In his Functional Analysis book (2nd ed) Rudin asserts on page 203 that 
$$
(1+|y|)^{2r} < (2n+2)^r (1 + y\_1^{2r}+...+y\_n^{2r}) 
$$
where $r$ is a positive integer and $|y|$ is the Euclidean norm of the real vector $y = (y\_1,...,y\_n)$. Can anyone see how to prove this?



simple inequality in Rudin
Recall the power mean inequality, which, in particular, tells you that


$$ \frac{\sum\_{i=1}^{m} x\_i}{m} \le \sqrt[s]{\frac{\sum\_{i=1}^{m} x\_i^s}{m}},$$


for any $s \ge 1$ and any non-negative $x\_i$. Rudin's inequality is a double application of this. First apply it with $m=2, s=2, x\_1=1, x\_2 =|y|$:


$$(1+|y|)^2 \le 2(1+\sum\_{i} y\_i^2).$$


Then apply it again, this time with $m=n+1$, $s=r$, $x\_i = y\_i^2$ for $1 \le i \le n$ and $x\_{n+1}=1$. This shall give you:


$$(1+\sum\_{i} y\_i^2)^r \le (n+1)^{r-1} (1 + y\_1^{2r}+...+y\_n^{2r}).$$


Now combine both inequalities.


