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- Here, I present question about the validity of Young's inequality.

Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To that end, we must demonstrate that ##u<0##, or equivalently,

$$ a^2+b^2-2ab<0$$.

Going through some straightforward algebraic manipulations, we then have

$$ab>\frac{1}{2}\left(a^{2}+b^{2}\right)$$.

Nevertheless, on recalling that ##a<0##, we then are led to conclude that

$$ab<\frac{1}{2}\left(a^{2}+b^{2}\right)$$.

Based on the above, I ask:

1. Would that last statement hold true by virtue of Young's inequality?

2. Is there any fallacious step in that given proof?

Thanks in advance.

$$ a^2+b^2-2ab<0$$.

Going through some straightforward algebraic manipulations, we then have

$$ab>\frac{1}{2}\left(a^{2}+b^{2}\right)$$.

Nevertheless, on recalling that ##a<0##, we then are led to conclude that

$$ab<\frac{1}{2}\left(a^{2}+b^{2}\right)$$.

Based on the above, I ask:

1. Would that last statement hold true by virtue of Young's inequality?

2. Is there any fallacious step in that given proof?

Thanks in advance.

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