Sum of r Powers is between Power of Maximum and r times Power of Maximum

Theorem

Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $a = \max \set {a_1, a_2, \ldots, a_r}$.

Then:

$a^n \le a_1^n + a_2^n + \cdots + a_r^n \le r a^n$

Proof

This proof is divided into $2$ parts:

Power of Maximum is not Greater than Sum of Powers

Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $a = \max \set {a_1, a_2, \ldots, a_r}$.

Then:

$a^n \le a_1^n + a_2^n + \cdots + a_r^n$

$\blacksquare$

Sum of $r$ Powers is not Greater than $r$ times Power of Maximum

Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $a = \max \set {a_1, a_2, \ldots, a_r}$.

Then:

$a_1^n + a_2^n + \cdots + a_r^n \le r a^n$

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 11$