Some generalizations of Carlson's inequalities (Q1125499)

Some generalizations of Carlson's inequality are obtained. For example, the following result is valid: Let \(\{a_n\}^\infty_{n=1}\) be a real sequence. If \(0<\alpha\leq 1\), then \[ \sum^\infty_{n=1}|a_n|\leq \sqrt{{\pi\over\alpha}} \Biggl(\sum^\infty_{n=1} n^{1-\alpha} a^2_n\Biggr)^{1/4} \Biggl(\sum^\infty_{n=1} n^{1+\alpha} a^2_n\Biggr)^{1/4}. \]