I am working on a homework question that has given me some difficulty.
The Question
Let $1<p\le2$ and let $f,g\in L^p(\Omega)$. Use Hanner's inequality,
$$2^p\left(||f||_p^p + ||g||^p_p\right)\ge(||f+g||_p+||f-g||_p)^p + \bigg|||f+g||_p-||f-g||_p\bigg|^p$$
to show that if $f_n$ weakly converges to $f$ in $L^p(\Omega)$, and $||f_n||_p \to ||f||_p$, then $||f_n-f||_p \to 0$.
My Work
I am aware that Hanner's inequality implies uniform convexity which implies the desired result. However the proofs I have seen that uniform convexity are very long and I don't imagine this is the scope of the question.
I tried substituting $g = f_n$ into Hanner, throwing away the second term on the left hand side and taking limits on both sides, however I am not sure the limit even exists on the left hand side.
Apply Hanner's inequality to $g=f_n$ for a fixed $n$ then take $\limsup_{n\to +\infty}$ to get (in view of the assumption of convergence of the norms) $$\tag{*} 2^{p+1}\lVert f\rVert_p^p\geqslant \limsup_{n\to +\infty}\left(\left(a_n+b_n\right)^p+\left\lvert a_n-b_n\right\rvert^p\right) $$ where $a_n=\lVert f+f_n\rVert_p$ and $b_n=\lVert f_n-f\rVert_p$. Note that sequence $(b_n)_{n\geqslant 1}$ is bounded; let $B:=\limsup_{n\to +\infty}b_n$ and $n_k\uparrow +\infty$ such that $b_{n_k}\to B$. Then by (*) and the weak convergence of $f+f_{n_k}$ to $2f$, we derive that $$ 2^{p+1}\lVert f\rVert_p^p\geqslant\left(\left(2\lVert f\rVert_p+B\right)^p+\left\lvert 2\lVert f\rVert_p-B\right\rvert^p\right). $$ Using the fact that the function $t\mapsto \left(2\lVert f\rVert_p+t\right)^p+\left\lvert 2\lVert f\rVert_p-t\right\rvert^p$ reaches its unique minimum at $t_0= 0$, we derive that the only possibility is $B=0$.