Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to find the the supremum of $\dfrac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}} $ in the unit ball centered at the origin . Here $x\neq y$, $x,y \in \mathbb R^n$. How do I proceed ? Thank you for your help.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Hint: By a modification of the triangular inequality you see that $$||x|^{2/3}-|y|^{2/3}|\leq |x-y|^{2/3}$$ for all $x\neq y$. So we have$$\dfrac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}}\leq 1$$ Is the value of the function $=1$ at some point?

share|improve this answer
    
Yup at $x=0, y=1$ . –  Theorem Jul 13 '12 at 10:51
    
So does this answer your question? –  Simon Markett Jul 13 '12 at 10:59
    
+1 for the maieutics. –  Did Jul 13 '12 at 11:31
    
@SimonMarkett : Yup :) . thanks a ton . –  Theorem Jul 13 '12 at 11:41
    
Btw it's $x=1$, $y=0$. @did I had to look up maieutics...but thx ;) –  Simon Markett Jul 13 '12 at 11:48

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.