<s>[INST] Upper Bound for $\frac{\ln(1+nx)}{1+x^2\ln(n)}\leq \frac{1+\ln(x)+\ln(2)}{x^2}$ I was reading a solution for an analysis problem and they argued that $$\frac{\ln(1+nx)}{1+x^2\ln(n)}\leq \frac{1+\ln(x)+\ln(2)}{x^2}$$ for $x\geq 1$ and $n\geq 1$,  $x$ a real number and $n$ a natural number. Why is that?
 [/INST] First the identity definitely doesn't hold for $n=1$ so let's assume $n\geq 2$, then we have
$$
\begin{align*}
\ln(1 + nx) &= \ln(nx) +\ln(1+1/nx) \\
&\leq \ln(nx) + 1/nx =\frac{ x^2 \ln (nx) + x/n }{x^2}.
\end{align*}
$$
Now
\begin{align*}
x^2\ln(nx) + x/n &= \left[ 1+x^2\ln(n) \right] \left( \frac{\ln(n)}{\ln(n) + x^{-2} } + \frac{\ln(x) }{\ln(n) + x^{-2} } +  \frac{1}{nx(\ln(n) + x^{-2} )} \right)
\end{align*}
giving 
\begin{align*}
\frac{ \ln( 1 + nx) }{ 1 + x^2 \ln(n) } &\leq \frac{1}{x^2}\left( \frac{x^2 \ln(n) }{x^2 \ln(n) + 1 } + \frac{x^2 \ln(x) }{x^2 \ln(n) + 1 } +\frac{x^2 }{nx ( x^2 \ln(n) + 1 )}\right) \\
&= \frac{\ln(n) }{x^2 \ln(n) + 1 } + \frac{\ln(x) }{x^2 \ln(n) + 1 } +\frac{1}{nx ( x^2 \ln(n) + 1 )} \\
&\leq
\frac{1}{x^2} \left( 1 + \frac{\ln(x)}{\ln(n)} + \frac{1}{n \ln (n) }\right)
\end{align*}
For $n \geq 3$ this last bound is actually tighter than the one you are trying to prove, and in fact your original inequality fails for $n=2$ but I don't have an immediate assertion of that fact. Will come back to it.

Just to finish the answer off for the case $n=2$ the original inequality is equivalent to
$$
\frac{ \ln(1 + 2x) }{ 1 + \ln(2 x) } \leq \frac{1 + x^2 \ln(2) }{x^2 } = \frac{1}{x^2} + \ln(2), \qquad x \geq 1.
$$
which is true for $x = 1$, but
$$
\lim_{x \rightarrow \infty}\frac{\ln(1 + 2x)}{1 + \ln(2x) } = 1, \qquad \lim_{x \rightarrow \infty} \frac{1}{x^2} + \ln(2) =\ln(2) < 1,
$$
and so the claim is false for $n=1, 2$.
</s>