I have the following function:

((-2 b m - 2 w1 - 5 w2 + 3 m w2 + 7 w3 - 3 m w3 + Sqrt[
 4 b^2 m^2 + (-2 w1 + w2 - 3 m w2 + w3 + 3 m w3)^2 + 
  4 b m (2 w1 + (5 - 3 m) w2 + (-7 + 3 m) w3)])/(6 (-1 + m) (w2 - w3)))

Subject to the following parameters restrictions:

w2 > w3 && w1 > 2 w2 - w3 && 0 < m < 1 && w3 > b > 0

I am trying to show that the previous function is strictly decreasing in m, given the range of the parameters. I tried to derivate the function, but the result is a very long expression. All the plots I did of the function gave me a strictly decreasing function.

Does anyone have an idea of how to solve this problem?

Thanks in advance,

Pau

expr = ((-2 b m - 2 w1 - 5 w2 + 3 m w2 + 7 w3 - 3 m w3 + 
     Sqrt[4 b^2 m^2 + (-2 w1 + w2 - 3 m w2 + w3 + 3 m w3)^2 + 
       4 b m (2 w1 + (5 - 3 m) w2 + (-7 + 3 m) w3)])/(6 (-1 + m) (w2 -
        w3)));
assumptions = w2 > w3 && w1 > 2 w2 - w3 && 0 < m < 1 && w3 > b > 0;
Reduce[
    D[expr, m] < 0 
    &&
    assumptions 
]

Simplify[%, Assumptions -> assumptions]