Compare P and Q: $a, b, c, d, m, n > 0$

[Albert1]

$a,b,c,d,m,n >0$

$ P=\sqrt {ab}+\sqrt {cd}$

$Q=\sqrt {ma+nc} \times \sqrt {\dfrac {b}{m}+\dfrac {d}{n}}$

$compare\,\, P \,\,and \,\,Q$

 

[caffeinemachine]

$a,b,c,d,m,n >0$

$ P=\sqrt {ab}+\sqrt {cd}$

$Q=\sqrt {ma+nc} \times \sqrt {\dfrac {b}{m}+\dfrac {d}{n}}$

$compare\,\, P \,\,and \,\,Q$

We show that $Q^4\geq P^4$.

This is same as showing that:

$(m/n)^2(ad)^2+(n/m)^2(bc)^2\geq 2abcd$.

This is clearly true by the AM-GM inequality.

Therefore $Q\geq P$.

 

[anemone]

My solution:

Spoiler

By applying Cauchy-Schwarz Inequality theorem to Q, we have:

\(\displaystyle Q=\sqrt {ma+nc} \times \sqrt {\dfrac {b}{m}+\dfrac {d}{n}} \ge \sqrt{ma}\sqrt{\frac{b}{m}}+\sqrt{nc}\sqrt{\frac{d}{n}} \ge \sqrt{ab}+\sqrt{cd}= P\)

and equality holds iff $\displaystyle \frac{a}{b}=\frac{c}{d}$.

 

[Albert1]

$a,b,c,d,m,n >0$

$ P=\sqrt {ab}+\sqrt {cd}$

$Q=\sqrt {ma+nc} \times \sqrt {\dfrac {b}{m}+\dfrac {d}{n}}$

$compare\,\, P \,\,and \,\,Q$

$Q^2=ab+cd+\dfrac {mad}{n}+\dfrac {nbc}{m}\geq ab+cd+2\sqrt {\dfrac {mad\times nbc}{mn}}=ab+cd+2\sqrt {abcd}=P^2$
$\therefore Q\geq P$

 

[CellCoree]

Both P and Q are expressions that involve square roots and positive variables, but they are not identical. P is the sum of two square roots, while Q is the product of two square roots. This means that they are not directly comparable in terms of numerical value. However, we can make some general observations about their similarities and differences.

First, both P and Q have the same variables, with the addition of the variable "m" in Q. This suggests that they may be related in some way, and that changing the value of m may have an impact on the relationship between P and Q.

Second, P and Q have different forms, with P being a sum and Q being a product. This means that they may behave differently when it comes to mathematical operations and simplification.

Finally, both P and Q have restrictions on their variables, with all variables being greater than 0. This suggests that they may have similar behaviors and outcomes when it comes to solving equations or analyzing their properties.

In conclusion, while P and Q cannot be directly compared in terms of numerical value, they share some similarities and differences that may be explored further through mathematical analysis and experimentation.