Maximum principle for certain generalized time and space fractional diffusion equations

We present an inequality for fractional derivatives related to the Leibniz rule that not only answers positively a conjecture raised by J. I. Diaz, T. Pierantozi, and L. Vzquez but also helps us to obtain a modern proof of the maximum principle for fractional differential equations. The inequality turns out to be versatile in nature as it can be used to obtain a priori estimates for many fractional differential problems.