Concavity in Fractional Calculus

Abstract

We discuss a concavity like property for functions $u$ satisfying $D^\alpha_{0^+} u\in C[0,b]$ with $u(0)=0$ and $-D^\alpha_{0^+} u(t)\ge 0$ for all $t\in[0,b]$. We develop the property for $\alpha\in(1,2]$, where $D^\alpha_{0^+}$ is the standard Riemann-Liouville fractional derivative.  We observe the property is also valid in the case $\alpha =1$. Finally, we show that under certain conditions, $-D^\alpha_{0^+} u(t)\ge 0$ implies $u$ is concave in the classical sense.