A best constant for Zygmund's conjugate function inequality

When the space $L{\log ^ + }L$ is given the Hardy-Littlewood norm the best constant in the corresponding version of Zygmund's conjugate function inequality is shown to be

$${\mathbf{K}} = \frac{{{1^{ - 2}} - {3^{ - 2}} + {5^{ - 2}} - {7^{ - 2}} + \cdots }}{{{1^{ - 2}} + {3^{ - 2}} + {5^{ - 2}} + {7^{ - 2}} + \cdots }}.$$

This complements the recent result of Burgess Davis that the best constant in Kolmogorov's inequality is ${{\mathbf{K}}^{ - 1}}$.