I want to show that $|x|^p,p \geq 1$ is convex, for this i have to prove the inequality $|(1-\lambda )x+\lambda y)|^p \leq (1-\lambda)|x|^p+\lambda |y|^p $ for $\lambda \in (0,1)$ Can anyone prove this inequality? I have proved the convexity using the composition of two convex functions, one of which was increasing but I am interested in a direct proof of this inequality.
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Is this over $\mathbb{R}$ or more generally over $\mathbb{R}^n$, or some inner-product space? – Julien Mar 20 '13 at 12:24
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IT would be great if you prove this for $\mathbb R^n$, but you are welcome to prove this in $\mathbb R$ – Mathematician Mar 20 '13 at 12:26
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I don't suppose you want the proof that uses differentiation. – Gautam Shenoy Mar 20 '13 at 12:27
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no I don't, proof using characterization is not needed, i want a direct proof of this inequality – Mathematician Mar 20 '13 at 12:29
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@julien This is not a direct proof of the inequality – Mathematician Mar 20 '13 at 12:30
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If you say it is not needed, do you have a proof in mind? – Julien Mar 20 '13 at 12:30
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can't we prove this inequality without showing that $|x|^p$ is convex, ofcourse after the proof it would be obvious that indeed it is. – Mathematician Mar 20 '13 at 12:32
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If you can prove it for $\mathbb{R}$, the $\mathbb{R}^n$ case (or more generally the inner product space case) follows instantly. Now iti suffices to prove it for $x>0$ and $y>0$. And the best way to do that is to observe that the function is twice differentiable with positive second derivative there. – Julien Mar 20 '13 at 12:33