Geodesic
Relations between moduli of smoothness in different metrics
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2009), pp. 17-25

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Two inequalities are proved in the paper. These inequalities extend and refine the well-known P. L. Yl'yanov's inequality concerning relations between moduli of continuity in various metrics. 
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M. K. Potapov; B. V. Simonov; S. Yu. Tikhonov. Relations between moduli of smoothness in different metrics. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (2009), pp. 17-25. http://geodesic.mathdoc.fr/item/VMUMM_2009_3_a3/