An inequality about a quadratic polynomial

Question

In a paper, I faced the following statement: $$(x_1 + x_2)^4 \leq 16(x_1^4+x_2^4)$$

Because there is no explanation, I thought that showing the inequality may be a trivial one. But, I failed to show the relationship.

Thus, why the statement above holds?

See for example https://math.stackexchange.com/q/2302845/42969 — Martin R, Dec 01 '22 at 10:53

score 3 · Accepted Answer · edited Dec 01 '22 at 11:26

3

As a hint: $$(x_1+x_2)\leq 2\max \{ x_1,x_2\}$$ can you take over ?

edited Dec 01 '22 at 11:26

Joshua Tilley

answered Dec 01 '22 at 11:00

Khosrotash

Lol, thank you for the simple and clear answer:) – minchul Dec 01 '22 at 11:15
How is this helpful? Do you intend to take the 4th power of both sides? But this destroys the inequality as $x_1, x_2$ may be negative. – Pomponazzo Dec 01 '22 at 11:28
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@Desperado $$|x_1+x_2| \leq (|x_1|+|x_2|) \leq 2 \max { (|x_1|, |x_2|}$$ – Li Kwok Keung Dec 01 '22 at 13:42

1 Answers1