### Proposition 5.5

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

### Proof

We only give a sketch of parts of the proof, leaving all the details to the reader.

Set so that . On the other hand, set

An explicit computation shows that we have

We express these quantities on purpose only in terms of b, P , Q and D, and will use the inequalities . When we want to find an upper bound for an expression in a, b, c, d, we consider this expression first as a function of the single parameter b which is constrained to satisfy . This is done by taking derivatives and looking at the value of the expression where the derivative vanishes and at the endpoints. We then have a function of P, Q and D only. In most cases, this expression does not even depend on P, and hence we again compute derivatives and look at endpoints to find the maximum as Q varies from 0 to . When the expression depends on P, we use the inequalities .

We give one example, the bound for bc. We set . Then

We find that the derivative in B of the RHS vanishes for

and for these values we have , the signs being independent. The study of the 4 functions

on the interval shows that the largest maximum is attained with both + signs for x=1, and is equal to 12, hence a local maximum of |bc| is attained for and is equal to . Furthermore, looking at the endpoints B=0 and , we see that bc is respectively 0 and , which is smaller that thus giving the desired inequality for |bc|. However, in addition we have , hence since a>0, if and only if .

When , we must have , hence in the expression for a the expression under the square root must be larger than , or equivalently which shows the lower inequality for b. Hence implies c<0. This implies that if bc<0, we must have b>0 and c<0. But in that case, the - sign must hold in the expression for c, and hence since b>0, in the 4 functions above the second sign must in fact be a - sign. If we look at the remaining two functions, we see that the local maximum of |bc| is now only . The endpoints are B=0 and B=P in this case, but then bc vanishes at both endpoints so the maximum is really for bc<0.

For the two inequalities involving , we get in fact a fourth degree equation in , but luckily this equation has B=P as double root, hence we are reduced to a second degree equation one again.

The details of this and of the other inequalities are left to the patient reader.