Geodesic
An extremal problem for classes of convolutions that do not increase variation
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (1982), pp. 3-7 
Nguyen Thi Thien Hoa. An extremal problem for classes of convolutions that do not increase variation. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (1982), pp. 3-7. http://geodesic.mathdoc.fr/item/VMUMM_1982_5_a0/
@article{VMUMM_1982_5_a0,
     author = {Nguyen Thi Thien Hoa},
     title = {An extremal problem for classes of convolutions that do not increase variation},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {3--7},
     year = {1982},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1982_5_a0/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We prove the following. Let $\Lambda_1$ and $\Lambda_2$ be variation-diminishing operators of the convolution type and $0<\varepsilon<1$. Then there exists $\widehat{h}$ such that $\|(\Lambda_2\circ\Lambda_1\varepsilon_{0,\widehat{h}})(\cdot)\|_{L_\infty(\mathbf R)} =\varepsilon$, where $\varepsilon_{0,h}(x)=\operatorname{sign}\sin\frac{\pi x}h$ and for every function $u_0(\cdot)$ with $\|u_0(\cdot)\|_{L_\infty(\mathbf R)}\leq1$ and $\|(\Lambda_2\circ\Lambda_1u_0)(\cdot)\|_{L_\infty(\mathbf R)}\leq\varepsilon$ we have $\|\Lambda_1u_0(\cdot)\|_{L_\infty(\mathbf R)}\leq\|\Lambda_1\varepsilon_{0,\widehat{\mathbf R}}(\cdot)\|_{L_\infty(\mathbf R)}$. This result generalizes a theorem of A. N. Kolmogorov on inequalities for the derivatives and some other like theorems.