Comonotone Second Jackson's Inequality

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Let 2s points yᵢ=-π≤ y₂ₛ<<y₁<π be given. Using these points, we define the points yᵢ for all integer indices i by the equality yᵢ=yᵢ₊₂ₛ+2π. We shall write f∈⁽¹⁾(Y) if f is a 2π-periodic function and f does not decrease on [yᵢ, yᵢ₋₁] if i is odd; and f does not increase on [yᵢ, yᵢ₋₁] if i is even. We denote Eₙ⁽¹⁾(f;Y) the value of the best uniform comonotone approximation. In this article the following Theorem -- the comonotone analogue of second Jackson's Inequality -- is proved. Theorem. If f∈⁽¹⁾(Y) Wʳ, r>2, then Eₙ⁽¹⁾(f;Y)≤ cnʳ, where c=c(r,Y)=const depending only on r and Y, Wʳ Sobolev space.