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
\((1a_1+2a_2+\cdots+na_n)^2?\)

Answer 1

\(\phi(x)=x^2\) 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: $$\left(\sum ia_i\right)^2\le\frac{n(n+1)}2$$