Question 123
Let $n \geqslant 2$ be an integer and $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying
(a) $x_{j}\gt -1$ for $j=1,2, \ldots, n$ and
(b) $x_{1}+x_{2}+\cdots+x_{n}=n$.
Prove the inequality
$$
\sum_{j=1}^{n} \frac{1}{1+x_{j}} \geqslant \sum_{j=1}^{n} \frac{x_{j}}{1+x_{j}^{2}}
$$
and determine when equality holds.
(a) $x_{j}\gt -1$ for $j=1,2, \ldots, n$ and
(b) $x_{1}+x_{2}+\cdots+x_{n}=n$.
Prove the inequality
$$
\sum_{j=1}^{n} \frac{1}{1+x_{j}} \geqslant \sum_{j=1}^{n} \frac{x_{j}}{1+x_{j}^{2}}
$$
and determine when equality holds.