Question 83
Let $a$, $b$, $c$ be non-zero distinct real numbers so that there exist functions $f$, $g : \mathbb{R}^+ \to \mathbb{R}$ so that
$$
a f(xy) + b f\left(\frac{x}{y}\right) = c f(x) + g(y)
$$
for all positive real $x$ and large enough $y$.
Prove that there exists a function $h : \mathbb{R}^+ \to \mathbb{R}$ so that
$$
f(xy) + f\left(\frac{x}{y}\right) = 2f(x) + h(y)
$$
for all positive real $x$ and large enough $y$.
$$
a f(xy) + b f\left(\frac{x}{y}\right) = c f(x) + g(y)
$$
for all positive real $x$ and large enough $y$.
Prove that there exists a function $h : \mathbb{R}^+ \to \mathbb{R}$ so that
$$
f(xy) + f\left(\frac{x}{y}\right) = 2f(x) + h(y)
$$
for all positive real $x$ and large enough $y$.