For simple polynomial and trigonometric inequalities (in real domain), it seems to be not that hard to program. It seems strange that Maple, after 28 years of development, still doesn't cover that - it is a high school problem.

My choice in the poll at the l.h.s. was Math Algorithms, and I thought to myself entering it - covering at least high school algebra, geometry, and trigonometry would be nice.

By the way, this particular algebraic inequality can be done as

Optimization:-Maximize(2*(a+b+c)-(a*b+b*c+a*c),
    a=-1..1,b=-1..1,c=-1..1);

                    [3., [a = 1., b = 1., c = 0.]]

as well as

maximize(2*(a+b+c)-(a*b+b*c+a*c),a=-1..1,b=-1..1,c=-1..1);

                                  3

A similar thing can be done in the original example as well,

Optimization:-Maximize(eqn,x=0..Pi,y=0..Pi,z=0..Pi,
    initialpoint=[x=0.1,y=0.1,z=0.2]);

   [4.00000000000000088, [x = 0., y = 0., z = 2.66786575027781580]]

That doesn't prove the inequalities though, because, as we know, both maximize and Optimization:-Maximize give only a local maximum, and global maximum might be different.

Global Optimization Toolbox would be better for such problems, but it has to be purchased separately. Last time I used it  (I think it was a version for Maple 10), it gave correct results for problems with about 50 variables or less, close to correct with 50-100 variables, and wasn't very good for problems with more than 100 variables. It may be improved since then, though.

Alec