Integral inequalities for self-reciprocal polynomials

Let n >= 1 be an integer and let P(n) be the class of polynomials P of degree at most n satisfying z(n) P(1/z) = P(z) for all z is an element of C. Moreover, let r be an integer with 1 <= r <= n. Then we have for all P is an element of P (n) : alpha(n)(r) integral(2 pi)(0) vertical bar P(e(it))vertical bar(2)dt <= integral(2 pi)(0) vertical bar P((r))(e(it))vertical bar(2)dt <= beta(n)(r) integral(2 pi)(0) vertical bar P(e(it))vertical bar(2)dt with the best possible factors beta(n)(r) = 1/2 Pi(r-1)(j-0) (n - j)(2). alpha(n)(r) = {Pi(r-1)(j=0) (n/2 - j)(2), if n is even, 1/2[Pi(r-1)(j=0) (n+1/2 - j)(2) + Pi(r-1)(j=0) (n-1/2 - j)(2)], if n is odd, This refines and extends a result due to Aziz and Zatgur (1997).