{
  "id": "2501.01854",
  "version": "v1",
  "published": "2025-01-03T15:08:15.000Z",
  "updated": "2025-01-03T15:08:15.000Z",
  "title": "On extending the class of convex functions",
  "authors": [
    "Shravan Mohan"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "In this brief note, it is shown that the function p^TW log(p) is convex in p if W is a diagonally dominant positive definite M-matrix. The techniques used to prove convexity are well-known in linear algebra and essentially involves factoring the Hessian in a way that is amenable to martix analysis. Using similar techniques, two classes of convex homogeneous polynomials is derived - namely, p^TW p2 and (p^k)^TW p^k - the latter also happen to be SOS-convex. Lastly, usign the same techniques, it is also shown that the function p^TW ep is convex over the positive reals only if W is a non-negative diagonal matrix. Discussions regarding the utility of these functions and examples accompany the results presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-03T15:08:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex functions",
      "diagonally dominant positive definite m-matrix",
      "brief note",
      "linear algebra",
      "martix analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}