A weak-type inequality for differentially subordinate harmonic functions

Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder for two harmonic functions and . That is, we prove the sharp weak-type inequality under the assumptions that , and the extra assumption that . Here is the harmonic measure with respect to and the constant is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.