If $f \in L^P(E)$, is $f^* = ||f||_p^{1-q} \cdot sgn(f) \cdot |f|^{p-1}$ the right expression for the conjugate of $f$?

Problem: For $p \in [1,\infty)$, $q$ the conjugate of $p$, and $f \in L^P(E)$m show that $$||f||_p = \max_{g \in L^q(E), ||g||_q \le 1} \int_E fg$$. Proof: Assuming that $||g||_q \le 1$, the Holder inequality and monotonicity of the integral tell us that $\int_{E} fg \le \int_E |fg| \le ||f||_p ||g||_q \le ||f||_p$, so that $\max_{||g||_q \le 1} \int_E fg \le ||f||_p$. But $f^* \in L^q(E)$ and satisfies $||f^*||_q$ and $\int_E f f^*$ = ||f||_p$...

I still don't know the right expression for $f^*$, but I know it satisfies the properties I mentioned in my proof.

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