On approximation of convex functions by rational ones in integral metrics (Q798903)

Let f be a real valued function on a bounded segment \(\Delta =[a,b]\) of the real axis and \(M(f)=Sup\{| f(x)|:x\in\Delta \}\). Let \(R_ n\) denote the set of all rational functions of degree not exceeding n, \(R_ n\) (f,p,\(\Delta)\) the best approximation of the function f by rational \(r\in R_ n\), in the metric of the space \(L_ p(\Delta)\), \(0<p<\infty\). Denoting by K(M,\(\Delta)\) the class of all convex functions on the segment \(\Delta\) such that M(f)\(\leq M\) the author obtains the following inequality which holds for every p, \(0<p<\infty\) and for \(n\in {\mathbb{N}}\), \(R_ n(M,p,\Delta)\leq Sup\{R_ n(f,p,\Delta);f\in K(M,\Delta)\}\).