Hardy-space threshold for univalent harmonic mappings

Determine whether every sense-preserving univalent harmonic mapping in the unit disk belongs to the harmonic Hardy space h^p for every p<1/α, where α=sup_{f∈S_H}|a_2|, and establish whether this exponent is sharp.

Background

The paper defines α as the supremum of the second analytic coefficient among normalized sense-preserving univalent harmonic mappings. Earlier work established progressively larger sufficient ranges of p for membership in hp, and sharp results are known for important subclasses such as convex and close-to-convex harmonic mappings.

A conjecture attributed to Nowak predicts the optimal general threshold p<1/α. The authors note that the conjecture was verified under an additional assumption on the analytic part, but remains unresolved for the entire class of univalent harmonic mappings.

References

These findings led her to conjecture that if f ∈ S_H, then f ∈ hp for all p < 1/α and that this order is sharp. In [8], this conjecture was verified under an additional assumption on the analytic part of f. However, for the entire class S_H, the conjecture remains open.

— Coefficients and Integral Mean Estimates for $K$-Quasiconformal Harmonic Mappings  (2608.28121 - Parashar et al., 28 Aug 2026) in Section 1.3, page 2