bandle, Alfred Wagner

The Asymptotic Shape of a Boundary Layer of Symmetric Willmore Surfaces of Revolution

Abstract

We consider the Willmore boundary value problem for surfaces of revolution over the interval [−1,1] where, as Dirichlet boundary conditions, any symmetric set of position α and angle arctanβ may be prescribed. Energy minimising solutions u _α,β have been previously constructed and for fixed β∈ℝ, the limit $\lim_{\alpha \searrow0 }u_{\alpha,\beta}(x) =\sqrt{1 - x^{2}}$ has been proved locally uniformly in (−1,1), irrespective of the boundary angle. Subject of the present note is to study the asymptotic behaviour for fixed β∈ℝ and α↘0 in a boundary layer of width kα, k>0 fixed, close to ±1. After rescaling $x\mapsto\frac{1}{\alpha}u_{\alpha,\beta }(\alpha(x-1)+1)$ one has convergence to a suitably chosen cosh on [1−k,1].

Hans-Christoph Grunau

A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

Abstract

A wide variety of articles, starting with the famous paper (Gidas, Ni and Nirenberg in Commun. Math. Phys. 68, 209–243 (1979)), is devoted to the uniqueness question for the semilinear elliptic boundary value problem −Δu=λu+u ^p in Ω, u>0 in Ω, u=0 on ∂Ω, where λ ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of Ω being a ball and, for more general domains, in the case λ=0. In (McKenna et al. in J. Differ. Equ. 247, 2140–2162 (2009)), we proposed a computer-assisted approach to this uniqueness question, which indeed provided a proof in the case Ω=(0,1)², and p=2. Due to the high numerical complexity, we were not able in (McKenna et al. in J. Differ. Equ. 247, 2140–2162 (2009)) to treat higher values of p. Here, by a significant reduction of the complexity, we will prove uniqueness for the case p=3.

Patrick J. McKenna, Filomena Pacella, Michael Plum, Dagmar Roth

Green Function Estimates Lead to Neumann Function Estimates

Abstract

It is shown that the Neumann function for −Δu+u=f on a bounded domain Ω⊂ℝⁿ can be estimated pointwise from below in a uniform way. The proof is based on known uniform estimates from below for the Green function.

Guido Sweers

Numerical Methods

Frontmatter

Deriving Inequalities in the Laguerre-Pólya Class from Properties of Half-Plane Mappings

Abstract

Newton, Euler and many after them gave inequalities for real polynomials with only real zeros. We show how to extend classical inequalities ensuring a guaranteed minimal improvement. Our key is the construction of mappings with bounded image domains such that existing coefficient criteria from complex analysis are applicable. Our method carries over to the Laguerre-Pólya class $\mathcal{L}$–$\mathcal{P}$ which contains real polynomials with exclusively real zeros and their uniform limits. The class $\mathcal{L}$–$\mathcal{P}$ covers quasi-polynomials describing delay-differential inequalities as well as infinite convergent products representing entire functions, while it is at present not known whether the Riemann ξ-function belongs to this class. For the class $\mathcal{L}$–$\mathcal{P}$ we obtain a new infinite family of inequalities which contains and generalizes the Laguerre-Turán inequalities.

Prashant Batra

Fundamental Error Estimate Inequalities for the Tikhonov Regularization Using Reproducing Kernels

Abstract

First of all, we will be concentrated in some particular but very important inequalities. Namely, for a real-valued absolutely continuous function on [0,1], satisfying f(0)=0 and $\int_{0}^{1}f'(x)^{2}\,dx<1$, we have, by using the theory of reproducing kernels

$$\int_0^1\left(\frac{f(x)}{1-f(x)}\right)^{\prime\,2}(1-x)^2\,dx \le\frac{\int_0^1f^{\prime\,2}(x)\,dx}{1-\int_0^1f^{\prime\,2}(x)\,dx}. $$

A. Yamada gave a direct proof for this inequality with a generalization and, as an application, he unified the famous Opial inequality and its generalizations.

Meanwhile, we gave some explicit representations of the solutions of nonlinear simultaneous equations and of the explicit functions in the implicit function theory by using singular integrals. In addition, we derived estimate inequalities for the consequent regularizations of singular integrals.

Our main purpose in this paper is to introduce our method of constructing approximate and numerical solutions of bounded linear operator equations on reproducing kernel Hilbert spaces by using the Tikhonov regularization. In view of this, for the error estimates of the solutions, we will need the inequalities for the approximate solutions. As a typical example, we shall present our new numerical and real inversion formulas of the Laplace transform whose problems are famous as typical ill-posed and difficult ones. In fact, for this matter, a software realizing the corresponding formulas in computers is now included in a present request for international patent. Here, we will be able to see a great computer power of H. Fujiwara with infinite precision algorithms in connection with the error estimates.

Luís P. Castro, Hiroshi Fujiwara, Saburou Saitoh, Yoshihiro Sawano, Akira Yamada, Masato Yamada

On the Approximation-Error of Some Numerical Methods for Obtaining the Optimal Deformable Model

Abstract

This paper deals with two approximation methods for obtaining the optimal deformable model: a discretization scheme by finite differences that generates an algorithm providing the approximating solution for energy-minimizing surfaces and a reconstruction method based on the Chebyshev discrete best approximation which approximates a plane deformable model represented by a finite set of points. Estimates for the approximation-error of these methods and results concerning their convergence or the topological structure of the set of unbounded divergence are presented, too.

Paulina Mitrea, Octavian Mircia Gurzău, Alexandru Ioan Mitrea

Geometric and Norm Inequalities

Frontmatter

The Longest Shortest Piercing

Abstract

We generalize an old problem and its partial solution of Pólya (Pólya in Elem. Math. 13, 40–41 (1958)) to the n-dimensional setting. Given a plane domain Ω⊂ℝ², Pólya asked in 1958 for the shortest bisector of Ω, that is for the shortest line segment l(Ω) which divides Ω into two subsets of equal area. He claimed that among all centrosymmetric domains of given area l(Ω) becomes longest for a disk. His proof, however, does not seem to be valid for domains that are not starshaped with respect to the center of Ω. In the present note we provide two proofs that it suffices to restrict attention to starshaped sets. Moreover we state and prove a related inequality in ℝⁿ. Given the volume of a measurable set Ω with finite Lebesgue measure, only a ball centered at zero maximizes the length of the shortest line segments running through the origin. In this sense the ball has the longest shortest piercing.

Bernd Kawohl, Vasilii V. Kurta

On a Continuous Mapping and Sharp Triangle Inequalities

Abstract

This is a survey on some recent results concerning the sharp triangle inequalities. Our results refine and generalize the corresponding ones in (Kato et al. in Math. Inequal. Appl. 10(2), 451–460 (2007)) and (Mitani et al. in J. Math. Anal. Appl. 10(2), 451–460 (2007)).

Tomoyoshi Ohwada

A Dunkl-Williams Inequality and the Generalized Operator Version

Abstract

C.F. Dunkl and K.S. Williams (Am. Math. Mon. 71, 53–54 (1964)) showed that for any nonzero elements x, y in a normed linear space $\mathcal{X}$

$$\biggl \Vert \frac{x}{ \Vert {x}\Vert }-\frac{y}{ \Vert {y}\Vert }\biggr \Vert \leq\frac{4 \Vert {x-y}\Vert }{ \Vert {x}\Vert + \Vert {y}\Vert }. $$

Recently, J. Pečarić and R. Rajić (J. Math. Inequal. 4, 1–10 (2010)) gave a refinement and, moreover, a generalization to operators $A,B \in\mathcal{B}(\mathcal{H})$ such that |A|, |B| are invertible as follows:

$$\bigl|A|A|^{-1}-B|B|^{-1}\bigr|^2 \le|A|^{-1}\bigl(p|A-B|^2+q\bigl(|A|-|B|\bigr)^2\bigr)|A|^{-1} $$

where p,q>1 with $\frac{1}{p}+\frac{1}{q}=1$.

In this note, we review some results concerning the Dunkl-Williams inequality and the generalization of the operator version of J. Pečarić and R. Rajić.

Kichi-Suke Saito, Masaru Tominaga

Generalized Convexity

Frontmatter

Jordan Type Representation of Functions with Generalized High Order Bounded Variation

Abstract

The aim of this work is to identify few classes of functions with generalized type of bounded variation for which a decomposition theorem of Jordan type holds. We refer especially to functions with nth order bounded variation with respect to a Tchebycheff system. The particular case of trigonometric Tchebycheff systems bring interesting results.

Gabriela Cristescu

On Vector Hermite-Hadamard Differences Controlled by Their Scalar Counterparts

Abstract

We present a new, in a sense direct, proof that the system of two functional inequalities

$$\biggl \Vert F \biggl(\frac{x+y}{2} \biggr) - \frac{1}{y-x} \int_{x}^{y} F(t)\,dt \biggr \Vert \leq\frac{1}{y-x} \int_{x}^{y} f(t)\,dt - f \biggl(\frac{x+y}{2} \biggr) $$

and

$$\biggl \Vert \frac{F(x) + F(y)}{2} - \frac{1}{y-x} \int_{x}^{y} F(t)\,dt \biggr \Vert \leq\frac{f(x) + f(y)}{2} -\frac{1}{y-x} \int_{x}^{y} f(t)\,dt $$

is satisfied for functions F and f mapping an open interval I of the real line ℝ into a Banach space and into ℝ, respectively, if and only if F yields a delta-convex mapping with a control function f.

A similar result is obtained for delta-convexity of higher orders with detailed proofs given in the case of delta-convexity of the second order, i.e. when the functional inequality

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-0348-0249-9_12/978-3-0348-0249-9_12_Equc_HTML.gif

holds true provided that x,y∈I, x≤y.

Roman Ger, Josip Pečarić

Functions Generating Strongly Schur-Convex Sums

Abstract

The notion of strongly Schur-convex functions is introduced and functions generating strongly Schur-convex sums are investigated. The results presented are counterparts of the classical Hardy–Littlewood–Pólya majorization theorem and the theorem of Ng characterizing functions generating Schur-convex sums. It is proved, among others, that for some (for every) n≥2, the function F(x ₁,…,x _n)=f(x ₁)+⋯+f(x _n) is strongly Schur-convex with modulus c if and only if f is of the form f(x)=g(x)+a(x)+c∥x∥², where g is convex and a is additive.

Kazimierz Nikodem, Teresa Rajba, Szymon Wąsowicz

Strongly Convex Sequences

Abstract

Let ω≥0 be a given number and I a subinterval of ℤ. We say that a sequence (f _k)_k∈I is ω-midconvex if

$$f_k \leq \frac{f_{k-1}+f_{k+1}}{2}-\omega \quad \mbox{for }k-1, k, k+1 \in I. $$

We give various characterizations of ω-midconvex sequences.

We also show that in a natural way one can derive from the above definition classical notions of convexity and strong convexity for functions defined on subintervals of ℝ.

Jacek Tabor, Józef Tabor, Marek Żołdak

Convexity and Related Inequalities

Frontmatter

Refinement of Inequalities Related to Convexity via Superquadracity, Weaksuperquadracity and Superterzacity

Abstract

This paper is about inequalities satisfied by functions called superterzatic and their relations to convex and to superquadratic functions. In analogy to inequalities satisfied by convex and by superquadratic functions that are reduced to equalities when f(x)=x, f(x)=x ², x≥0 respectively, the inequalities satisfied by superterzatic functions reduce to equalities when f(x)=x ³, x≥0.

In particular, we deal here with the generalization of the inequality

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-0348-0249-9_15/978-3-0348-0249-9_15_Equa_HTML.gif

x,y≥0, q≥3, that reduces to equality for q=3.

Shoshana Abramovich, Slavica Ivelić, Josip Pečarić

On Two Different Concepts of Subquadraticity

Abstract

In the recent years, subquadratic functions have been investigated by several authors. However, two different concepts of subquadraticity have been considered. Based on a simple modification of the geometric notion of concave functions a function f:[0,∞[ →ℝ is called subquadratic if, for each x≥0, there exists a constant c _x∈ℝ such that the inequality

$$ f(y)-f(x)\leq c_x(y-x)+f\big(|y-x|\big) $$

is valid for all nonnegative y.

Related to the concept of quadratic functions, a function f:ℝ→ℝ is said to be subquadratic if it fulfils the inequality

$$ f(x+y)+f(x-y)\leq 2f(x)+2f(y) $$

for all x,y∈ℝ. In the present paper, the connections between these two concepts are described and a third inequality related to these concepts is studied.

Attila Gilányi, Csaba Gábor Kézi, Katarzyna Troczka-Pawelec

Connections Between the Jensen and the Chebychev Functionals

Abstract

This work is devoted to the study of connections between the Jensen functional and the Chebychev functional for convex, superquadratic and strongly convex functions. We give a more general definition of these functionals and establish some inequalities involving them. The entire discussion incorporates both the discrete and the continuous approach.

Flavia Corina Mitroi

Other Equations and Inequalities

Frontmatter

Functional Inequalities and Equivalences of Some Estimates

Abstract

The purpose of the chapter is to deal with some old and a few new functional inequalities which are motivated by well-known estimates on the real line or on an interval which involve the exponential function. We are concerned with the following six functional inequalities:

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-0348-0249-9_18/978-3-0348-0249-9_18_Equa_HTML.gif

and

$$ (1+y)f(x) \leq f(x+y).$$

Włodzimierz Fechner

On Measurable Functions Satisfying Multiplicative Type Functional Equations Almost Everywhere

Abstract

Using the so-called “almost” variant of a well-known generalization of Steinhaus’ theorem, first we prove a general result on the multiplicative type functional equation (3), then we solve functional equations (1) and (2) originated from statistics under such conditions.

Antal Járai, Károly Lajkó, Fruzsina Mészáros

On the L 1 Norm of the Weighted Maximal Function of Walsh-Marcinkiewicz Kernels

Abstract

The L ¹ norm of the maximal function of Walsh-Marcinkiewicz kernel is infinite. Thus, we have to use some weight function to “pull it back” to the finite.

The main aim of this chapter is to investigate the integral of the weighted maximal function of the Walsh-Marcinkiewicz kernels. We give a necessary and sufficient conditions for that the weighted maximal function of the Walsh-Marcinkiewicz kernels is in L ¹. For our motivation we refer the readers to the papers (Gát in Acta Acad. Paedagog. Agriensis Sect. Mat. 30, 55–66 (2003); Mező and Simon in Publ. Math. (Debr.) 71(1–2), 57–65 (2007); Nagy in JIPAM. J. Inequal. Pure Appl. Math. 9(1), 1–9 (2008)).

Károly Nagy

Quasimonotonicity as a Tool for Differential and Functional Inequalities

Peter Volkmann

Titel: Inequalities and Applications 2010
herausgegeben von: Catherine Bandle
Attila Gilányi
László Losonczi
Michael Plum
Verlag: Springer Basel
Electronic ISBN: 978-3-0348-0249-9
Print ISBN: 978-3-0348-0248-2
DOI: https://doi.org/10.1007/978-3-0348-0249-9