Checking the sign value of the derivatives of a complicated function

Question

Consider the following function:

e[w_] := 1/(2 w (w + a i W) (1 + a i w W)) (2 w^2 + 2 a i w^3 W + a^2 i^2 (-1 + 2 w^2) W^2 - \[Sqrt](a i W (4 (i + lambda) w^3 + 4 a i w^2 (1 + 2 i + 2 lambda + (i + lambda) w^2) W + 4 a^2 i^2 w (1 + i + lambda + 2 (i + lambda) w^2) W^2 + a^3 i^3 (1 + 4 (i + lambda) w^2) W^3)))

I'm trying to do two things. First, I would like to verify whether $e$ is an increasing and concave function of $w$ under the assumption of $a \in [0,1]$, $i \in [0,1]$, $\lambda \in [0,1]$, $w>0$, $W>0$. That is, $\frac{\partial e}{\partial w}>0$, $\frac{\partial^2 e}{\partial w^2}<0$.

Here is my code for this:

Clear["Global`*"]; 
e[w_]:= 1/(2 w (w + a i W) (1 + a i w W)) (2 w^2 + 2 a i w^3 W + a^2 i^2 (-1 + 2 w^2) W^2 - \[Sqrt](a i W (4 (i + lambda) w^3 + 4 a i w^2 (1 + 2 i + 2 lambda + (i + lambda) w^2) W + 4 a^2 i^2 w (1 + i + lambda + 2 (i + lambda) w^2) W^2 + a^3 i^3 (1 + 4 (i + lambda) w^2) W^3)));
Assuming[a > 0 && a < 1 && i > 0 && i < 1 && lambda > 0 && lambda < 1 && w > 0 && W > 0, FullSimplify@Reduce[e'[w] > 0]]
Assuming[a > 0 && a < 1 && i > 0 && i < 1 && lambda > 0 && lambda < 1 && w > 0 && W > 0, FullSimplify@Reduce[e''[w] < 0]]

Second, I would like to find $w$ that solves $$\frac{\partial e}{\partial w}=\frac{e}{w}$$ Here is my code for this:

Assuming[a > 0 && a < 1 && i > 0 && i < 1 && lambda > 0 && lambda < 1 && w > 0 && W > 0, FullSimplify@Solve[e'[w] == e[w]/w, w]]

These codes are running forever. Any help please?

Answer 1

Function e[w]only depends on two parameters a i W and i+lambda

e[w_] := (2 w^2 + 2 a i w^3 W + 
     a^2 i^2 (-1 + 
        2 w^2) W^2 - \[Sqrt](a i W (4 (i + lambda) w^3 + 
          4 a i w^2 (1 + 2 i + 2 lambda + (i + lambda) w^2) W + 
          4 a^2 i^2 w (1 + i + lambda + 2 (i + lambda) w^2) W^2 + 
          a^3 i^3 (1 + 4 (i + lambda) w^2) W^3))) /. {W -> aiw/(a i), 
    lambda -> ip\[Lambda] - i} // Simplify 

Its possible now to plot the regions

RegionPlot3D[e'[w] > 0 , {aiw, 0.1, 10}, {ip\[Lambda], 0.1, 10}, {w, 0.1, 20},Evaluated -> True, PlotLabel -> "e'[w]>0",AxesLabel -> {"a i W", "i+lambda", "w"}]

RegionPlot3D[e''[w] < 0 , {aiw, 0.1, 10}, {ip\[Lambda], 0.1, 10}, {w, 0.1, 20}, Evaluated -> True, PlotLabel -> "e''[w]<0",AxesLabel -> {"a i W", "i+lambda", "w"}]

addendum

second question

cond = Evaluate[e'[w] - e[w]/w];
ContourPlot3D[ cond  , {aiw, 0.1, 10}, {ip\[Lambda], 0.1, 10}, {w,0.1, 10}, AxesLabel -> {"a i W", "i+lambda", "w"},Contours -> { 0} ]

It looks like cond==0can't be fullfilled in this parameterrange!