Some Estimates For Real Functionals In Finite-Dimensional Spaces

In this paper, we explore some real functionals in finite-dimensional spaces which satisfy the condidions /a/, /b/, /c/, /d/, /e/, /f/, and /g/, formulated in the theorems. Thanks to the conditions of Theorem 1 we can assert f(Sigma(k)(i=1) alpha(i)chi(i)) <= Sigma(k)(i=1)alpha(i)f(x(i)), where x(i) = (x(1)(i), x(2)(i), ..., x(n)(i)), alpha(i) >= 0, Sigma(n)(i=1)alpha(i) = 1, i.e. the functional fis convex. In the next two theorems we assert, that the functional is concave, i.e. f((Sigma(k)(i=1) alpha(i)chi(i)) >= Sigma(k)(i=1)alpha(i)f(x(i)), where x(i) = (x(1)(i), x(2)(i), ..., x(n)(i)), alpha(i) >= 0, Sigma(n)(i=1)alpha(i) = 1. Analagous results we have about general convexity in seminormed spaces and seminormed algebras in [3]. About the general concavity in finite-dimensional spaces we have some estimates in [4]. Such results had been used in the geometry of the Banach spaces-[1],2]. These results can be applied in the mentioned areas. The given example after Theorem 2 explains how could be apply these results.