On the Maksa-Volkmann functional inequality |f(x+y)|≥|f(x)+f(y)| when the range of f is a space of functions

Rădulescu, Marius and Rădulescu, Sorin



P. Volkmann functional inequality \left\vert f\left(x+y\right) \right\vert \geq \left\vert f\left( x\right) +f\left( y\right) \right\vert is extended to functions f:G\rightarrow \mathfrak{F}\left( X,E\right) where G is an
additive group and \mathfrak{F}\left( X,E\right) is the space of functions from a set X to a linear normed space E. As a corollary one proves that an operator T:C\left( X,K\right) \rightarrow C\left( X,K\right) which satisfies the functional inequality \left\vert T\left( f+g\right) \right\vert \geq \left\vert T\left( f\right) +T\left( g\right) \right\vert , f,g\in C\left( X,K\right) is additive. Here we denoted by X a compact topological space, K is \mathbb{R} or \mathbb{C} and C\left( X,K\right) is the linear space of continuous functions defined on X with values in K.

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Rădulescu, Marius, Rădulescu, Sorin