If p , q and r $\displaystyle \in$ positive real numbers , with at least one of them less than unity , prove that (1-p)(1-q)(1-r)>1-p-q-r

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- Jan 14th 2009, 11:37 PMmathaddictproving
If p , q and r $\displaystyle \in$ positive real numbers , with at least one of them less than unity , prove that (1-p)(1-q)(1-r)>1-p-q-r

- Jan 15th 2009, 12:24 AMNonCommAlg
after expanding the LHS, your inequality becomes: $\displaystyle pq+qr+rp > pqr.$ dividing by $\displaystyle pqr$ gives us: $\displaystyle \frac{1}{p}+\frac{1}{q}+\frac{1}{r} > 1,$ which is true because we have that at least one of $\displaystyle \frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ is bigger than 1.

- Jan 15th 2009, 06:51 AMmathaddict
- Jan 15th 2009, 07:43 AMIsomorphism