We prove the inequality
$$ \overline{\omega}(t)\le \underset{s>0}{\operatorname{inf}}\left(\omega \left(\frac{s}{2}\right)+\frac{\omega (s)}{s}t\right), $$
where ω(t) is a function of the modulus-of-continuity type and \( \overline{\omega}(t) \) is its smallest concave majorant.
The consequences obtained for Jackson’s inequalities in C2πare presented.
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