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Possible Duplicate:
Does $|x|^p$ with $0 satisfy the triangular inequality on $\mathbb{R}$?

Is the function

$f(t)= t^{\alpha},\quad \alpha\in (0,1)$

a subadditive function?

My teacher said categorically that this is true. But I'm not so sure.

EDIT: $~~~~~0\leq t\leq 1$

  • 0
    Try with $0\le t\le 1$.2012-06-05

1 Answers 1

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By homogeneity, in the inequality $(x+y)^{\alpha}\leq x^{\alpha}+y^{\alpha}$ just deal with the case $y=1$. Let $f(t):=t^{\alpha}+1-(t+1)^{\alpha}$. The derivative has the sign of $t^{\alpha-1}-(t+1)^{\alpha-1}$ which is non-negative since $\alpha<1$. Hence $f(t)\geq f(0)=0$ and we are done.

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    That's true. Thanks!2012-06-05