Estimation of distribution tails for normalized and self-normalized sums

A combinatorial inequality is derived. This inequality is applied to obtain new estimates for probabilities of large deviations of normalized and self-normalized sums of independent and dependent positive random values. As a consequence, an estimate from above is derived for the strong law of large numbers. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 77–87.This research was supported in part by the Ministry of Education of Russia, grant E00-1.0-45, and by the Russian Foundation for Basic Research, grant 02-01-01099a.Translated by V. A. Egorov.     

A bernstein type inequality for derivatives of rational functions on two intervals

In this paper we establish an inequality for derivatives of rational functions with a fixed denominator generalizing V. S. Videnskii's inequality to the case of two intervals. To prove its asymptotic exactness, we use a new representation of Akhiezer-Zolotarev fractions with the least deviation from 0 on two intervals. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 508–514, October, 1999.     

On a geometric inequality

An inequality is obtained connecting the characteristics of a figureG with the characteristics of similar images of a convex figureg contained inG and containingG. In the case wheng is a disk the well-known inequality of Bonnesen is obtained.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 49–52.     

On the theorems of Turan,Amrein-Berthier,and Zigmund

A rather sharp inequality of Turan's lemma type is obtained. Its applications to some uniqueness theorems are discussed. No proofs are given. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 117–123. Translated by V. Vasyunin.     

Hardy-Sobolev Inequalities in a Cone

The attainability of the exact constant in the Hardy-Sobolev inequality is established in an arbitrary cone in ℝ ⁿ . Bibliography: 17 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 39–46.     

A multidimensional analog of a theorem due to zygmund

Zygmund proved an inequality describing the dependence of the modulus of continuity of the adjoint function on that of the original function lying in the space of 2π-periodic continuous functions. The present article contains estimates of partial moduli of continuity of the adjoint function of several variables in the spaceC. Examples show that these estimates are sharp. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 717–727, May, 1997. Translated by A. M. Chebotarev     

Pseudoparabolic variational inequalities without initial conditions

We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a variable t. Under certain conditions imposed on the coefficients of the inequality, we prove theorems on the unique existence of a solution for a class of functions with exponential growth as t → ∞. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 919–929, July, 1998.     

Systems of parabolic variational inequalities without initial conditions

We consider a parabolic variational inequality without initial conditions. We construct a class of existence and uniqueness for a solution of this inequality. This class is defined by the exponential decrease or increase of solutions as t→−∞, depending on the coefficients of the inequality. L’viv University, L’viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 540–547, April, 1997.     

Sharp inequality for deviation of Rogozinski sums and the second continuity modulus in the space of periodic continuous functions

The sharp constant (uniformly in n) is found in a Jackson-type inequality involving the Rogozinski sums of order n and the second modulus of continuity with step π/(n+1). Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 26–45. Translated by S. Yu. Pilyugin.     

On certain nonlinear pseudoparabolic variational inequalities without initial conditions

We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variablet. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior ast→−∞. Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 328–337, March, 1999.     

A generalization of the Cramér-Rao inequality

The classical Cramér-Rao inequality is proved for a wider class of loss functions. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 154–161.     

Operators (not) similar to a contraction: Pisier's counterexample via singular integrals

The main difference of this exposition of Pisier's counterexample from the original one is that in the proof of the key inequality probabilistic considerations (like Brownian motion) are replaced by certain standard constructions from the theory of Calderòn-Zygmund operators. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 79–95. Translated by S. V. Kislyakov.     

The one-dimensional character of an extremum point of the friedrichs inequality in spherical and plane layers

Some properties of extremum points of the Friedrichs inequality in special domains Ω are studied. Bibliography: 11 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 171–190.     

Numerical solution of the zero-head seepage problem by discretization of the variational inequality

The problem of zero-head seepage through a cutoff is reduced to solving a variational inequality which is discretized by the finite element method. The discrete variational inequality is solved by a two-layer iterative process. A rate of convergence bound is obtained for the approximate solution and the optimal parameters of the two-layer iterative process are determined. A numerical experiment supports the theoretical results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 45–56, 1986.     

On an integral inequality

In this paper we deduce an integral inequality which is an analog of a known two-parameter inequality of Hardy and Littlewood ([1], Theorem 382). A need for inequalities of a similar type arises, for example, in studying the imbedding of the functional spaces B _p ^l in the space L_q if this study leads to a basis of the method of integral representations of functions.Translated from Matematicheskie Zametki, Vol. 6, No. 2, pp. 139–148, August, 1969.     

Two inequalities for parameters of a cellular algebra

Two inequalities are proved. The first is a generalization for cellular algebras of a well- known theorem about the coincidence of the degree and the multiplicity of an irreducible representation of a finite group in its regular representation. The second inequality that is proved for primitive cellular algebras gives an upper bound for the minimal subdegree of a primitive permutation group in terms of the degrees of its irreducible representations in the permutation representation.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 82–95.     

A Necessary Condition for Interpolation by Functions of the Lipschitz Class

For a weak modulus of continuity and the corresponding Lipschitz class, a necessary condition for interpolation by analytic functions is given in the form of Dyn'kin's inequality. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 39–42.     

Mutual estimates of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-norms and the Bellman function

In this paper, we describe the range of the L^p-norm of a function under fixed L^p-norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two L^p-norms as well as such nontrivial inequalities as the Gehring inequality or the reverse H?lder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5 Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 81–138.     

Hardy's inequalities in function spaces containing derivatives of noninteger order

A theorem on Hardy's inequality in function spaces containing derivatives of noninteger order is proved. Translated fromMatematichcskie Zametki, Vol. 63, No. 5, pp. 673–678, May, 1998. The author wishes to thank Professor V. A. Kondrat'ev for his attention to this work.     

On singular integrals related to the Littlewood-Paley inequality for arbitrary intervals

The proof of the inequality mentioned in the title requires the knowledge of the fact that operators of a certain class are Calderón-Zygmund singular integral operators. We slightly extend this class. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 113–119.