Negative results in form-preserving approximation of higher order (Q2387806)

Let \(f\) be \(q\)-convex on the closed interval \([-1, 1]\) and also let \(f\) belong to the class of functions possessing an absolutely continuous \((r-1)\)st derivative on \([-1, 1]\). Let \(E_n^{q} f\) denote the value of the best \(q\)-convex polynomial approximation of such an \(f\). The authors formulate a proposition involving an inequality for an estimation of \(E_n^{q} f\) in terms the \(L_{\infty}\) norm of \(f^r\). The said proposition is known to be valid in the cases: (i) \(q=1; r=1, r\geq 2\), (ii) \(q>1; r=1, 2\), and (iii) \(q=2, r>2\). For \(q=3, r\geq 4\) the proposition is an open problem. Also it is known that for \(q\geq 4, r>2\) the said inequality in the Proposition is not valid. In this paper the authors give a refinement on the known result for the case \(q\geq 4, r>2\). They further give two theorems to study the cases (A) \(q>3, r<(q-1)\) and (B) \(q\geq 4, r\geq (q-1)\).