An extremal problem for classes of convolutions that do not increase variation

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.