Pith Number

pith:4MHXH6PF

pith:2026:4MHXH6PF2QNFV566YLJ3DZJK4G

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Hadamard product of convex functions and Jackson operator

J. Sok\'o\l, K. Piejko, K. Tr\c{a}bka-Wi\c{e}c\a{l}aw

Jackson's difference operator for convex univalent functions equals the Hadamard product of two power series even for complex q.

arxiv:2605.18412 v1 · 2026-05-18 · math.CV

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\pithnumber{4MHXH6PF2QNFV566YLJ3DZJK4G}

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Record completeness

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3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

Jackson's difference operator for convex univalent functions in |z|<1 with complex parameter q can be considered as a Hadamard product of two power series.

C2weakest assumption

The functions under consideration are analytic, convex, and univalent in the open unit disk, and the complex q is chosen so that the difference operator remains well-defined and the Hadamard product representation holds.

C3one line summary

The paper studies properties of the Jackson q-difference operator for convex univalent functions with complex q, expressed as a Hadamard product.

References

11 extracted · 11 resolved · 0 Pith anchors

[1] A. W. Goodman, Univalent Functions, II, Mariner Publishing Co.: Tampa, Florida (1983) 1983

[2] F. H. Jackson, Onq-functions and certain difference operator, Trans. Royal Soc. Edinburgh 46(1908) 253–281 1908

[3] F. H. Jackson, Onq-definite integrals, Quarterly J. Pure and Appl. Math. 41(1910) 193–203 1910

[4] K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. Jap., 2(1)(1934-35) 129–135 1934

[5] K. Piejko, J. Sok´ o l, On convolution andq-calculus, Boletin Soc. Mat. Mexicana, 26(2)(2020) 349–359 2020

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:05:59.456861Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e30f73f9e5d41a5af7dec2d3b1e52ae1a1a1545d040d75cc133732c5d2301f3f

Aliases

arxiv: 2605.18412 · arxiv_version: 2605.18412v1 · doi: 10.48550/arxiv.2605.18412 · pith_short_12: 4MHXH6PF2QNF · pith_short_16: 4MHXH6PF2QNFV566 · pith_short_8: 4MHXH6PF

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/4MHXH6PF2QNFV566YLJ3DZJK4G \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: e30f73f9e5d41a5af7dec2d3b1e52ae1a1a1545d040d75cc133732c5d2301f3f

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "1dabaed36baf6f2655d84fc242da602ea7f41784ea8fb16b1f705033c1b641d4",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/publicdomain/zero/1.0/",
    "primary_cat": "math.CV",
    "submitted_at": "2026-05-18T13:51:36Z",
    "title_canon_sha256": "34d15821a499a89e9c5dc1ff5ca05c456559929467b5d729b25f429228f4ebc9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.18412",
    "kind": "arxiv",
    "version": 1
  }
}