{"paper":{"title":"Explicit solutions of the invariance equation for means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alfred Witkowski, Janusz Matkowski, Monika Nowicka","submitted_at":"2015-01-10T14:44:26Z","abstract_excerpt":"Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in a recent paper of P. Kahlig and J. Matkowski, and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02356","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}