Why can't the powerful Reduce and Solve functions handle this type of parametric inequality problem?

Questions
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1

Reduce[{x> -1, (1 - m + x) (E^x - 1/(1 + x))>0, 0 < m < 1}, x, Reals]

Why can’t the powerful Reduce and Solve functions handle this type of parametric inequality problem?

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2

Reducing the complicated inequality first perhaps gives the solution you seek:

Reduce[{x > -1, Reduce[(1 - m + x) (E^x - 1/(1 + x)) > 0, {}, Reals], 
  0 < m < 1}, x, Reals]
(*  0 < m < 1 && (-1 < x < -1 + m || x > 0)  *)
3

Excellent! The problem is solved by nesting an additional Reduce function. Great job, and thank you very much for this method!

4

A little change of variable solves the problem too:

Reduce[{x > -1,
    (1 - m + x) (E^x - 1/(1 + x)) > 0,
    0 < m < 1} /. x -> y - 1,
  Reals] /. y -> x + 1

I wonder why Reduce works with y but not with x.

5

This use of x works for me (I guess I didn’t try it, because I would have chosen to post this version, not the other one):

Reduce[{x > -1, (1 - m + x) (E^x - 1/(1 + x)) > 0, 0 < m < 1}, {}, Reals]

Also this, which is closer to your code, Gianluca:

Reduce[{x > -1, (1 - m + x) (E^x - 1/(1 + x)) > 0, 0 < m < 1}, Reals]

It’s the leaving off of the variable in the exponential that is the key. For instance, adding y as an explicit variable to your solution cause Reduce[] to fail:

Reduce[{x > -1, (1 - m + x) (E^x - 1/(1 + x)) > 0, 0 < m < 1} /. 
   x -> y - 1, y, Reals] /. y -> x + 1