Let $ x(t) \in \mathbb{R}$ be a continuous and derivable function. Then, for any time instant $ t \geq t_0$ \begin{equation} \label{11.44} \frac{1}{2} D^{\alpha} x^2(t) \leq x(t) D^{\alpha} x(t), \ \ \forall \alpha \in (0, 1), \end{equation} that $ D^\alpha$ is caputo derivative. Proving that expression is true, is equivalent to prove that
$ x(t) D^{\alpha} x(t) – \frac{1}{2} D^{\alpha} x^2(t) \geq 0, \ \ \forall \alpha \in (0, 1)$ .
After simplicity we have: \begin{equation}\label{11.67} \biggl[\frac{y^2(t_0)}{2\Gamma (1- \alpha) (t- t_0)^\alpha}\biggr]+ \frac{\alpha}{2\Gamma (1- \alpha)} \int_{t_0}^{t} \frac{[x(t)-x(\tau)]^2}{(t-\tau)^{\alpha+1}} \,d\tau \geq 0. \end{equation} If we prove that the function $ f(\tau)=\frac{[x(t)-x(\tau)]^2}{(t-\tau)^{\alpha+1}}$ is integrable, then the theorem is proved. Can this result be obtained?
