Anyway here is a picture of the situation, using the suggested substitution.
Red = forbidden zone, Blue+Green = allowed zone (in a triangle over x,y).

The minimum is fine, the maximum is non-feasible and over the triangle
the maximum is in the border, isolated. As the triangle is an open one
there is no maximum, just a supremum. As the participants already said. 

f:=sqrt((x+1)*(y^2+2)*(z^3+3))+sqrt((y+1)*(z^2+2)*(x^3+3))+
   sqrt((z+1)*(x^2+2)*(y^3+3));
g:=subs(z = 3-x-y, %);

myRange:= x=0 .. 4.1, y=0 .. 4.1:
piecewise( 0 < (3-x-y) , 1, ""):
  plot3d(g*%, myRange, axes=boxed, shading=zhue, orientation=[-145,80]): 
  p1:=%:
piecewise( 0 <= -(3-x-y) , 1, ""):
  plot3d(g*%, myRange, axes=boxed, color="LightSalmon"): 
  p2:=%:
plots[display](p1, p2);

I agree with Markiyan ... so why dont you just provide some examples ?

@toandhsp you already have the answer in http://www.mapleprimes.com/questions/202347-How-Can-I-Find-The-Greatest-Value-Of#comment209293

sorry, you are right (it is the minimum)

for me it works (though I modified it a bit): using the assumptions
to simplify and evalc for F only:

restart; Digits:=15;
theAssumptions:= 0 < T, 0 < k, 0 < ki, 0 < w;

Gp:=1/exp(sqrt(w*I)):
Cf:=((T*w*I+1)/(Tf*w*I+1)):
Cpi:=(k*w*I+ki)/(w*I):
L:=Cf*Cpi*Gp: #L:=evalc(L):
#F:=subs(Ms=2,Tf=(k*T^(alfa)/Mn)^(1/alfa),evalc(abs(1+L)^2-1/Ms^2));
F:=subs(Ms=2,Tf=(k*T^(alfa)/Mn)^(1/alfa),(abs(1+L)^2-1/Ms^2)):
F:=subs(Mn=10,alfa=2/10,(F));


          |           (T w I + 1) (k w I + ki) I        |2
     F := | 1 - --------------------------------------- |  - 1/4
          |                  5                     1/2  |
          |     (1/100000 I k  T w + 1) w exp((w I)   ) |


F:=evalc(F):
F:= simplify(F) assuming theAssumptions:
collect(%, exp):
combine(%, exp):
simplify(%, size):
F:=collect(%, exp);

Fw:=diff(F,w):
Fk:=diff(F,k):
Ft:=diff(F,T):
Sys:='{F,Fw,Fk,Ft}'; #indets(%, symbol);

fsolve(Sys, 
  {w=0 .. infinity,k=0 .. infinity,ki=0 .. infinity,T=0 .. infinity});

  {T = 0.100306935191953, k = 4.76741548387957, 
   ki = 74.6566837362003, w = 17.5607002958772}

b:= a means that you assign 0 to b, since a is already assigned a value.

Compare to

  restart;
  b := a;
  a := 2;
  b; a;

Here b is assigned to a.

In your case for a Matrix you assign the Matrix, change 1 of its entries,
and thus Maple knows it.

Actually it is the Laplace transform of 1/((x+u)*(x+v)) in -t*I.

Writing u and v in terms of indexed a and b and assuming them to be real just obfuscates the task.

I think the output says: if x = u or x = v then one can not integrate over the singularity, being a pole. The 'message' about u < 0 (or that for v) is due to the integration w.r.t. x along the positive axis.

And I would agree the 'message' is puzzling. But a valid answer: it says a) assuming u (or v) is real (by using orders) and then b) about signs (it would be more nice to refer to Re and Im parts). However the setting is not quite friendly for the machine. And ignores any Math, leaving it to the CAS..

just want to mention, that (for me) that does not change the plotted graph (using Digits = 15) and giving the same value for m2. Luckily.

NB: Digits = n does not say 'exact for n Digits'. So acer's post says (for me): 10 Digits are not enough to get the value for MeijerG in that case

plot([Re(c),Im(c)],snr=-5..10, color=[red,blue]);

Additionally I suggest to avoid ''with(RealDomain)" except you really know what you are doing, in most cases it is better to use plot(  Re(c), ... )

something like http://homalg.math.rwth-aachen.de/ ?

I guess he means without specifying m,n

"combine", and MultiSeries is de-activated

'Evalf[3]' shows that these are rational functions. Now using only the denominator that means to find roots of polynomials. They are of degree 20 (and coefficients are of magnitude 10^360, but one scale by that).

Is there a reason why there are solutions over the Reals?

And are there reasons why solutions are only isolated points?(I have some doubts about it, Maple was not able to 'proof' that the solutions are complex zero dimensional).

Perhaps it makes sense to look at the task before that equations: how do they come into play?

Well, it may be a practical problem. But actually all that is in
Maple's local help as well (*). So it would already help to provide
some shortcut to be copied into Maple's help. Actually I doubt
that one needs a fullblown online help, identical to a local one.

But I would expect that at least the *input* is displayed a text,
which can be copied. For the *output* I would expect that those
images have an alternate text which like like 'lprint' of that.

(*) I mean: which can be called in the local installation.