Name of $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$)?

Question

I checked these

What is the difference between square of sum and sum of square?

Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.

It is easy to see $p$-th power ($p\ge 1$) version, i.e., $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$), holds as well using the argument by Quang Hoang in Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. Is there a name for this inequality so that I can just quote? (It might be an elementary result but people around me bother to put "from Cauchy--Schwarz,..." when it is clearly Cauchy--Schwarz, so.)

Answer 1

I do not believe there is a name for your specific inequality (which I rewrite as following): $$ |x|^p+|y|^p\le \big(|x|+|y|\big)^p, \ p \ge1 \iff \boxed{\ \ \left(|x|^p +|y|^p\right)^{\frac{1}{p}} \le |x|+|y|, \quad p \ge 1 \ \ } $$ However, it can be viewed as a special case of multiple more general statements, such as Jensen, AMGM, Hölder, and probably many other inequalities after appropriate substitution and/or change of variables. The closes call would probably be the generalized mean inequality: $$ M_j\left( x_1, \dots, x_n \right) \leq M_i\left( x_1, \dots, x_n \right) \quad \text{ whenever } \quad j<i. \label{*} \tag{*} $$

Here $M_k \left( x_1, \dots, x_n \right)$ is so-called power mean, which is defined as $$ M_k(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^k \right)^{\frac{1}{k}}. $$

! In particular, assuming $n=2$, $j = 1$, and $i = p$, and denoting $\left(x_1, \dots, x_n \right) := \left(\,\chi, \gamma \right)$, we get $$ \begin{aligned} M_1\big(\left|\,\chi\right|, \left|\gamma\right|\big) & = \dfrac{1 }{2}\big(\left|\,\chi\right| + \left|\gamma\right|\big) = \dfrac{\left|\,\chi\right| }{2} + \dfrac{\left|\gamma\right| }{2}, \\ M_p\big(\left|\,\chi\right|,\left|\gamma\right|\big) & = \left( \dfrac{1}{2} \left(\left|\,\chi\right|^p+\left|\gamma\right|^p\right)\right)^{\frac{1}{p}} = \Bigg( \left(\frac{\left|\,\chi\right|}{2}\right)^p + \left(\frac{\left|\gamma\right|}{2}\right)^p \Bigg)^{\frac{1}{p}}.\\ \end{aligned} $$ By $\eqref{*}$, we have $$ \dfrac{\left|\,\chi\right| }{2} + \dfrac{\left|\gamma\right| }{2} \le \left( \bigg(\frac{\left|\,\chi\right|}{2^{\frac{1}{p}}}\bigg)^p + \bigg(\frac{\left|\gamma\right|}{2^{\frac{1}{p}}}\bigg)^p \right)^{\frac{1}{p}} . \label{**} \tag{**} $$ Denoting $ x := 2^{-\frac{1}{p}}\chi, \ \ y := 2^{-\frac{1}{p}}\gamma$ and raising both sides of $\eqref{**}$ to the power $p$, we get
$$ \left|x\right|^{p}+\left|y\right|^{p} \le \big(\left|x\right|+\left|y\right|\big)^{p}. $$

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To summarize, I believe that (strictly speaking) there is probably no name for your inequality. However, that the generalized mean is as close as you can get to your inequality, although some formula conversion is still required.

Answer 2

I think this should be called an special case of Minkowski's Inequality.

1 - For finite sequences, the Minkowski Inequality states that

$$ \left(\sum_{k=1}^n |x_k + y_k |^p \right) ^{1/p} \le \left(\sum_{k=1}^n |x_k|^p \right)^{1/p} + \left(\sum_{k=1}^n |y_k|^p\right)^{1/p} $$

Your inequality then follows if $x_1 = x, x_i = 0, \; i \ge 2$, and $y_2 = y, y_i = 0, i \ne 2$.

2 - More Generally, the case above can be easily extended to infinite sequences. The most general result is that, if $f,g : \Omega \to \Bbb{R}$ are measurable functions in a measure space $(\Omega, \Sigma, \mu)$, then the inequality

$$ \| f+g \|_{L^p (d\mu)} \le \|f\|_{L^p(d\mu)} + \|g\|_{L^p(d\mu)} $$

Where $ \|f\|_{L^p(d\mu)}^p = \int_ {\Omega} |f(x)|^p d \mu (x) $

3 - There was a little lie at topic 2, because there is yet another generalization of this result, called Minkowski's Inequality for Integrals. In the link above, you might check a proof of this result, based on these previously presented ones.