A note on the connection between non-additive entropy and $h$-derivative

In order to study as a whole a wide part of entropy measures, we introduce a two-parameter non-extensive entropic form with respect to the $h$-derivative, which generalizes the conventional Newton--Leibniz calculus. This new entropy, $S_{h,h'}$, is proved to describe the non-extensive systems and recover several types of well-known non-extensive entropic expressions, such as the Tsallis entropy, the Abe entropy, the Shafee entropy, the Kaniadakis entropy and even the classical Boltzmann--Gibbs one. As a generalized entropy, its corresponding properties are also analyzed.