I am trying to show the below inequality $$|x|^{a_1} + |y|^{a_2} \leq C(|x-y|^{a_1} + |y|^{\max(a_1,a_2)}) $$ for constants ($a_1>0,a_2$>0) and $C$ is some other constant. What is the quickest way to show this? I thought of doing it case by case and applying Jensen's inequality. For example, assume $1<a_1 \leq a_3$. Then \begin{equation*} |x|^{a_1} + |y|^{a_2} \leq (|x-y| + |y|)^{a_1} + |y|^{a_2} \leq 2^{a_1}\frac{|x-y|^{a_1} + |y|^{a_1}}{2} + |y|^{a_2} \leq 2^{a_1-1}(|x-y|^{a_1}) + (2^{a_1-1} |y|^{a_1}+ |y|^{a_2}) \end{equation*} Then what? Does this approach make sense? It looks like I might have to consider a lot of cases. Is there a quicker way? Thanks!
Edit 1:
As pointed out in the comments, the inequality fails for some cases. I made a mistake when formulating the question. The actual inequality that I want to show is given below-
$$\bigg(\int_{\Omega}(|x|^{a_1} + |y|^{a_2})^{c}d\mu\bigg)^{1/c} \leq C\bigg(\int_{\Omega} \big(|x-y|^{a_1} + |y|^{\max(a_1,a_2)}\big)^c d\mu \bigg)^{1/c}$$ for some positive constant $c$. Any leads are appreciated. Thanks.
Edit 2: Please see equation (A.2) in page 25 of this paper. This inequality shows up there and that is precisely what I want to understand. Hope this provides the context.