A mathematical preprint gives a general Hardy-space guarantee for K-quasiconformal harmonic mappings: every mapping in the class SH(K) belongs to h^p when the positive exponent p is below the reciprocal of a paper-defined quantity, alpha_K. It also places the same broad class inside a weighted harmonic Bergman space below a related cutoff. More explicit coefficient and growth bounds are reported for narrower quasi-subordinate and odd subclasses.
That qualification matters because alpha_K is not assigned an exact numerical value in the supplied analysis. The general Hardy result therefore cannot be reduced here to one decimal threshold. The review also says the exponent has not been shown to be sharp, so the theorem identifies a guaranteed range rather than the largest possible range.
The paper defines SH(K) through mappings that are K-quasiregular and univalent, meaning one-to-one, in the domain it denotes by D. Its wider program covers coefficient estimates, growth and integral-mean bounds, and membership in Hardy and weighted-Bergman spaces for harmonic mappings in the unit disk.
A parameter controls the broad results
Alongside the Hardy statement, the paper places SH(K) in a weighted harmonic Bergman space whenever the positive exponent is below the ratio of beta plus two to alpha_K. Here beta is the weight parameter, p is the space exponent, and alpha_K is the coefficient quantity used in the general bound. The supplied analysis does not state that this range is optimal.
The proofs use integration along radial lines in the disk and invoke Lemma B as an integral-mean tool. The conclusions are thus derived from the properties built into the mapping classes and their coefficients, rather than from a sample of observed functions.
More specific classes bring more specific estimates
The quasi-subordination results impose a structured relationship between analytic functions. One function is formed by taking a comparison function, composing it with an analytic map that stays inside the unit disk and is zero at the origin, and multiplying by an analytic factor whose modulus is at most one. This framework defines the narrower class used for several of the paper's coefficient and growth results.
For that quasi-subordination subclass, coefficient estimates cover every integer index from 2 upward. The paper also gives a growth estimate that bounds the size of a mapping at points inside the disk by a function of the quasiconformal parameter and the point's radius. These estimates are stated to be sharp within the class treated by the theorem.
The broader quasi-subordinate class receives Hardy-space membership for positive exponents below one-half. It also belongs to the weighted harmonic Bergman space for exponents below beta plus two, divided by two, when beta is greater than minus one. The supplied analysis does not say that either cutoff is sharp.
The odd subclass has its own coefficient and growth estimates. They apply from coefficient index 1 onward and include a strict bound on the difference between the magnitudes of the paired coefficients, along with explicit bounds for each coefficient sequence. The theorem sets the coefficient constant lambda at 1.1305 and gives a growth bound inside the disk that is stated to be sharp for the specified extremal function.
For odd mappings in SH(S,K), the paper guarantees Hardy-space membership for positive exponents below one. The corresponding weighted-Bergman guarantee holds for exponents below beta plus two. The theorem statement does not establish that these ranges are optimal.
What the theorems leave unresolved
The broad Hardy-space theorem is presented as progress toward the underlying problem, not as a complete determination of the sharp exponent. The supplied analysis leaves open whether the range below one-half holds for every K-quasiconformal harmonic mapping. The class-specific results also do not establish the same conclusions for all univalent harmonic mappings outside the subclasses studied.
The authors describe the research as purely theoretical and not associated with data. There are no participants, observations or sampled units behind the bounds. The document is identified in its front matter as arXiv:2608.28121v1, dated 28 Aug 2026, and remains a preprint in the supplied metadata.
Paper data and sources
Original title: Coefficients and Integral Mean Estimates for $K$-Quasiconformal Harmonic Mappings
Authors: Jasbir Parashar, Saminathan Ponnusamy, A. Sairam Kaliraj
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text