Lemma 6.4

Let be a reduced form such that . We have the following inequalities:



Then , and

and since we have and hence the RHS is an increasing function of R, so the inequality must be true at least for , thus giving

In particular, this implies

We also obtain from above


thus giving the inequalities and , or in other words and . The same inequalities can be obtained directly from (5).

Hence we obtain:

Now an easy computation shows that the maximum of subject to x, y nonnegative and is attained for

and is equal to , thus giving all but the last inequality of the lemma.

For the last inequality using and , we obtain

thus proving all the inequalities of the lemma.

Note that in the first four inequalities, we have used only the bound for A+B and not the stronger bound for . However it can be checked that this would not improve the final bounds. However, this is not the case for the next two inequalities.