International Journal of Mathematics and Mathematical Sciences 
Volume 32 (2002), Issue 2, Pages 65-71
doi:10.1155/S0161171202202306

On the maximum value for Zygmund class on an interval

Huang Xinzhong,1 Oh Sang Kwon,2 and Jun Eak Park2

 1Department of Applied Mathematics, Huaqiao National University, Quanzhou, Fujian 362011, China
2Department of Mathematics, Kyungsung University, Pusan 608-736, China
Received 2 February 2002

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.