Eventually areally meanp-valent functions

Theorems concerning areally meanp-valent functions are extended to eventually areally meanp-valent functions. In particular, suppose is eventually areally meanp-valent in the unit disc,b, c are positive integers,a≧max {p−1, 0}. If |a _n|≦Cn ^α for alln=bm+c,m=1, 2, …, then |a _n|≦C′n ^α for alln. This is a marked extension of results due to Goluzin and to Hayman.     

On holomorphic functions attaining their norms

We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon-Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k, it cannot be approximated by norm attaining polynomials with degree less than k. For , a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm.     

On a uniqueness theorem for functions with a sparse spectrum

We present an example of a set { } satisfying the following two conditions: (1) there exists a nonzero positive singular measure on the unit circle { } with spectrum in Λ; (2) if the spectrum of f∈L¹ { } is contained in Λ and f vanishes on a set of positive measure, then f=0. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 7–14. Translated by A. B. Aleksandrov.     

On a theorem of Privalov and normal functions

A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

     

On Kellog's theorem for discontinuous green functions

Translated from Matematicheskie Zametki, Vol. 53, No. 3, pp. 151–153, January, 1993.     

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.     

Carathéodory theorem for matrix-valued functions and the maximal jump of spectral functions in extension problems

Translated from Matematicheskie Zametki, Vol. 48, No. 3, pp. 3–11, September, 1990.     

On the uniqueness of maximal functions

The uniqueness theorem for the one-sided maximal operator has been proved.     

On weak type inequalities for dyadic maximal functions

We obtain sharp estimates for the localized distribution function of the dyadic maximal function , given the local L¹ norms of ? and of G○? where G is a convex increasing function such that G(x)/x→+∞ as x→+∞. Using this we obtain sharp refined weak type estimates for the dyadic maximal operator.     

On uniqueness theorem for a class of functions analytic in a halfplane

We obtain a uniqueness theorem for a class of analytic functions of exponential type in a halfplane.     

On maximal functions for Mikhlin-Hörmander multipliers

Given Mikhlin-Hörmander multipliers , with uniform estimates we prove an optimal bound in L^p for the maximal function and related bounds for maximal functions generated by dilations. These improve the results in [M. Christ, L. Grafakos, P. Honzík, A. Seeger, Maximal functions associated with multipliers of Mikhlin-Hörmander type, Math. Z. 249 (2005) 223-240].     

On the maximal ergodic theorem for certain subsets of the integers

It is shown that the set of squares {n ²|n=1, 2,…} or, more generally, sets {n ^t|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL ²-functions. This gives an affirmative answer to a problem considered by A. Bellow [Be] and H. Furstenberg [Fu]. The previous result extends to polynomial sets {p(n)|n=1, 2,…} and systems of commuting transformations. We also state density conditions for random sets of integers in order to be “good sequences” forL ^p-functions,p>1.     

On maximal functions in Orlicz spaces

Let and be the functions having the representations and , where is a positive continuous function such that and is quasi-increasing. Then the maximal function is a function in Orlicz space for all if and only if there exists a positive constant such that for all .

     

On the corona theorem for almost periodic functions

LetAP ⁺ (R ⁿ ) denote the Banach algebra of all continuous almost periodic functions onR ⁿ whose Bohr-Fourier spectrum is contained in an additive semi-group [0, ) ⁿ . We show that the maximal ideal space ofAP ⁺ (R ⁿ ) may have a nonempty corona and we characterize all for which the corona is empty. Analogous results are established for algebras of almost periodic functions with absolutely convergent Fourier series.     

On the mean value theorem for semidifferentiable functions

The G-semidifferentiability concept was introduced in the reference Giannessi (J Optim Theory Appl 60:191–241, 1989) and it furnishes a general scheme for treating generalized derivatives. We prove that, in this context, it is possible to obtain a mean value theorem. By exploiting this result we deduce conditions for a function to be lipschitzian, C-decreasing or quasiconvex.