In this paper we provide some (not new) estimates on distances from our two previous papers together with some new estimates. Namely some estimates on distances in spaces of harmonic functions in the unit ball and the upper half space are provided. New estimates concerning mixed norm spaces and general weighted Bergman spaces are obtained and discussed.

В работе существенно усилен результат, полученный в [1], где для непрерывной функции F(х), дифференцируемой по направлениям в R n\{0} были получены условия альфа-достижимости области, определяемой неравенством F(х)<0. Полученные условия также являются полезными и в случае ноль-достижимых (то есть звездообразных) областей. This paper continues the study of α-accessible domains in R n in nonsmooth case. A domain Ω⊂R n, 0ϵΩ, is α-accessible (see [2], [3]), [0; 1); if for every point pϵ∂Ω there exists a number r = r(p)>0 such that the cone K +(p,α,r)={xϵR n : ||x-p||≤r, (x-p,p/||p||)≥||x-p||cos(απ/2)} is included in Ω'=R n\Ω. α-accessible domains are starlike domains with respect to the inner point zero and satisfy cone condition which is important for applications,such as the theory of integral representations of functions, imbedding theorems, the questions of the boundary behavior of functions, the solvability of Dirichlet problem. Appropriate conditions of α-accessibility of domain, defined by the inequality F(x)<0 for a continuous function F in R n are obtained in [1]. There were obtained and some sufficient conditions of α-accessibility. In this article, these sufficient conditions have been significantly strengthened. Here is an example of their use. Conditions, obtained in the theorem, and consequences to it are also sufficient for the starlikeness of the set (the case when α=0). In the smooth case the criterion for the α-accessibility of domain was obtained in [2].

Motivated by the work of P. Lindqvist, we study eigenfunctions of the one-dimensional p-Laplace operator, the sinp functions, and prove several inequalities for these and p-analogues of other trigonometric functions and their inverse functions. Similar inequalities are given also for the p-analogues of the hyperbolic functions and their inverses.

In [1] W. Gross constructed the example of an entire function of infinite order whose set of asymptotic values is equal to the extended complex plain. We obtain an analog of Gross' result for functions, analytic in planar domains of arbitrary connectivity with isolated boundary fragment.