Someone's Inequality ? $(a + b)^2 \le 2(a^2 + b^2) $

Question

For real $a, b$ then $(a + b)^2 \le 2(a^2 + b^2) $

This fairly trivial inequality crops up a lot in my reading on (Lebesgue) integration, is it named after someone ? It extends rather obviously for positive reals to $a^2 + b^2 \le (a + b)^2 \le 2(a^2 + b^2) $.

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Proof (if you need it):

$0 \le (a - b)^2 = a^2 + b^2 -2ab \implies 2ab \le a^2 + b^2$
$(a + b)^2 = a^2 + b^2 + 2ab$ which by previous $ \le 2(a^2 + b^2) $.

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Application:

If $f, g$ are positive functions then $(f + g)^2$ is integrable $\iff$ $f^2, g^2$ are integrable since $f^2 + g^2 \le (f + g)^2 \le 2(f^2 + g^2) $ pointwise.

Answer 1

That inequality probably does not have a name as it is so basic. In any case it can be viewed as a special case of Young's inequality.

Answer 2

This follows from the Cauchy–Schwarz inequality $$ |\langle \mathbf{u},\mathbf{v}\rangle| ^2 \leq \langle \mathbf{u},\mathbf{u}\rangle \cdot \langle \mathbf{v},\mathbf{v}\rangle, $$ with $ \mathbf{u} = (1,1)$ and $ \mathbf{v} = (a,b) $ in $\mathbb R^2$.

Answer 3

This inequality can as well be seen as a particular case of the equivalence between $p$-norms in $\Bbb R^n$. Indeed for $1<p\leq q<\infty$, it holds $$\|x\|_q \leq \|x\|_p \leq n^{1/p-1/q}\|x\|_q$$ In the particular case $n=2$, $p=1$ and $q=2$ we get $$(|a|+|b|)=\|(a,b)\|_1\leq 2^{1-1/2}\|(a,b)\|_2 =\sqrt{2(a^2+b^2)},$$ which is even slightly tighter as $|a+b| \leq |a|+|b|$.

I should nevertheless point out that the equivalence between $p$-norms is proved using the Cauchy-Schwarz (or more generally the Hölder) inequality mentioned by @Ihf

Answer 4

Given real variables, the algebraic identity

$$ (a-b)^2 + (a+b)^2 = 2(a^2+b^2) $$

implies both the inequality

$$ (a+b)^2 \le 2(a^2+b^2) $$

and

$$ (a-b)^2 \le 2(a^2+b^2). $$

You asked

This fairly trivial inequality crops up a lot in my reading on (Lebesgue) integration, is it named after someone ?

As is typical, I don't think the identity or the inequalities have a given name.

However, the identity is included in my collection of "Special Algebraic Identities".