# Is a four-variable function less than one?

I have the following function of four variables

f=Sqrt[(1 + w - x + 4 y z) - Sqrt[4 (-w + x) + (1 + w - x + 4 y z)^2]]/Sqrt[2]


I want to show that $$f(w,x,y,z)<1$$ under the conditions $$x\geq0,w\geq 1, y>0, z>0$$; then, I use NMaximize[] command to find the maximum value of $$f(w,x,y,z)$$, that is

NMaximize[{f, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}]


If the maximization of $$f(w,x,y,z)$$ under the previous conditions gives a value less than one, I show that $$f(w,x,y,z)<1$$. However, the result of the previous code is

{1., {w -> 1.42498, x -> 0.00149322, y -> 0., z -> 0.0000601358}}


Then, Mathematica carries out the numerical maximization with $$y=0$$ even though I'm using the condition $$y>0$$ in NMaximize[]. Then, my questions are:

(1) There exists some way to force Mathematica to carry out the numerical maximization with $$y>0$$ ?

and

(2) Is there another simpler way to show $$f(w,x,y,z)<1$$ (subject to $$x\geq0,w\geq 1, y>0, z>0$$) with Mathematica?

• The expression is not always real-valued in that domain. In[189]:= f /. {w -> 1, x -> 4, y -> 1/100, z -> 100} Out[189]= I Dec 6, 2023 at 21:15
• Evaluate Maximize[{f, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}] which returns Maximize::wksol: Warning: there is no maximum in the region in which the objective function is defined and the constraints are satisfied; a result on the boundary will be returned. and {1, {w -> 2, x -> 0, y -> 1, z -> 0}} This indicates that function was increasing as it approached the boundary and as you get arbitrarily close to boundary (z == 0 in this instance) the result will be arbitrarily close to 1 Look at inst = FindInstance[{f == 1 - 10^-20, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}, 1] Dec 6, 2023 at 21:37
• One other way to ask: Reduce[x>=0&&w>=1&&y>0&&z>0&&Sqrt[(1+w-x+4y z)-Sqrt[4(-w+x)+(1+w-x+4y z)^2]]/Sqrt[2]>=1,{w,x,y,z}] instantly returns False
– Bill
Dec 6, 2023 at 21:51

The command

FindInstance[{f >= 1, x >= 0, w >= 1, y > 0, z > 0}, {w, x, y, z}]


returns {} which, if one trusts Mathematica, proves that $$f$$ is strictly less than 1.

I don't personally trust Mathematica, for what it's worth.