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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).

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.)

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.)

This is known to be Carlson's inequality from 1935 (for $t_k\geq 0$, and not all $t_k$ are $0$). The constant $\pi^2$ is indeed optimal, 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).

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.)

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).

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.)

This is known to be Carlson's inequality from 1935 (for $t_k\geq 0$, and not all $t_k$ are $0$). The constant $\pi^2$ is indeed optimal, 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).

This is known to be Carlson's inequality from 1935 (for $t_k\geq 0$, and not all $t_k$ are $0$). The constant $\pi^2$ is indeed optimal, 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).

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.)