A majorant problem

Let f ( z ) = ∑ k = 0 ∞ a k z k , a 0 ≠ 0 be analytic in the unit disc. Any infinite complex vector θ = ( θ 0 , θ 1 , θ 2 , … ) such that | θ k | = 1 , k = 0 , 1 , 2 , … , induces a function f θ ( z ) = ∑ k = 0 ∞ a k θ k z k which is still analytic in the unit disc. In this paper we study the problem of maximizing the p -means: ∫ 0 2 π | f θ ( r e i ϕ ) | p d ϕ over all possible vectors θ and for values of r close to 0 and for all p < 2 . It is proved that a maximizing function is f 1 ( z ) = − | a 0 | + ∑ k = 1 ∞ | a k | z k and that r could be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets of f 1 . This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood. One consequence of our paper is that for p < 2 the extremal function for the Hardy-Littlewood problem should be − | a 0 | + ∑ k = 1 ∞ | a k | z k . We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.