Relations between upper bounds of absolute values of functions and their higher derivatives

In this paper we consider functions $f(t)$, $-\infty<t<+\infty$, which are $n$ times continuously differentiable with a given convex modulus of continuity of the $n$-th derivative. For a certain class of periodic functions we establish a relationship between upper bounds of the absolute values of a function and its $n$-th derivative.