Geodesic
Bernstein-type inequalities in a Banach space
Matematičeskie zametki, Tome 17 (1975) no. 6, pp. 925-937 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove in a general scheme of Banach spaces, that when a Jackson inequality of an appropriate order holds, Bernstein-type inequalities and their sharpenings in the sense of Stechkin and Nikolskii are equivalent to one another. 
@article{MZM_1975_17_6_a10,
     author = {K. Sherer},
     title = {Bernstein-type inequalities in {a~Banach} space},
     journal = {Matemati\v{c}eskie zametki},
     pages = {925--937},
     year = {1975},
     volume = {17},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_6_a10/}
}
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K. Sherer. Bernstein-type inequalities in a Banach space. Matematičeskie zametki, Tome 17 (1975) no. 6, pp. 925-937. http://geodesic.mathdoc.fr/item/MZM_1975_17_6_a10/