Inequalities for fractional derivatives via the Marchaud derivative

We study the Marchaud fractional derivative. We pay special attention to when the Marchaud fractional derivative is equal to the well-known Caputo or Riemann-Liouville fractional derivative. Conditions when this equality held were given in the interesting paper of Vainikko (2016). Several recent papers have used results from that paper in discussing inequalities that are useful in the study of stability. We have found some gaps in the proofs in the Vainikko paper but we give a proof of the most useful parts;in fact we also prove that equality holds under a more general condition. We use this equality to prove various inequalities for fractional derivatives, including a maximum principle, often under weaker conditions than previously given. In particular we prove strong versions of inequalities for differentiable convex functions that are useful in studying stability by Lyapunov's method. For more information see https://ejde.math.txstate.edu/Volumes/2025/20/abstr.html