I just wonder if someone might give a hint. I've thought to apply Minkowski inequality, but I couldn't find out the way.
Given $2 \leq p<\infty$. Show that for any real-valued functions $f,g \in L^p(\mathbb{R})$, the following holds: \begin{align*} 2 \left( \left\|\frac{f}{2} \right\|_{L^p}^p+ \left\|\frac{g}{2} \right\|_{L^p}^p\right)\leq \left\|\frac{f-g}{2} \right\|_{L^p}^p+ \left\|\frac{f+g}{2} \right\|_{L^p}^p\leq \frac{1}{2}\left(\|f\|_{L^p}^p+\|g\|_{L^p}^p \right). \end{align*}
Proof: For the right hand side I realized the following: \begin{align*} \left\|\frac{f-g}{2} \right\|_{L^p}^p+ \left\|\frac{f+g}{2} \right\|_{L^p}^p &\leq \frac{1}{2^p}(\|f\|_{L^p}^p+\|g\|_{L^p}^p+\|f\|_{L^p}^p+\|g\|_{L^p}^p ) \quad \textrm{ by Minkowski inequality} \\ &=\frac{1}{2^{p-1}}(\|f\|_{L^p}^p+\|g\|_{L^p}^p) \leq \frac{1}{2}(\|f\|_{L^p}^p+\|g\|_{L^p}^p). \end{align*} For the left hand side, we can observe that $$ f=\frac{f+g}{2}+\frac{f-g}{2} \quad \textrm{ and } \quad g=\frac{g+f}{2}-\frac{f-g}{2}. $$ Hence, \begin{align*} \left\|\frac{\frac{f+g}{2}+\frac{f-g}{2}}{2} \right\|_{L^p}^p+ \left\|\frac{\frac{f+g}{2}-\frac{f-g}{2}}{2} \right\|_{L^p}^p&\leq \frac{2}{2^p}\left( \left\|\frac{f-g}{2} \right\|_{L^p}^p+ \left\|\frac{f+g}{2} \right\|_{L^p}^p \right) \quad \textrm{by Minkowski inequality} \\ & \leq \frac{1}{2}\left( \left\|\frac{f-g}{2} \right\|_{L^p}^p+ \left\|\frac{f+g}{2} \right\|_{L^p}^p\right) \end{align*} which is equivalent to $$ 2 \left( \left\|\frac{f}{2} \right\|_{L^p}^p+ \left\|\frac{g}{2} \right\|_{L^p}^p\right)\leq \left\|\frac{f-g}{2} \right\|_{L^p}^p+ \left\|\frac{f+g}{2} \right\|_{L^p}^p. $$
All you have to do is apply the right-hand side that you successfully proved to $F=\frac{f+g}2$ and $G=\frac{f-g}2$.