On the maximum value for Zygmund class on an interval (Q1854128)

Summary: We prove that if \(f(z)\) is a continuous real-valued function on \({\mathbb R}\) with the properties \(f(0)=f(1)=0\) and that \(\|f\|_{z}= \inf_{x,t} |f(x+t)-2f(x)+f(x-t)/t|\) is finite for all \(x,t\in {\mathbb R}\), which is called Zygmund function on \({\mathbb R}\), then \(\max_{x\in[0,1]} |f(x)|\leq (11/32)\|f\|_z\). As an application, we obtain a better estimate for the Zygmund bound in the Zygmund class.