Let $X,Y>0$, $p>2$ and $q>p-1$. Then does the following inequality hold: $$ (|X|^{p-2}+|Y|^{p-2})(X^{1-q}+Y^{1-q})\leq C X^{p-1-q}+ D Y^{p-1-q}, $$ for some positive constants $C$ and $D$ may deped on $p,q$?
It seems Youngs's inequality may be useful. But I am unable to use it properly.
Can somebody please help?
Thanks.
Since $x,y>0$ we can drop the absolute value bars. For $p:=q:=3$ the left side yields \begin{align*} (x+y)(x^{-2}+y^{-2})=x^{-1}+xy^{-2}+x^{-2}y+y^{-1} \end{align*} whereas the right side turns out to be \begin{align*} Cx^{-1}+Dy^{-1}. \end{align*}
For a fixed $y\in\mathbb{R}$ we have that $xy^{-2}$ will be greater than $Cx^{-1}+Dy^{-1}$ for each $C,D>0$ for all $x\geq M$ (where $M$ depends on your choice of $y,C,D$), since $x$ is growing faster that $x^{-1}$ as $x\rightarrow\infty$.
On the other hand for a fixed $y$ we have that $x^{-2}y$ will be greater than $Cx^{-1}+Dy^{-1}$ for each $C,D>0$ for all $0<x\leq m$ for the same reason.
Hence this inequality does not hold on the whole quarter space.