Geodesic
Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces
Matematičeskie zametki, Tome 63 (1998) no. 3, pp. 332-342 
V. F. Babenko; V. A. Kofanov; S. A. Pichugov. Additive inequalities for intermediate derivatives of differentiable mappings of Banach spaces. Matematičeskie zametki, Tome 63 (1998) no. 3, pp. 332-342. http://geodesic.mathdoc.fr/item/MZM_1998_63_3_a1/
@article{MZM_1998_63_3_a1,
     author = {V. F. Babenko and V. A. Kofanov and S. A. Pichugov},
     title = {Additive inequalities for intermediate derivatives of differentiable mappings of {Banach} spaces},
     journal = {Matemati\v{c}eskie zametki},
     pages = {332--342},
     year = {1998},
     volume = {63},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_63_3_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Suppose that $X$ and $Y$ are real Banach spaces, $U\subset X$ is an open bounded set star-shaped with respect to some point, $n,k\in\mathbb N$, $k, and $M_{n,k}(U,Y)$ is the sharp constant in the Markov type inequality for derivatives of polynomial mappings. It is proved that for any $M\ge M_{n,k}(U,Y)$ there exists a constant $B>0$ such that for any function$f\in C^n(U,Y)$ the following inequality holds: $$ |\kern -.8pt|\kern -.8pt|f^{(k)}|\kern -.8pt|\kern -.8pt|_U\le M|\kern -.8pt|\kern -.8pt|f|\kern -.8pt|\kern -.8pt|_U+B|\kern -.8pt|\kern -.8pt|f^{(n)}|\kern -.8pt|\kern -.8pt|_U. $$ The constant $M=M_{n-1,k}(U,Y)$ is best possible in the sense that $M_{n-1,k}(U,Y)=\inf M$, where $\inf$ is taken over all $M$ such that for some $B>0$ the estimate holds for all $f\in C^n(U,Y)$.

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