For a non-zero function f in H1 , the classical Hardy space on the unit disc, we put Sf= {g E H1 : argf(i8 ) = argg(ei0) a.e. 0}. The intersection of Sf and the unit sphere in H1 is just a set of solutions of some extremal problem in H1 It is known that Sf can be represented in the form Sf = S x g0, where β is a Blaschke product and g0 is a function in H1 with S90 = {Λ· g0 : Λ> O}. Also it is known that the linear span of Sf is of finite dimensional if and only if β is a finite Blaschke product, and when β is a finite Blaschke product, we can describe completely the set Sβ and the zeros of functions in Sβ. In this paper, we study the set of zeros of functions in Sβ when β is an infinite Blaschke product whose set of singularities is not the whole circle. Especially we study the behavior of zeros of functions in Sβ in the sectors of the form: Δ = { reiQ : 0 < r <_ 1, c1 < 0 < c2} on which the zeros of B has no accumulation points, and establish a convergence order theorem of zeros in Δ of functions in Sβ
