On oscillating polynomials
An algebraic polynomial that has all its zeros in a finite interval \([a,b]\) is called oscillating on \([a,b]\). In the present paper the authors prove the following Theorem. Let \(P\) be an oscillating on \([-1,1]\) polynomial of degree \(n\) with symmetric zeros, and let \(P\) attain its maximal absolute value at the endpoints. Then \(P'\) attains its maximal absolute value at the endpoints too. In fact, the authors prove more than that: for every oscillating polynomial \(P\) on \([-1,1]\) such that \(|P(-1)|=\|P\|\), each local extremum of \(P'\) from the first half is smaller in absolute values than \(|P'(-1)|\). As an application of the method developed in the paper the authors show that the classic Zolotarev polynomials have maximal derivative at one of the endpoints.
scientific article; zbMATH DE number 5817259
scientific article