Optimal young's inequality and its converse: a simple proof

We give a new proof of the sharp form of Young's inequality for convolutions, first proved by Beckner [Be] and Brascamp-Lieb [BrLi]. The latter also proved a sharp reverse inequality in the case of exponents less than 1. Our proof is simpler and gives Young's inequality and its converse altogether. Submitted: March 1997, Final version: April 1997     

A converse to the Jensen inequality,its matrix extensions and inequalities for minors and eigenvalues

We obtain a scalar inequality, converse to the Jensen inequality. We also derive an operator converse to the Jensen inequality. As special cases, we obtain inequalities, similar to the Kantorovich one as well as some operator generalizations of them. Using some exterior algebra, we prove a generalization of the Sylvester determinant theorem. We also deduce some determinant analogs of the additive and multiplicative Kantorovich inequalities.     

A generalization of Lomonosov's inequality and its applications to invariant subspaces

In this article we generalize Victor Lomonosov's famous inequality so as to be applicable to a wider class of functions. Then using it we prove that the adjoint of an algebra with a compactness property which is weaker than the PS property, employed by Victor Lomonosov, has nontrivial invariant subspaces.

     

On a dual Hardy-Hilbert’s inequality and its generalization

Summary This paper deals with a dual Hardy-Hilbert’s inequality with a best constant factor involving the beta function, which is an extension of the Hilbert’s inequality with the form of (p,q)-parameter. We also consider its extended form and an equivalent inequality.     

Integral analog of one generalization of the Hardy inequality and its applications

Under certain conditions on continuous functions μ, λ, a, and f, we prove the inequality

A generalization of trigonometric convexity and its relation to positive harmonic functions in homogeneous domains

We consider functions which are subfunctions with respect to the differential operator

A new generalization of Aczél’s inequality and its applications to an improvement of Bellman’s inequality

A new generalized version of Aczél’s inequality is proved. This is a unified generalization of some known results. Moreover, the result is applied to the improvement of the well-known Bellman’s inequality.     

Application of Michael's theorem and its converse to sublinear operators

A theorem of Michael on continuous selectors and its converse are used in this article to study subdifferentials of continuous sublinear operators with values in a cone of lower semicontinuous functions. It is proved that such operators are subdifferentiable (i.e., have nonempty subdifferentials) if their domains are separable Banach spaces. Sublinear operators that are not subdifferentiable are found.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 67–75, July, 1992.     

A capacitary weak type inequality for Sobolev functions and its applications

In this paper a capacitary weak type inequality for Sobolev functions is established and is applied to reprove some well-known results concerning Lebesgue points, Taylor expansions in the -sense, and the Lusin type approximation of Sobolev functions.

     

A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind

This paper presents a new discrete Gronwall inequality. Using the inequality, we prove convergence and error estimate of the numerical solutions of the second weakly singular Volterra integral equation, where discrete equation is derived by Novot's quadrature formula.     

On a conjecture regarding the extrema of Bessel functions and its generalization

It was conjectured by Á. Elbert in J. Comput. Appl. Math. 133 (2001) 65-83 that, given two consecutive real zeros of a Bessel function of order ν, j_ν,κ and j_ν,κ+1, the zero of the derivative between such two zeros j_ν,κ′ satisfies . We prove that this inequality holds for any Bessel function of any real order. In addition to these lower bounds, upper bounds are obtained. In this way we bracket the zeros of the derivative. It is discussed how similar relations can be obtained for other special functions which are solutions of a second order ODE; in particular, the case of the zeros of is considered.     

A fundamental inequality for meromorphic functions in an angular domain and its application

This paper is to study the characteristic functions of meromorphic functions for an angular domain. We show an inequality which is similar to the inequality in the second fundamental theorem of Nevanlinna, and estimate its remainder. Finally, as an application we obtain a sufficient and necessary condition of Borel directions. Supported by the National Natural Science Foundation of China.     

Best constants in the Hardy-Rellich inequalities and related improvements

We consider Hardy-Rellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. Rellich-Sobolev inequalities). We discuss also the optimality of these inequalities in the sense that we establish (in most cases) that the constants appearing there are the best ones. Next, we investigate the polyharmonic operator (Rellich and higher order Rellich inequalities); the difficulties arising in this case come from the fact that (generally) minimizing sequences are no longer expected to consist of radial functions. Finally, the successively use of the Rellich inequalities lead to various new higher order Rellich inequalities.     

Spectrum and scattering in a “spin-boson” model with not more than three photons

A model of radiative decay with a fixed atom and not more than three photons is studied. A spectral analysis of the Hamiltonian is made. This is done by means of scattering theory in a pair of spaces with a specially chosen embedding. The existence of wave operators and their asymptotic completeness are proved. The constructions are based on a detailed analysis of the resolvent of the Hamiltonian.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 63–81, April, 1995.     

Variation of constants,vector Lyapunov functions,and comparison theorem

We obtain a new comparison result and discuss its relation to known results. A simple application to stability theory is also given to indicate the usefulness of the comparison result.