Recognitions:
Homework Help

## Equality holds in Minkowski's Inequality when

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.
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 Recognitions: Homework Help 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
 when one function is a constant multiple of the other, ie f = ag or somethng like that

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