Carlson’s and Related Inequalities

Abstract

Let a ₁,a ₂,... be a sequence of nonnegative real numbers not all equal to zero, and let $\sum\limits_{k = 1}^{ + \infty } {{k^2}a_k^2 < + \infty }$. Then

$${\left( {\sum\limits_{k = 1}^{ + \infty } {{a_k}} } \right)^4} \leqslant \pi 2\left( {\sum\limits_{k = 1}^{ + \infty } {a_k^2} } \right)\left( {\sum\limits_{k = 1}^{ + \infty } {{k^2}a_k^2} } \right),$$

(1.1)

where π² is the best possible constant.