How to find the minimum value of $\dfrac{a_1^2+a_2^2+\dots+a_n^2}{(a_1+a_2+\dots+a_n)^2}$?

Question

Motivation: To find the maximum value of $c$ in 2.4-1 Lemma (Linear combinations) of the book Kreyszig - Introductory Functional Analysis with Applications for $X=\mathbb{R^n}$ which reduces to solving the following question:

What is the minimum value of $\dfrac{a_1^2+a_2^2+\dots+a_n^2}{(a_1+a_2+\dots+a_n)^2}$ for $a_i \ge 0$?

For $n=2$, it is easy: let $x=a_1/a_2$ and the minimum occurs when $x=1$ or $a_1=a_2$.

For $n>2$, I tried the techniques that I had learnt from multi-variable Calculus but it gets quite complicated with no light ahead!

Answer 1

Let $\vec{a} = \begin{bmatrix}a_1 & a_2 & \cdots & a_n\end{bmatrix}^T$ and $\vec{1} = \begin{bmatrix}1 & 1 & \cdots & 1\end{bmatrix}^T$. Using the Cauchy-Schwarz Inequality, we have

\begin{align*} \|\vec{a}\|^2 \|\vec{1}\|^2 &\ge (\vec{a} \cdot \vec{1})^2 \\ (a_1^2+a_2^2+\cdots+a_n^2)(1^2+1^2+\cdots+1^2) &\ge (a_1\cdot 1+a_2\cdot 1+\cdots+a_n\cdot 1)^2 \\ n(a_1^2+a_2^2+\cdots+a_n^2) &\ge (a_1+a_2+\cdots+a_n)^2 \\ \dfrac{a_1^2+a_2^2+\cdots+a_n^2}{(a_1+a_2+\cdots+a_n)^2} &\ge \dfrac{1}{n} \end{align*}

Equality holds iff $\vec{a}$ and $\vec{1}$ are parallel, i.e. $a_1 = a_2 = \cdots = a_n$. (Also, we need the $a_i$'s to be non-zero.)

Answer 2

Notice that the expression $$\frac{a_1^2 + \ldots + a_n^2}{(a_1 + \ldots + a_n)^2}$$ is invariant under $a_i \mapsto \lambda a_i$ for any $\lambda \neq 0$. So we are just looking for the maximum of $a_1 + \ldots + a_n$ on the sphere. This is the projection of the vector $(a_1, \ldots, a_n)$ onto the line through the origin and $(1,\ldots,1)$, and so is maximized when $a_i = \frac{1}{\sqrt{n}}$ for all $i$. So the minimum of the original expression must be $\frac{1}{n}.$