Arithmetic Average of Second Chebyshev Function/Lemma 2

Lemma
Let $x \ge 3$ be a real number.

Then:


 * $-2 \map \ln {x + 1} \le \paren {x - 1} \map \ln {x - 1} - x \ln x$

Proof
Define a function $g : \openint 1 \infty \to \R$ by:


 * $\map g x = \paren {x - 1} \map \ln {x - 1} - x \ln x + 2 \map \ln {x + 1}$

We have that $g$ is differentiable with:

We can see that $\map {g'} x \ge 0$ for $x \ge 3$.

So from Real Function with Positive Derivative is Increasing:


 * $g$ is increasing.

So for $x \ge 3$ we have:

so for $x \ge 3$ we have: