A modulus of continuity for a class of quasismooth functions

Abstract:

Let $Z$ be the class of real-valued functions $f(x)$, defined and continuous on the closed interval $I = [ - 1,1]$, such that $f( - 1) = f(1) = 0$ and \[ |f(\xi ) - 2f\{ (\xi + \eta )/2\} + f(\eta )| \leqslant |\xi - \eta |\] for all $\xi$ and $\eta$ in $I$. We show that $\omega (h) = h{\log _2}\left \{ {2eK/(h{{\log }_2}e)} \right \}$ is a modulus of continuity on $Z$, if $K = {\sup _{f \in Z}}{\max _{x \in I}}\left | {f(x)} \right |$.