On the second moduli of continuity

We prove an inequality for the second moduli of continuity of continuous functions. Applying this inequality, we construct a nonnegative nonincreasing continuous function $\omega$ on $[0,+\infty)$ that vanishes at zero and is such that the function $\omega(\delta)/\delta^2$ decreases on $(0,+\infty)$ while $\omega$ is not asymptotically (as $\delta\to0$) equivalent to the second modulus of continuity of any continuous function.