This can be done by the change Abs[y] -> a^2, Abs[z] -> b^2
in order to obtain a polynomial in a
and b
as follows.
Maximize[{1/
32 (8 - Sqrt[(-8 - 36 Abs[y] (1 + 2 Sqrt[Abs[z]]) -
27 (Abs[y] + 2 Abs[y] Sqrt[Abs[z]])^2)^2 -
64 (1 + Abs[y] + 2 Abs[y] Sqrt[Abs[z]])] +
36 Abs[y] (1 + 2 Sqrt[Abs[z]]) + 27 (Abs[y] + 2 Abs[y] Sqrt[Abs[z]])^2),
0 < Abs[y] < 1 && 0 < Abs[z] < 1 && Sqrt[Abs[y]] > 0 &&
Sqrt[Abs[z]] > 0} /.{Sqrt[Abs[y]]-> a,Sqrt[Abs[z]] -> b,Abs[y]->a^2,Abs[z] -> b^2}, {a, b}]
(*{1/4, {a -> 0, b -> Indeterminate}}*)
and the warning (not an error)
Maximize::natt: The maximum is not attained at any point satisfying the given constraints.
The result means that b
may be an arbitrary value (of course, 0<b&&b<1
.
Making use of Minimize
instead of Maximize
, one obtains
(*{1/32 (359 - 35 Sqrt[105]), {a -> 1, b -> 1}}*)
and a similar warning.
It remains to notice that the polynomial, being a continuous function, takes each value between its minimum value and maximum value on a (compact) set a>=0&&a<=1&&b>=0&&b<=1
. Numeric calculations with NMaximize
and NMinimize
with Method->"DifferentialEvolution
confirm those results.
Summarizing the above, one may conclude that the range under consideration is $(\frac{1}{32} \left(359-35 \sqrt{105}\right),\frac 1 4)$.