On one property of the modulus of continuity for periodic functions of higher orders

Abstract

UDC 517.5

For the moduli of continuity of $2\pi$-periodic functions $\omega_k(f,h)$ of order $k = 1,2,\ldots, $ we prove the inequalities
$$
\omega_k(f,\pi)\leq\frac{2^k}{C_k^{[\frac{k}{2}]}}\frac{1}{\pi}\int\limits_0^{\pi}\omega_k(f,h)dh,
$$
for even $k.$
The inequalities are exact in the spaces $C_{2\pi}$ and $L_1[-\pi,\pi]$.