Monic Polynomials with Extremal Properties Original Research Article

We denote by αn, m and βn, m, n ≥ m + 2, m ≥ 0, the magnitudes [formula] extended over all the monic polynomials of degree n, where ||·||p = (∫1−1 |·|pdx)1/p, for p = 1,2, respectively. For p = 1 we give an asymptotically exact estimate of the values of αn, m for m = o([formula]) and we point out the polynomials which attain this estimate; for p = 2 we obtain 2−2n + 2m + 2αn−m ≤ βn, m ≤ 2−2n + 3m + 2αneO(m2/n), with 12 < αn−m < αn < 3/2.