We will also use some inequalities derived from the above ones. Our approach will be based
on the following ideas: If a sharp inequality in x is valid for x greater than a large value M , if we want to use that inequality for, say, √3 x, the inequality we derive will only be valid (without
further arguments) for x larger than M 3. It is likely that M 3 is a very large number, sometimes being out of reach for computer checking of various relations. One way of dealing with this problem is to weaken a little bit the initial sharp inequality, and try to balance this loss by a smaller "starting point". This approach might lead us to inequalities which better fit the particular problems we are facing.