Wang et al. coefficient conjecture for all K-quasiconformal harmonic mappings

Prove Conjecture B for every normalized sense-preserving univalent K-quasiconformal harmonic mapping f=h+g in S_H^0(K), namely establish |a_n|−|b_n|≤n, |a_n|≤A(n,k), and |b_n|≤B(n,k) for all n≥2, with equality attained by the K-quasiconformal harmonic Koebe-type function P_k.

Background

Conjecture B proposes sharp coefficient bounds for normalized K-quasiconformal harmonic mappings, expressed through the explicit functions A(n,k) and B(n,k), where k=(K−1)/(K+1). The extremal candidate is the K-quasiconformal harmonic Koebe-type function P_k.

Prior work verified the conjecture for several geometric subclasses. The present paper proves it for the quasi-subordination subclass S_Hq0(S,K), but does not resolve it for the full class S_H0(K), so the general conjecture remains an unresolved problem.

References

One of these conjectures reads as follows. Conjecture B. Suppose that f = h + g ∈ S0H (K) is of the form (1). Then (6) |an| − |bn| ≤ n, |an| ≤ A(n, k), and |bn| ≤ B(n, k), for n = 2, 3, . . . , where A(n, k) and B(n, k) are defined by (4) and (5), respectively. Equality holds for the function P k given by (3).

— Coefficients and Integral Mean Estimates for $K$-Quasiconformal Harmonic Mappings  (2608.28121 - Parashar et al., 28 Aug 2026) in Conjecture B, Section 1.6, pages 5–6