To the mean-value theorem

New versions of the mean-value theorem for real and complex-valued functions are presented.     

On the mean-value theorem for analytic functions

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved. Volyn University, Lutsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1143–1147, August, 1997.     

A generalized mean-value theorem

We give another proof of Seymour and Zaslavsky's theorem: For every familyf ₁,f ₂,...,f _n of continous functions defined on [0, 1], there exists a finite setF[0, 1] such that the average sum off _k overF coincides with the integral off _k for everyk=1, 2,...,n.     

A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L[f], we prove that there exists a unique two variable mean M[f] such that

     

Reduction of matrix polynomials to a special quasitriangular form

We find necessary and sufficient conditions for polynomials with matrix coefficients over an arbitrary field to be reducible via a similarity transformation to block triangular form with regular diagonal blocks of maximum degree. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

The mean-value theorem for elliptic operators on stratified sets

In this paper, an analog of the mean-value theorem for harmonic functions is proved for an elliptic operator on the stratified set of “stratified” spheres whose radius is sufficiently small. In contrast to the classical case, the statement of the theorem has the form of a special differential relationship between the mean values over different parts of the sphere. The result is used to prove the strong maximum principle.     

Uniform approximation by polynomials on compacta of special form

A lower estimate of the least deviations is obtained and polynomials of the best uniform approximation are found for some functions given on compact sets of the complex plane containing complete preimages Q ^?1(v _j ) of several points v _j ?? ? for some polynomial Q(z) of complex variable.     

An elementary proof of a converse mean-value theorem

We present a new converse mean value theorem, with a rather elementary proof.     

The Bolzano mean-value theorem and partial differential equations

We study the existence of solutions to abstract equations of the form $0=Au+F(u)$, $u\in K?E$, where A is an abstract differential operator acting in a Banach space E, K is a closed convex set of constraints being invariant with respect to resolvents of A and a perturbation F satisfies a certain tangency condition. Such problems are closely related to the so-called Poincaré–Miranda theorem, being the multi-dimensional counterpart of the celebrated Bolzano intermediate value theorem. In fact our main results should be regarded as infinite-dimensional variants of Bolzano and Miranda–Poincaré theorems. Along with single-valued problems we deal with set-valued ones, yielding the existence of the so-called constrained equilibria of set-valued maps. The abstract results are applied to show existence of (strong) steady state solutions to some weakly coupled systems of drift reaction–diffusion equations or differential inclusions of this type. In particular we get the existence of strong solutions to the Dirichlet, Neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type. Certain aspects of the Bernstein theory for bvp for second order ODE are studied, too. No assumptions concerning structural coupling (monotonicity, cooperativity) are undertaken.     

On the finite-increment theorem for complex polynomials

For an arbitrary polynomial P of degree at most n and any points z ₁ and z ₂ on the complex plane, we establish estimates of the form $$ \left| {P(z_1 ) - P(z_2 )} \right| \geqslant d_n \left| {P'(z_1 )} \right|\left| {z_1 - \zeta } \right| $$ , where ζ is one of the roots of the equation P(z) = P(z ₂), and d _n is a positive constant depending only on the number n.     

A Hahn-Banach theorem for integral polynomials

We study the problem of extendibility of polynomials over Banach spaces: when can a polynomial defined over a Banach space be extended to a polynomial over any larger Banach space? To this end, we identify all spaces of polynomials as the topological duals of a space spanned by evaluations, with Hausdorff locally convex topologies. We prove that all integral polynomials over a Banach space are extendible. Finally, we study the Aron-Berner extension of integral polynomials, and give an equivalence for non-containment of .

     

Pietsch's factorization theorem for dominated polynomials

We prove that, like in the linear case, there is a canonical prototype of a p-dominated homogeneous polynomial through which every p-dominated polynomial between Banach spaces factors.