# Regarding validity of expressions involving inequalities

This expression is valid:

$$\left|\log(1+x) - \log(1 - x)\right| \leq \frac{\left|x-y\right|}{1+|x-y|},\ \ 0

How we can prove using Mathematica (either graphically or using $$x = y$$ and $$x \neq y$$)

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 *)
• Sorry...It was typing mistake from my side. Feb 12, 2021 at 18:02
• It is actually,Abs[Log[1 + x] - Log[1 - y]] <= Abs[x - y]/(1 + Abs[x - y]) Feb 12, 2021 at 18:02
• @meraj please correct it in the question by editing it. Jul 12, 2021 at 21:35
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 *)

Just for fun (noting that in general the inequality is false):

Manipulate[
ContourPlot[f[x, y], {x, 0, 1}, {y, 0, 1}, Contours -> {0},