A majorant problem

Let f(z)= ￠ ‘k=0 ￠ akzk, a0 ￠ ‰ 0 be analytic in the unit disc. Any infinite complex vector =( 0, 1, 2, ￠ € |) such that | k|=1, k=0,1,2, ￠ € |, induces a function f (z)= ￠ ‘k=0 ￠ ak kzk which is still analytic in the unit disc.