Convergence in $L_1$ and uniformly bounded in $L^2$ implies convergence in $L^p$, $1<p<2$.

Question

I am studying a qual and I do not know how to do the following question.

Let $(f_n)$ be a sequence of Lebesgue measurable functions on $\mathbb{R}$ that converges to $f$ in $L^1$. Suppose in addition that $f_n\in L^2$ and there exists a constant $M$ such that $\|f_n\|_2<M$ for all $n$. I have already known $f\in L^2$. But I would like to prove that $f_n\rightarrow f$ in $L^p$, where $1<p<2$.

I have tried using Holder's inequality but without success. Any help will be appreciated.

Answer 1

Hint: Hölder says $$ \begin{align} \int_A|f(x)|^p\,\mathrm{d}x &=\int_A|f(x)|^{2-p}|f(x)|^{2(p-1)}\,\mathrm{d}x\\ &\le\left(\int_A|f(x)|\,\mathrm{d}x\right)^{2-p} \left(\int_A|f(x)|^2\,\mathrm{d}x\right)^{p-1}\tag{1} \end{align} $$ That is, $$ \|f\|_p^p\le\|f\|_1^{2-p}\|f\|_2^{2(p-1)}\tag{2} $$