Prove $\frac 1{x+1}<\ln\left(\frac{x+1}x\right)<0.5\left(\frac 1x+\frac 1{x+1}\right)$

I looking for help with proving the following inequality. Any relevant logarithmic identities would be great. Tried differentiating and taking limits and I'm lost as to how to approach this.

$$\frac 1{x+1}<\ln\left(\frac{x+1}x\right)<0.5\left(\frac 1x+\frac 1{x+1}\right)$$

Solution 1:

Let us recall a property of integrals, so simple one could think nothing good could come out of it, and yet:

For every interval $(a,b)$ with $a\leqslant b$ and functions $u$ and $v$ such that $u\geqslant v$ on $(a,b)$, $$\int_a^bu(x)\mathrm dx\geqslant\int\limits_a^bv(x)\mathrm dx.$$ In particular, if $u\geqslant c$ on $(a,b)$, for some constant $c$, then $\displaystyle\int_a^bu(x)\mathrm dx\geqslant c(b-a)$.

In the present case, one may start from the identity $\log((x+1)/x)=\int\limits_x^{x+1}u(t)\mathrm dt$ with $u(t)=1/t$ and note that $u(t)\geqslant1/(x+1)$ for every $t$ in the interval $(x,x+1)$. This should give one inequality.

For the other inequality, one may prove that $v(s)=u(x+1/2+s)+u(x+1/2-s)$ is an increasing function of $s$ for $s$ in $(0,1/2)$ and note that $\log((x+1)/x)=\int_0^{1/2}v(s)\mathrm ds$.

Can you reach that point and finish the proof from there?