We consider a variant of the online semi-definite programming problem: The decision space consists of positive semi-definite matrices with bounded diagonal entries and bounded Γ- trace norm, which is a generalization of the trace norm defined by a positive definite matrix Γ. To solve this problem, we propose a follow-the-regularized-leader algorithm with a novel regularizer, which is a generalisation of the log-determinant function parameterized by the matrix Γ. Then we apply our algorithm to online binary matrix completion (OBMC) with side information and online similarity prediction with side information, and improve mistake bounds by logarithmic factors. In particular, for OBMC our mistake bound is optimal.