A survey of useful inequalities in fractional calculus

Abstract

Abstract We present a survey on inequalities in fractional calculus that have proven to be very useful in analyzing differential equations. We mention in particular, a “Leibniz inequality” for fractional derivatives of Riesz, Riemann-Liouville or Caputo type and its generalization to the d-dimensional case that become a key tool in differential equations; they have been used to obtain upper bounds on solutions leading to global solvability, to obtain Lyapunov stability results, and to obtain blowing-up solutions via diverging in a finite time lower bounds. We will also mention the weakly singular Gronwall inequality of Henry and its variants, principally by Medved, that plays an important role in differential equations of any kind. We will also recall some “traditional” inequalities involving fractional derivatives or fractional powers of the Laplacian.