Let \(1\leq p\leq \infty\), \(L_ p\) be the space of 1-periodic functions f(x) with the norm \(\| f\|_ p=(\int^{1}_{0}| f(x)| \;pdx)^{1/p}\) \(1\leq p<\infty\) and \(L_{\infty}=c\) be the space of continuous 1-periodic functions with the norm \[ \| f\|_ c=\sup | f(x)|, \] \(\omega (\delta,f)_ p\) is called the modulus of continuity \(\omega (\delta,f)_ p=\sup \{\| f(x+t)-f(x)\|_ p:\) \(0\leq t\leq \delta \}\) \(0\leq \delta \leq 1/2\). It is proved that for a function \(f\in L_ p\) \((2