Upper bound for ratio as a function of terms on the LHS
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I've recently been stumped over the following inequality. Suppose $x,y ge 0$ and $a > 1.$ Do there exist constants $c_1, c_2, c_3 in [0,infty)$ which do not depend on $x$ or $y$ such that $$frac{a + frac{x}{2y}}{a-frac{1}{2}} le c_1x + c_2y + frac{c_3}{y}$$
Thanks in advance for your help.
real-analysis calculus algebra-precalculus inequality
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
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add a comment |
$begingroup$
I've recently been stumped over the following inequality. Suppose $x,y ge 0$ and $a > 1.$ Do there exist constants $c_1, c_2, c_3 in [0,infty)$ which do not depend on $x$ or $y$ such that $$frac{a + frac{x}{2y}}{a-frac{1}{2}} le c_1x + c_2y + frac{c_3}{y}$$
Thanks in advance for your help.
real-analysis calculus algebra-precalculus inequality
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
$endgroup$
add a comment |
$begingroup$
I've recently been stumped over the following inequality. Suppose $x,y ge 0$ and $a > 1.$ Do there exist constants $c_1, c_2, c_3 in [0,infty)$ which do not depend on $x$ or $y$ such that $$frac{a + frac{x}{2y}}{a-frac{1}{2}} le c_1x + c_2y + frac{c_3}{y}$$
Thanks in advance for your help.
real-analysis calculus algebra-precalculus inequality
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
$endgroup$
I've recently been stumped over the following inequality. Suppose $x,y ge 0$ and $a > 1.$ Do there exist constants $c_1, c_2, c_3 in [0,infty)$ which do not depend on $x$ or $y$ such that $$frac{a + frac{x}{2y}}{a-frac{1}{2}} le c_1x + c_2y + frac{c_3}{y}$$
Thanks in advance for your help.
real-analysis calculus algebra-precalculus inequality
real-analysis calculus algebra-precalculus inequality
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
asked Jan 16 at 20:39
Jack BurkeJack Burke
524
524
add a comment |
add a comment |
1 Answer
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$begingroup$
Consider the sequence $(x_n,y_n)=(n,frac{1}{n})$. Plugging in gives you:
$$frac{a + frac{n^2}{2}}{a-frac{1}{2}} le c_1n + c_2frac{1}{n} + c_3n$$
Sending $nto infty$ yields a contradiction (why?).
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
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1 Answer
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1 Answer
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active
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votes
$begingroup$
Consider the sequence $(x_n,y_n)=(n,frac{1}{n})$. Plugging in gives you:
$$frac{a + frac{n^2}{2}}{a-frac{1}{2}} le c_1n + c_2frac{1}{n} + c_3n$$
Sending $nto infty$ yields a contradiction (why?).
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
$endgroup$
add a comment |
$begingroup$
Consider the sequence $(x_n,y_n)=(n,frac{1}{n})$. Plugging in gives you:
$$frac{a + frac{n^2}{2}}{a-frac{1}{2}} le c_1n + c_2frac{1}{n} + c_3n$$
Sending $nto infty$ yields a contradiction (why?).
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
$endgroup$
add a comment |
$begingroup$
Consider the sequence $(x_n,y_n)=(n,frac{1}{n})$. Plugging in gives you:
$$frac{a + frac{n^2}{2}}{a-frac{1}{2}} le c_1n + c_2frac{1}{n} + c_3n$$
Sending $nto infty$ yields a contradiction (why?).
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
$endgroup$
Consider the sequence $(x_n,y_n)=(n,frac{1}{n})$. Plugging in gives you:
$$frac{a + frac{n^2}{2}}{a-frac{1}{2}} le c_1n + c_2frac{1}{n} + c_3n$$
Sending $nto infty$ yields a contradiction (why?).
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
answered Jan 16 at 20:46
F. ConradF. Conrad
1,300412
1,300412
add a comment |
add a comment |
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