On Comparison Theorems for Generalized Moduli of Continuity

In this note we will present sharpened versions of two theorems of Harold Shapiro on comparison between generalized moduli of continuity. By specialization of the measures defining the moduli of continuity in question our theorem gives a sharp form of the Jackson and Bernstein theorems. In particular our theorem implies the known fact that the order of best approximation by trigonometric polynomials for any continuous and periodic function f satisfies E n (f) = O ((log n) −1 ) if and only if the modulus of continuity of f is O (|log t| −1 ).