Let $0 < p < q \leq \infty$ and suppose $ E\subset \mathbb{R}^N$ with $m(E)=1$ (where $m$ is the Lebesgue measure). I am asked to find necessary and sufficient conditions for: $ ( \int_E{|f|}^pdx)^{1/p} = (\int_E{|f|}^qdx)^{1/q}$ I know how to prove that that LHS $\leq$ RHS, this is done with Jensen's inequality in its measure-theoretic form, considering the function $\phi(t)=t^{q/p}$, which is convex, because $p. This is how it works:
$(\int_E{|f|^pdx})^{q/p}=\phi(\int_E{|f|^p}dx) \leq \int_E{\phi(|f|^p)dx}= \int_E{|f|^qdx }$
I was guessing that the necessary and sufficient condition required in the excercize is that $f(x) = 0$ almost everywhere (EDIT as observed in the comments, the condition is also satisfied if $f$ is such that $|f|$ is constant a.e.), but I'm stuck trying to prove it. I tried manipulating the exponents, but nothing seems to work; maybe some result on the inverse implication in Jensen's inequality could help, but I can't find any.
I hope someone can point me in the right direction, thanks in advance!