Metric property of a real polynomial

For x ∈ R , let M ( P , δ ) be the measure m { x : | P ′ ( x ) / ( n P ( x ) ) | ⩾ δ } ( δ > 0 ), where P is an arbitrary polynomial of positive degree n . We prove thatsup Q M ( Q , δ ) = π / δ in the class of all real polynomials Q . As a corollary, we obtain the estimate of the derivative of a rational function, improving the known results of Gonchar and Dolzhenko, and prove that the inequality m { x : r ′ ( x ) / r ( x ) ⩾ n } ⩽ 2 π ( r is any real rational function of degree n ), conjectured by Borwein, Rakhmanov and Saff, is valid, at the least, for even and for odd functions r .