Possible Duplicate:
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
In the condition $a,b \in[0,\infty)$, $1\le p<\infty$,
How can I conclude this inequality? $(a+b)^p \le 2^{p-1} (a^p + b^p)$
Possible Duplicate:
Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
In the condition $a,b \in[0,\infty)$, $1\le p<\infty$,
How can I conclude this inequality? $(a+b)^p \le 2^{p-1} (a^p + b^p)$
you could solve it using Hölder inequality. $ (a+b)^p \leq 2^{p-1}(a^p+b^p) \Leftrightarrow (a+b) \leq (1+1)^{1- \frac{1}{p}} (a^p +b^p)^{\frac{1}{p}} $ and then apply Hölder.
Hint: $f(x)=x^p$ and convexity
$f(\frac a2 + \frac b2) \leq \frac 12 f(a) + \frac 12 f(b)$