What's the idea behind this inequality? $e^{\frac{\ln x}{x-1}}\geq\frac{e}{3}(1+\frac{1}{\sqrt{x}}+\frac{1}{x})$
This inequality's image (left hand substracted right hand)

which shows a perfect approximation between (1,1.5), where it nearly keeps unchanged.
I wonder how this inequality was created, did it use some series expanding or some famous inequalities?

With substitution $x = \frac{1}{y^2}$, the inequality is written as
$$\mathrm{e}^{\frac{-2y^2\ln y}{1-y^2}} 
\ge \frac{\mathrm{e}}{3}( 1 + y + y^2).$$
Denote $f(y) = \mathrm{e}^{\frac{-2y^2\ln y}{1-y^2}}$.
Then the inequality above is written as $f(y) \ge f(1) + f'(1)(y-1) + \frac{1}{2}f''(1)(y-1)^2$.
The RHS is the second order Taylor approximation of LHS.
This approximation is nice around $y=1$ because $f'''(1) = 0$ and $f'''(y)$ is small near $y=1$.
