Assume that
$1a_1^2+2a_2^2+\cdots+na_n^2 = 1,$

where the $a_j$ are real numbers. As a function of $n$, what is the maximum value of
$(1a_1+2a_2+\cdots+na_n)^2?$

Question

Assume that
$1a_1^2+2a_2^2+\cdots+na_n^2 = 1,$

where the $a_j$ are real numbers. As a function of $n$, what is the maximum value of
$(1a_1+2a_2+\cdots+na_n)^2?$

anonymous · Answer

$\phi(x)=x^2$ is concave, so by Jensen's inequality: $$\phi\left(\frac{\sum ia_i}{\sum i}\right)\le\frac{\sum i\phi(a_i)}{\sum i}\\left(\frac{\sum ia_i}{n(n+1)/2}\right)^2\le\frac{\sum ia_i^2}{n(n+1)/2}$$

now we're told that $\sum ia_i^2=1$ so: $$\left(\sum ia_i\right)^2\le\frac{n(n+1)}2$$