H\"older Functionals and Quotients

We describe an inequality of finite or infinite sequences of real numbers and their quotients. More precisely, we compare the quotient of H\"older functionals of two sequences of numbers with the sum of their quotients. In the last section we investigate the `wideness' of the inequality, i.e. we show that both the inequality can converge into an equality, and the difference between the two sides of the inequality can be arbitrary large.