In my [previous MO question](https://mathoverflow.net/questions/262008/seeking-proofs-infinite-series-inequalities), the inequality was about a specific series and nicely answered by Cherng-tiao Perng. After testing with a few more numerical infinite sums, I came to realize that perhaps more is true.



> 
> Does the following hold true for any convergent series?
>  $$\left(\sum\_{k\geq1}t\_k\right)^4
> <\pi^2\left(\sum\_{k\geq1}t\_k^2\right)\left(\sum\_{k\geq1}k^2t\_k^2\right).$$
>  Is the constant $\pi^2$ optimal?
> 
> 
> 



An attempt to generalize the previous inequality
This is known to be Carlson's inequality from 1935 (for $t\_k\geq 0$, and not all $t\_k$ are $0$). The Swedish mathematician Fritz Carlson (1888-1952) also proved the optimality of the constant $\pi^2$. For an elegant elementary proof of the inequality see G. H. Hardy, A note on two inequalities, J. London Math. Soc. 11, 167-170, 1936 (DOI <https://doi.org/10.1112/jlms/s1-11.3.167> or <http://onlinelibrary.wiley.com/doi/10.1112/jlms/s1-11.3.167/abstract>).


*Note*: This inequality has found many modern day applications and generalizations, see the book "Multiplicative Inequalities of Carlson Type and Interpolation" by L. Larsson et al., World Scientific, 2006. (DOI <https://doi.org/10.1142/6063>; there you will find a free sample chapter with the classical proofs of Carlson and Hardy.)


