Definition 2.3. The Riemann-Liouville fractional integral operator of order $\alpha \geq 0$, for a function $f \in C_\mu,(\mu \geq-1)$ is defined as $$ \begin{aligned} J^\alpha f(t) & =\frac{1}{\Gamma(\alpha)} \int_0^t(t-\tau)^{\alpha-1} f(\tau) d \tau ; \quad \alpha>0, t>0, \\ J^0 f(t) & =f(t), \end{aligned} $$ where $\Gamma(\alpha):=\int_0^{\infty} e^{-u} u^{\alpha-1} d u$.
Theorem 3.1. Let $f$ and $g$ be two synchronous functions on $[0, \infty[$. Then for all $t>0, \alpha>0$, we have: $$ J^\alpha(f g)(t) \geq \frac{\Gamma(\alpha+1)}{t^\alpha} J^\alpha f(t) J^\alpha g(t) . $$
Question: does there exist a reverse of this result, i.e. something of the form: $$ J^\alpha(f g)(t) \leq C J^\alpha f(t) J^\alpha g(t) . $$