Littlewood‐Type Problems on [0,1]

We consider the problem of minimizing the uniform norm on [0, 1] over non‐zero polynomials p of the form p ( x ) = ∑ j = 0 n a j x jwith  | a j | ⩽ 1 ,a j ∈ C ,where the modulus of the first non‐zero coefficient is at least δ > 0. Essentially sharp bounds are given for this problem. An interesting related result states that there are absolute constants c 1 > 0 and c 2 > 0 such that exp ⁡ ( − c 1 n ) ⩽ inf 0 ≠ p ∈ F n∥ p ∥ [ 0 , 1 ] ⩽ exp ⁡ ( − c 2 n ) ,for every n ⩾ 2, where F n denotes the set of polynomials of degree at most n with coefficients from {−1, 0, 1}. This Chebyshev‐type problem is closely related to the question of how many zeros a polynomial from the above classes can have at 1. We also give essentially sharp bounds for this problem. Inter alia we prove that there is an absolute constant c > 0 such that every polynomial p of the form p ( x ) = ∑ j = 0 n a j x j , with  | a j | ⩽ 1 , | a 0 | = | a n | = 1 ,a j ∈ C ,has at most c n real zeros. This improves the old bound c n log ⁡ n given by Schur in 1933, as well as more recent related bounds of Bombieri and Vaaler, and, up to the constant c , this is the best possible result. All the analysis rests critically on the key estimate stating that there are absolute constants c 1 > 0 and c 2 > 0 such that | f ( 0 ) | c 1 / a ⩽ exp ⁡ ( c 2 / a ) ∥ f ∥ [ 1 − a , 1 ] ,for every f ∈ S and a ∈ (0, 1], where S denotes the collection of all analytic functions f on the open unit disk D := {z ∈ C: ∣z∣ < 1} that satisfy | f ( z ) | ⩽ 1 1 − | z |for  z ∈ D .