Inspired by the accuracy and efficiency of the $\gamma$-norm of a matrix, which is closer to the original rank minimization problem than nuclear norm minimization (NNM), we generalize the $\gamma$-norm of a matrix to tensors and propose a new tensor completion approach within the tensor singular value decomposition (t-svd) framework. An efficient algorithm, which combines the augmented Lagrange multiplier and closed-resolution of a cubic equation, was developed to solve the associated nonconvex tensor multi-rank minimization problem. Experimental results show that the proposed approach has an advantage over current state of the art algorithms in recovery accuracy.