  In this note we introduce a new family of entropy powers which are related to
generalized entropies, called Sharma-Mittal entropies, and we prove their
concavity along diffusion processes generated by $L^2$-Wasserstein gradient
flows of corresponding entropy functionals. This result extends the result of
Savar\'e and Toscani on the concavity of R\'enyi entropy powers (IEEE Trans.
Inf. Theory, 2014) and reveals a connection to R\'enyi entropy power
inequalities by Bobkov and Marsiglietti (IEEE Trans. Inf. Theory, 2017).
