We consider in this work a classical online scheduling problem with release times on a single machine. The quality of service of a job is measured by its stretch, which is defined as the ratio of its response time over its processing time. Our objective is to schedule the jobs non-preemptively in order to optimize the maximum stretch. We present both positive and negative theoretical results. First, we provide an online algorithm based on a waiting strategy which is (1 + √ 5−1 2 ∆)-competitive where ∆ is the upper bound on the ratio of processing times of any two jobs. Then, we show that no online algorithm has a competitive ratio better than 1 + √ 5−1 2 ∆. The proposed algorithm is asymptotically the best algorithm for optimizing the maximum stretch on a single machine.