bibkey: miller1978starlike authors: Sanford S. Miller, Petru T. Mocanu, and Maxwell O. Reade year: 1978 title: Starlike integral operators doi: 10.2140/pjm.1978.79.157 url: https://msp.org/pjm/1978/79-1/pjm-v79-n1-p13-p.pdf claim: “Classical starlike integral preservation motivates the local positive-real first-contact argument inside the all-composition disk consumer.” strata_touched:

• D5/S3/AnalyticClosure/Polylogarithm/CompositionZeroFree license: citation-only triage: anchor

Classical integral preservation

Verified locator

DOI: 10.2140/pjm.1978.79.157 Source URL: https://msp.org/pjm/1978/79-1/pjm-v79-n1-p13-p.pdf Pacific Journal of Mathematics 79(1), 157-168 (1978). Theorem 2 on printed page 162 states preservation of the normalized starlike class under the displayed integral operator for positive beta and nonnegative gamma. Lemma A on page 158 supplies its positive-real differential-inequality background. The title, author line and theorem were checked in the primary PDF; the DOI identity was checked against Crossref. No proof text or Lean code is transplanted.

Use and exact scope

The source consumer contains the following local implication: for analytic $P$ on the unit disk, $\mathrm{Re}P(0)>0$, positive real part of $Q$, and $z{P}^{\prime }=1-PQ$, the real part of $P$ stays positive. The proof attains a minimum contact radius on a compact subdisk. Angular minimality sets the imaginary part of $z{P}^{\prime }$ to zero; the inward radial derivative sets its real part nonpositive. The differential equation then forces $P=0$ at contact and $z{P}^{\prime }=1$, a contradiction.

For the normalized series the local application takes $P=g/h$ with $h$ already nonzero. The conclusion proves $g$ nonzero; that fact is never an input. Induction uses the actual source recurrence and its removable extension. The order can be greater than one; no univalence statement for the unnormalized multiple polylogarithm is asserted.

This is attribution for a classical method, not a claim of new classical integral-preservation mathematics. It is not an independently exported or frozen generic theorem. The new source-specific formal construction is assessed as repository-derived, with this reference acknowledged.