Does $|x|^{p_1}+|y|^{p_2}\ge(|x|+|y|)^{\max\{p_1,p_2\}}$ or $\frac{|x|^{p_1}}{(|x|+|y|)^{\min\{p_1,p_2\}}}\le 1$ hold true?

Question

Let $x, y\in\mathbb{R}^*$ and let $p_1, p_2 >1$. I'm trying to prove that at least one of the inequalities, up to some positive constant, holds true: $$1)\qquad |x|^{p_1} +|y|^{p_2}\ge (|x|+|y|)^{\max\{p_1, p_2\}};$$

$$2)\qquad \begin{cases} \frac{|x|^{p_1}}{(|x|+|y|)^{\min\{p_1, p_2\}}}\le 1\\ \frac{|y|^{p_2}}{(|x|+|y|)^{\min\{p_1, p_2\}}}\le 1 \end{cases}.$$

I couldn’t prove any of it so far, could someone please help me with that? Or instead give a counterexample?

Thank you in advance!

Answer 1

1. inequality fails by taking $p_1=2$, $p_2=3$, $x=2$, and sending $y\to 0$. Inequality reads $4+\text{small} \ge (2+\text{small})^3$.

2. if one holds then clearly the other as well. But now in the first, take $p_1=3$, $p_2=2$, $y=1$ and send $x\to\infty$.

PS to save you the trouble of asking in the future, it's trivial to graphically find counterexamples using Desmos.

Answer 2

The inequality $1)$ fails for $x=y=1.$ The first inequality $2)$ fails for $x=2 $, $y\to 0^+$ and $p_2<p_1.$