Question 148
Let $a \gt 0$ be a real number and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying
$$
f\left(x_{1}\right) + f\left(x_{2}\right) \geq a f\left(x_{1} + x_{2}\right), \forall x_{1}, x_{2} \in \mathbb{R} .
$$
Prove that
$$
f\left(x_{1}\right) + f\left(x_{2}\right) + f\left(x_{3}\right) \geq \frac{3 a^{2}}{a+2} f\left(x_{1} + x_{2} + x_{3}\right), \forall x_{1}, x_{2}, x_{3} \in \mathbb{R}.
$$
$$
f\left(x_{1}\right) + f\left(x_{2}\right) \geq a f\left(x_{1} + x_{2}\right), \forall x_{1}, x_{2} \in \mathbb{R} .
$$
Prove that
$$
f\left(x_{1}\right) + f\left(x_{2}\right) + f\left(x_{3}\right) \geq \frac{3 a^{2}}{a+2} f\left(x_{1} + x_{2} + x_{3}\right), \forall x_{1}, x_{2}, x_{3} \in \mathbb{R}.
$$