Geodesic
On the second moduli of continuity
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 150-152.

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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. 
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S. V. Konyagin. On the second moduli of continuity. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 150-152. http://geodesic.mathdoc.fr/item/TM_2010_269_a11/

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