I am trying to bound $(\frac{a}{x}+\frac{b}{y})(x+y)$ from above and below in terms of $a,b$.
I've bounded it below with the caushy schwartz inequality to:
$$(\frac{a}{x}+\frac{b}{y})(x+y) \geq (\sqrt a+\sqrt b)^2$$
But not sure how to bound it above. Any ideas?
$a$ and $b$ are positive.
The expression $F=\left( \frac{a}{x} + \frac{b}{y} \right) (x+y) $ is not bounded from above for large $x$, $y$. More precisely, for every $M$ and $N$, there are $x$ and $y$ such that $x,y > N$ such that $f>M$. To see this, just expand it to obtain $$ F=\frac{a y}{x}+a+\frac{b x}{y}+b $$ Let $M$ and $N$ be given numbers. Without loss of generality, assume that they are positive. Choose $x = N+1$ and $y> xM/a$ to see that $F>M$.