© Hindawi Publishing Corp. ON THE MAXIMUM VALUE FOR ZYGMUND CLASS ON AN INTERVAL

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. 2000 Mathematics Subject Classification: 30C62, 30C75. 1. Introduction and the main results. A continuous real-valued function f(x) on R is said to belong to the class Λ∗(R) if there exists a constant C such that ∣∣f(x+t)−2f(x)+f(x−t)∣∣ ≤ C|t|, (1.1) for all x,t ∈ R. This class introduced by Zygmund [8] is called Zygmund class, and we denote the infimum of the values C in (1.1) by ‖f‖z. Gardiner and Sullivan [6