Re: inequality

OK, I think I know how to prove this. We add a term in the middle:

(|x|^r x - |y|^r y)(x - y) > 1/2 (|x|^r + |y|^r) |x - y|^2 > c |x - y|
^{r + 2} (*)

The second inequality in (*) is quite simple, just use triangle and
Young's inequalities. As for the first one, we have

(|x|^r + |y|^r) |x - y|^2 = (|x|^r + |y|^r) (|x|^2 + |y|^2 - 2xy) =
|x|^{r + 2} + |y|^{r + 2} + { |x|^2 |y|^r + |y|^2 |x|^r } - 2xy(|x|^r
+ |y|^r)

Now with the aid of Young's inequality again, the terms in curly
brackets are estimated by

|x|^{r + 2} + |y|^{r + 2}

without any constant (this is important!) Combining these results, we
get the first inequality in (*).

P. K.
.

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