A compendium of inequalities -- how can it be used?

Posted on 2007-11-06 11:53 By JacquesC ( Maple Rating 4

1728)

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This morning, I saw the announcement of a paper on the ArXiv, titled Inequalities for Integer and Fractional Parts. I found the results very pretty, even though many of them are rather weird.

But then it struck me: this ought to be useful. In fact, it ought to be useful when doing experimental mathematics, something for which Maple is rather helpful. So why the blog post? Because of the next realization: I don't really know how I would 'integrate' this knowledge into Maple. Whatever ideas I come up with to do this seem less than half-baked. So I decided to see if the collective wisdom here could be applied to come up with this "how".

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Inequalities

Comment on 2007-11-06 15:29 from Robert Israel ( Maple Rating 4

1893)

I wonder how many of these inequalities are true. For example, I randomly chose Theorem 21 which says if x > 0,

$x/(x + floor(x)) + x/(x+frac(x)) >= 5/2$

This is wrong: the left side actually varies between 4/3 and 3/2. Moreover, Maple can be persuaded to assert this:

> _EnvTry := hard:
  x := a + b:
  Q:= x/(x+b) + x/(x+a):
  is(Q > 3/2) assuming a >= 0, b > 0;

false

Note that the expression is symmetric in a and b; you can take a = frac(x) and b = floor(x) or vice versa, and at least one of these is strictly positive.

I doubt that a table of inequalities would be very useful in itself, because the one you want will hardly ever be in the table (though it might be a consequence of one or more items there). I think this should be seen in the context of the need to improve Maple's ability to decide whether nonlinear inequalities are true in connection with the "assume" facilities. For example:

> is(a^2 + b + 1 >= a) assuming a >= 0, b >= 0;

unimpressive results

Comment on 2007-11-12 02:14 from Joe Riel ( Maple Rating 4

1367)

The theorems seem mostly worthless. Besides the error previously mentioned, the few I checked are overly conservative. Consider theorem 1:

floor(x)/(3*x+frac(x)) + frac(x)/(3*x+floor(x)) ≥ 4/15

The real limit should be 2/7, not 4/15. Similarly for theorem 6, the right side should be 4/3/x, not the conservative 1/x.

Theorem 32 part 2 fails:

solve(y2 := abs(cos(floor(x))) + abs(cos(frac(x))) - abs(cos(x));
fsolve(y2, x=2.9..3);
                                2.966118521
limit(y2, x=3, left);
                           -cos(2) + cos(1) + cos(3)
evalf(%);
                                 -0.0335433542

So does Theorem 25, at multiple locations for x < 0:

y := abs(sin(floor(x))) + abs(sin(frac(x))) + abs(cos(x)):
plot([1,y], x=-10..4);
limit(y, x=-2, left);evalf(%);
                                sin(3) - cos(2)

                                 0.5572668446