# Equality holds in Minkowski's Inequality when

by benorin
Tags: equality, holds, inequality, minkowski
 HW Helper P: 1,025 I would like to determine necessary and sufficient conditions for equality to hold in Minkowski's Inequality in $L^{p}(X,\mu)$. For $1\leq p\leq \infty,$ we have $\forall f,g\in L^{p}(X,\mu)$ $$\left\{\int_X \left| f+g\right| ^{p} d\mu\right\} ^{\frac{1}{p}} \leq \left\{\int_X \left| f\right| ^{p} d\mu\right\} ^{\frac{1}{p}} + \left\{\int_X \left| g\right| ^{p} d\mu\right\} ^{\frac{1}{p}}$$ here I wish to allow f and g to be complex. Any help would be nice.
 HW Helper P: 1,025 So I think I have all the pieces, but I lack your brain power. Minkowski's Inequality is proved (in Rudin's text) via Holder's Inequality, namely For $\frac{1}{p}+\frac{1}{q}=1$, [itex]1< p< \infty,1