<s>[INST] Show that $(1+x)^{1-x}(1-x)^{1+x}<1$ If $x$ is a positive proper fraction. Prove that $$(1+x)^{1-x}(1-x)^{1+x}<1$$
Actually this question has two parts I can't do the $1^{st}$ part but the $2^{nd}$ part was quite easy with respect to the $1^{st}$ one. The $2^{nd}$ was to show that $$a^bb^a<(\frac{a+b}{2})^{a+b} $$ I observe that if $1^{st}$ is true then if I will put $$x=\frac{a-b}{a+b}$$ the equation of part $1^{st}$ will take the form $$a^bb^a<(\frac{a+b}{2})^{a+b} $$ and hence proved but I am unable to prove that $(1+x)^{1-x}(1-x)^{1+x}<1$
 [/INST] Since $\log$ is concave,
$$
\begin{align}
\frac{1-x}2\log(1+x)+\frac{1+x}2\log(1-x)
&\le\log\left(\frac{1-x}2(1+x)+\frac{1+x}2(1-x)\right)\\
&=\log\left(1-x^2\right)
\end{align}
$$
Therefore,
$$
\color{#090}{(1+x)^{1-x}(1-x)^{1+x}}\le\color{#C00}{\left(1-x^2\right)^2}
$$

</s>