I have this condition $$ 0<\mu F(x, t) \leq t f(x, t), \text { for a.e } x \in \Omega, \quad 0<|t| \leq \gamma. $$ where $F(x,t)=\int_0^t f(x,s) ds$.
I want to deduce that there exists $c>0$ such that
$$ F(x, t) \geq c_{0}|t|^{\mu}, \quad \text { for a.e } x \in \Omega,|t| \leq \gamma.$$
I do this \begin{align*} \mu F(x,t)\leq t f(x,t)&\Rightarrow \dfrac{\mu}{t}\leq \dfrac{f(x,t)}{F(x,t)}\\ &\Rightarrow \int \dfrac{\mu}{t}\leq \int \dfrac{f(x,t)}{F(x,t)}\\ & \Rightarrow \mu\ln(|t|)+c\leq \ln(|F(x,t)|) \\ &\Rightarrow c'|t|^{\mu}\leq |F(x,t)| \end{align*}
How to delete the absulut value from F(x,t) ?