1
$\begingroup$

Given $f(x,y),g(x,y)$, positive functions of $x,y\in \mathbb R$. can we write $\limsup_{y\longrightarrow \infty}\; \sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)}$

in terms of a product of $f(x,y), g(x,y)$ instead of $\frac{f(x,y)}{g(x,y)}$?

(Note: You can assume all limits exist and nonzero.)

Progress: I believe that $\sup(1/g(x,y))=1/\inf g(x,y)$, so

$\sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)} \leq \sup_{x\in \mathbb R} f(x,y) . \sup_{x\in \mathbb R} 1/g(x,y)$ $\leq \sup_{x\in \mathbb R} f(x,y) .\frac{1}{\inf_{x\in \mathbb R} g(x,y)}$

so:

$\limsup_{y\longrightarrow \infty}\; \sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)}\leq \limsup_{y\longrightarrow \infty}\;\sup_{x\in \mathbb R} f(x,y) .\limsup_{y\longrightarrow \infty}\;\frac{1}{\inf_{x\in \mathbb R} g(x,y)}$

and here I got stuck!

  • 0
    So no help!!!...2011-12-27

0 Answers 0