A best constant for Zygmund’s conjugate function inequality

Abstract

When the space L log + L L{\log ^ + }L is given the Hardy-Littlewood norm the best constant in the corresponding version of Zygmund’s conjugate function inequality is shown to be \[ K = 1 − 2 − 3 − 2 + 5 − 2 − 7 − 2 + ⋯ 1 − 2 + 3 − 2 + 5 − 2 + 7 − 2 + ⋯ . {\mathbf {K}} = \frac {{{1^{ - 2}} - {3^{ - 2}} + {5^{ - 2}} - {7^{ - 2}} + \cdots }}{{{1^{ - 2}} + {3^{ - 2}} + {5^{ - 2}} + {7^{ - 2}} + \cdots }}. \] This complements the recent result of Burgess Davis that the best constant in Kolmogorov’s inequality is K − 1 {{\mathbf {K}}^{ - 1}} .