The maximum of a quasismooth function

Abstract:

Let Z be the class of real-valued functions, defined and continuous on the closed interval $I = [ - 1,1]$, such that $f( \pm 1) = 0$ and $|f(\xi ) - 2f\{ (\xi + \eta )/2\} + f(\eta )| \leqslant |\xi - \eta |$ for all $\xi$ and $\eta$ in I. Let $K = {\sup _{f \in Z}}{\max _{x \in I}}|f(x)|$. We will prove that $13/10 \leqslant K \leqslant 1014/779 < 1.301669$.