Sharp Lp-bounds for a Small Perturbation of Burkholder's Martingale Transform

Let {d(k)}(k >= 0) be a complex martingale difference in L-p[0, 1], where 1 < p < infinity, and {epsilon(k)}(k >= 0) be a sequence in {+/- 1}. We obtain the following generalization of Burkholder's famous result. If tau is an element of [-1/2, 1/2] and n is an element of Z(+), then parallel to Sigma(n)(k=0) ((epsilon k)(tau))d(k)parallel to(Lp([0,1),C2)) <= ((p* - 1)(2) + tau(2))(1/2)parallel to Sigma(n)(k=0) d(k)parallel to(Lp([0,1),C)), where ((p* -1)(2) + tau(2))(1/2) is sharp and p* - 1 = max{p - 1, 1/(p - 1)}. For 2 <= p < infinity the result is also true with sharp constant for vertical bar tau vertical bar <= 1.