On the maximum value for Zygmund class on an interval

Abstract

We prove that if f(z) is a continuous real-valued function on ℝ with the properties f(0)=f(1)=0 and that ‖f‖ z =infx,t|f(x+t)−2f(x)+f(x−t)/t|is finite for all x,t∈ℝ, which is called Zygmund function on ℝ, then maxx∈[0,1]|f(x)|≤(11/32)‖f‖z. As an application, we obtain a better estimate for Skedwed Zygmund bound in Zygmund class.