I'm trying to prove the following inequality $$ \int_{\mathbb{R}} \frac{\lambda}{exp\{-x\} + \lambda} f^2(x) dx \leq log(\frac{1}{\lambda})^{-1} \int_{\mathbb{R}} (1+x^2)^{m} f^{2}(x)dx, $$ where $m$ can be any possible integer and $\lambda\in (0,1)$.
I try to prove the above inequality by showing on $\mathbb{R}$, the following inequality holds \begin{equation} \frac{\lambda}{exp\{-x\} + \lambda} \leq log(\frac{1}{\lambda})^{-1} (1+x^2)^{m}. \end{equation}
To do this, I split the $\mathbb{R}$ in to $S = \{x : log(\frac{1}{\lambda})^{-1} (1+x^2)^{m} \geq 1\}$ and $S^{c}$ be $S$'s Complement in $\mathbb{R}$. Then, the inequality holds over $S$. However, I have difficulty to prove the inequality also holds on $S^{c}$. I have tried to take the regular approach like take the first/second derivative of the function, but it turns out very hard to deal with when the function simultaneously has polynomial and exponential part.
I also use software to check if the inequality holds by picking multiple combination of $\lambda$ and $m$ and all of them seems the inequality holds.
Is there any trick or approach that I can prove the inequality without dealing with the annoying derivative?
Any help is appreciated.
