Let $f:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function . If for any $\gamma>1$ and $|\xi|\geq r$ for any $r>0$ we have $0<\gamma F(x,\xi)\leq\xi f(x,\xi)$ where $\displaystyle F(x,\xi)=\int_0^\xi f(x,\tau)\,d\tau$ for all $x\in\mathbb{R}^n$ , then prove that $$F(x,\xi)\geq r^{-\gamma}\min\{F(x,r),F(x,-r)\}|\xi|^\gamma-\max_{|\xi|\leq r}F(x,\xi)-\min\{F(x,r),F(x,-r)\}$$ for all $x\in\mathbb{R}^n \ , \ \xi\in\mathbb{R}$ .
It is easy to see $\displaystyle F(x,\xi)\leq\max_{|\xi|\leq r}F(x,\xi)$ for all $|\xi|\leq r$ . Also $\left(\dfrac{|\xi|}{r}\right)^\gamma-1\geq0$ for $|\xi|\geq r$ since $\gamma>1$ . But to combine these two , how the minimum is contributing there I can't understand . Any help is appreciated .