The two-dimensional Matkowski--Sutô equation with holomorphic and strictly increasing generators
Abstract
We study the two-dimensional Matkowski--Sutô equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of Tóth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Daróczy and Páles, persists under holomorphy but not under monotonicity.
Related
- Increasing stability for the inverse potential problem of the fractional Helmholtz equation
- Holomorphic normal forms of six-dimensional totally nondegenerate CR manifolds in C^5
- Using the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with two explicit levels
- Rigidity of symmetric holomorphic functions on infinite-dimensional sequence spaces
- Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions
BibTeX
@misc{okamura2026twodimensional,
title = {{The two-dimensional Matkowski--Sutô equation with holomorphic and strictly increasing generators}},
author = {Kazuki Okamura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11102},
eprint = {2609.11102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11102},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The two-dimensional Matkowski--Sutô equation with holomorphic and strictly increasing generators
AU - Kazuki Okamura
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11102
UR - https://arxiv.org/abs/2609.11102
AB - We study the two-dimensional Matkowski--Sutô equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of Tóth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Daróczy and Páles, persists under holomorphy but not under monotonicity.
ER -