Existence of constant for inequality hold.

Question

Prove or disprove the following statements. There exists $C>0$ and $\alpha > 1$ such that for all $x,y,z \geq 0$ we have $${x^5} + {y^7} + {z^9} \geqslant C{\left( {{x^2} + {y^2} + {z^2}} \right)^\alpha }.$$

Answer 1

Such numbers do not exist. Put $y=z=0$ to get $x^{5} \geq C x^{2\alpha}$. This can hold for all $x \geq 0$ if and only if $\alpha=\frac 5 2$. To see this consider the cases $\alpha > \frac 5 2$ and $\alpha < \frac 5 2$. In the first case write the inequality as $x^{2\alpha -5} \leq \frac 1 C$ and let $x \to \infty$ to get a contradiction. Similarly, the second case leads to a contradiction if we let $x \to 0$. This proves that $\alpha=\frac 5 2$. Now repeat the argument taking $x=z=0$. You will see that the inequality can hold iff $\alpha=\frac 7 2$. Hence, there is no $\alpha$ for which the inequality holds for all $x,y,z$.