Polynomials with graphs of maximum length (Q1966209)

Let \(g\) and \(G\) be continuous functions defined on an interval \([a,b]\) such that \(g<G\) that is, \(g(x) < G(x)\) for all \(x\in [a,b]\). Let us also denote the set of all polynomials \(p\) of degree not greater than \(n\) and satisfying \(g \leq p\leq G\) by \(P_n (g,G)\). \textit{E. P. Dolzhenko} and \textit{E. A. Sevast'yanov} [Vestn. Mosk. Univ., Ser. I 1994, No. 3, 49-59 (1994; Zbl 0884.41008)] have constructed functions \(g\) and \(G\) for \(n=2\) such that the graph length of each of the two snakes is less than 0.76 times that of a certain function from \(P_2(g,G)\). The author is able to show interalia that 0.625 is the smallest ratio possible for the foregoing result.