Regarding validity of expressions involving inequlaities

Your claim is invalid.

Look at your expression:

Abs[Log[1 + x] - Log[1 - x]] <= Abs[x - y]/(1 + Abs[x - y])

Assume x==y, then the right hand side is zero. The left hand side is e.g. for x == y == 0.5

Abs[Log[1 + x] - Log[1 - x]]/. {x -> .5, y -> .5}
(* 1.09861 *)

Clear["Global`*"]

The inequality is not valid over the specified ranges.

ineq = Abs[Log[1 + x] - Log[1 - x]] <= Abs[x - y]/(1 + Abs[x - y]);

Plot3D[Evaluate[List @@ ineq],
 {x, 0, 1}, {y, 0, 1},
 PlotLegends -> "Expressions", ClippingStyle -> None,
 AxesLabel -> Automatic]

ineq /. {x -> 1/2, y -> 1/2}

(* False *)