Renyi's alpha-divergences are known to play an important role in the quantification of trade-off relations in all sorts of problems in classical information theory, which makes it important to find their right quantum generalizations. One such generalization has been in use for some time by now, and has found an operational interpretation in the problems of the quantum Chernoff- and Hoeffding bounds, for the parameter values alpha between 0 and 1. Recently, another definition of the quantum Renyi divergences has been put forward by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203 (2013) , and Wilde, Winter, Yang, arXiv:1306.1586, which, among others, have the attractive property of encompassing many of the popular quantum divergences as special cases, including the min- and the max-relative entropy, the collision relative entropy and the relative entropy. Here we show that these new Renyi divergences also have an operational interpretation, for the values alpha>1, in the strong converse problem of quantum hypothesis testng. These results suggest that the operationally relevant definition of the quantum Renyi divergences depends on the parameter alpha: for alpha<1, the right choice seems to be the traditional definition, whereas for alpha>1, the right choice is the newly introduced version. We also discuss a subadditivity property of these new Renyi divergences, that leads to particularly simple proofs for the achievability parts of composite coding problems.