Some Estimates For Real Functionals In Finite-Dimensional Spaces

Abstract

In this paper, we explore some real functionals in finite‐dimensional spaces which satisfy the conditions /a/, /b/, /c/, /d/, /e/, /f/,and /g/, formulated in the theorems. Thanks to the conditions of Theorem 1 we can assert f( ∑ i = 1kαixi)≤ ∑ i = 1kαif(xi, where xi = (x1i,x2i,…,xni),αi>0, ∑ i = 1nαi = 1, i.e. the functional fis convex. In the next two theorems we assert, that the functional is concave, i.e. f( ∑ i = 1kαixi)≥ ∑ i = 1kαif(xi, where xi = (x1i,x2i,…,xni),αi>0, ∑ i = 1nαi = 1,Analogous 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‐[l], [2]. These results can be applied in the mentioned areas. The given example after Theorem 2 explains how could be apply these results.