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If x,y are positive real numbers and m, ...

If x,y are positive real numbers and m, n are positive integers, then prove that `(x^(n) y^(m))/((1 + x^(2n))(1 + y^(2m))) le (1)/(4)`

Text Solution

Verified by Experts

Using `A.M. ge G.M`., we have
`(1 + x^(2n))/(2) ge 1 (1 xx x^(2n))^(1//2)`
or `(1 + x^(2n))/(2) ge x^(n) " " (1) `
and `(1+ y^(2m))/(2) ge (1 xx y^(2m)) ^(1//2)`
or `(1+ y^(2m))/(2) ge y^(m) " " (2)`
Multiplying (1) and (2)
`((1 + x^(2n)) (1 + y^(2m)))/(4() ge x^(n) y^(m)`
`implies (x^(n) y^(m))/((1 + x^(2n)) (1 + y^(2m))) le (1)/(4)`.
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