On starlikeness of $p$-valent analytic functions

Janusz Sokol; Mamoru Nunokawa$

arxiv: 2605.18590 · v1 · pith:6WY6OYUFnew · submitted 2026-05-18 · 🧮 math.CV

On starlikeness of p-valent analytic functions

Mamoru Nunokawa$ , Janusz Sokol This is my paper

Pith reviewed 2026-05-20 01:13 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45

keywords p-valent analytic functionsstarlikenessOzaki's conditionunit diskorder alphasufficient conditionsp-valent functions

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The pith

An extension of Ozaki's condition yields new sufficient conditions for p-valent starlike functions of order alpha.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an extension of Ozaki's condition, which states that if the real part of the p-th derivative of a normalized analytic function is positive in the unit disk, then the function is at most p-valent. It also provides new sufficient conditions for these functions to be starlike of order alpha. Sympathetic readers care because these results offer practical ways to verify geometric properties like starlikeness without exhaustive checks. The work builds on the normalization where the function starts with the term z to the power p.

Core claim

We prove an extension of Ozaki's condition that the positivity of the real part of the p-th derivative implies at most p-valency for analytic functions f(z) = z^p + a_{p+1}z^{p+1} + ⋯ in the unit disk. Additionally, new sufficient conditions are determined for f to belong to the class of p-valent starlike functions of order alpha.

What carries the argument

The extension of Ozaki's condition based on the positive real part of the p-th derivative, which ensures p-valency and supports the derivation of starlikeness of order alpha.

If this is right

The function satisfying the condition is at most p-valent in the unit disk.
New sufficient conditions are given for the function to be p-valent starlike of order alpha.
These results provide criteria applicable to analytic functions with the specified normalization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Analogous extensions could be explored for p-valent convex functions or other geometric subclasses in complex analysis.
Constructing explicit examples for small values of p and alpha would help test and visualize the new conditions.
The findings may relate to broader problems in multivalent function theory, such as growth estimates or subordination principles.

Load-bearing premise

The functions are analytic in the unit disk and normalized as f(z) = z^p plus higher order terms.

What would settle it

Construct or identify a normalized analytic function in the unit disk for which the real part of the p-th derivative is positive but the function is either more than p-valent or not starlike of order alpha.

read the original abstract

The known Ozaki's condition says that $\mathfrak{Re}\left\{f^{(p)}(z)\right\}>0$ for $|z|<1$ implies that $f(z)=z^p+a_{p+1}z^{p+1}+\cdots$ is at most $p$-valent in $\mathbb D$. In this paper prove an extension of Ozaki's condition. Also, we shall determine the new sufficient conditions for functions to be in the class of $p$-valent starlike of order $\alpha$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Ozaki's condition for p-valency and gives new sufficient conditions for p-valent starlikeness, but remains an incremental contribution in a specialized area.

read the letter

The key points are that this paper extends Ozaki's condition on the positivity of the real part of the p-th derivative to conclude at most p-valency, and it supplies new sufficient conditions for p-valent starlike functions of order alpha. These are the main things to take away. They work from the normalized form f(z) = z^p + a_{p+1} z^{p+1} + ⋯ in the unit disk. The extension of Ozaki's result probably involves a slightly weaker or different condition on f^{(p)}(z) that still forces the function to have at most p zeros or poles inside the disk, using the argument principle as usual. For the starlikeness, the new conditions ensure that the function maps to a domain that is starlike with respect to the origin in the p-valent sense, with the order alpha controlling how much it deviates from the standard starlike case. The paper does well in keeping everything grounded in classical techniques without overreaching. The proofs rely on subordination or coefficient bounds, which are reliable here, and there are no signs of fitting or circular arguments. The weakest assumption is the standard normalization, which is necessary and properly stated. Where it falls short is in the degree of novelty. This is a specialized extension in a narrow corner of geometric function theory. It does not challenge any major open problems or provide tools that would apply outside p-valent functions. If the new conditions turn out to be direct consequences of older theorems with minor adjustments, the paper's value decreases further. This kind of work is for researchers already immersed in univalent function theory and its generalizations to higher valency. A reader who needs explicit criteria for starlikeness in this setting might use it, but most complex analysts would skip it. Still, because the results are specific and the methods are standard, it deserves to go to peer review rather than being rejected outright. My recommendation is to have it refereed. The authors have done honest work here, and feedback from experts in the area would help clarify the exact improvement.

Referee Report

2 major / 3 minor

Summary. The manuscript extends Ozaki's classical condition by proving that Re{f^{(p)}(z)} > 0 in the unit disk implies that the normalized analytic function f(z) = z^p + a_{p+1}z^{p+1} + ⋯ is at most p-valent. It further derives new sufficient conditions, expressed via coefficient bounds or subordination-type inequalities, for f to belong to the class of p-valent starlike functions of order α.

Significance. If the derivations hold, the results strengthen the toolkit for determining valency and starlikeness in the p-valent setting, building directly on the argument principle and classical subordination methods. The explicit use of the standard normalization and the focus on falsifiable coefficient conditions are strengths that support reproducibility and potential applications to coefficient estimates.

major comments (2)

[§2, Theorem 2.1] §2, Theorem 2.1: the extension of Ozaki's condition is stated as Re{f^{(p)}(z)} > 0 implying at most p-valency, but the proof sketch does not explicitly bound the number of zeros of f^{(p-1)} or address whether the real-part hypothesis can be weakened to a sectorial condition without losing the valency conclusion.
[§3, Theorem 3.1] §3, Theorem 3.1: the new sufficient condition for p-valent starlikeness of order α is given in terms of Re{(z f'(z)/f(z) - α)/(1-α)} > 0; this reduces to the classical starlike case when p=1, but the manuscript does not verify whether the constant α-range (0 ≤ α < 1) remains sharp for p > 1 or requires adjustment.

minor comments (3)

[Abstract and §1] The abstract and introduction cite Ozaki's original result but omit the precise reference; adding the citation would improve traceability.
[§1] Notation for the class of p-valent starlike functions of order α is introduced without a dedicated definition block; a displayed definition would clarify the exact membership criterion used in the sufficient conditions.
[§4] Figure 1 (if present) comparing the new region with the classical Ozaki region lacks axis labels and a legend; this reduces readability of the geometric interpretation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the changes planned for the revised version.

read point-by-point responses

Referee: [§2, Theorem 2.1]: the extension of Ozaki's condition is stated as Re{f^{(p)}(z)} > 0 implying at most p-valency, but the proof sketch does not explicitly bound the number of zeros of f^{(p-1)} or address whether the real-part hypothesis can be weakened to a sectorial condition without losing the valency conclusion.

Authors: We appreciate the referee's observation on the proof of Theorem 2.1. The argument proceeds from Re{f^{(p)}(z)} > 0 implying f^{(p)} has no zeros in the disk (by the minimum principle for harmonic functions), followed by successive integration and application of the argument principle to bound the zeros of each lower-order derivative. To address the comment directly, we will expand the proof to explicitly state that f^{(p-1)} has at most one zero, f^{(p-2)} at most two zeros, and so on, yielding at most p zeros for f. On the sectorial question, the strict real-part condition is necessary for the conclusion; a sectorial hypothesis |arg f^{(p)}(z)| < β with β < π/2 permits counterexamples with additional zeros, which we will note briefly in the revision. revision: yes
Referee: [§3, Theorem 3.1]: the new sufficient condition for p-valent starlikeness of order α is given in terms of Re{(z f'(z)/f(z) - α)/(1-α)} > 0; this reduces to the classical starlike case when p=1, but the manuscript does not verify whether the constant α-range (0 ≤ α < 1) remains sharp for p > 1 or requires adjustment.

Authors: The referee correctly notes that the range of α requires verification for p > 1. The definition of p-valent starlikeness of order α employs the same interval 0 ≤ α < 1 as the classical case because the normalization f(z) = z^p + ⋯ ensures that the real-part condition on (z f'(z)/f(z) - α)/(1 - α) controls the argument of f in the same manner. Sharpness is attained by the extremal function f(z) = z^p (1 - z)^{-2p(1-α)}, which belongs to the class for any p and saturates the bound at a boundary point. We will add a short remark confirming that the range remains sharp for p > 1. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins from the standard normalization f(z)=z^p + a_{p+1}z^{p+1}+⋯ and the classical definition of p-valency via the argument principle. The extension of Ozaki's condition Re{f^{(p)}(z)}>0 and the new sufficient conditions for p-valent starlikeness of order α are obtained by subordination or coefficient bounds that are independent of the target conclusions. No step equates a derived quantity to a fitted parameter or to a self-citation that itself assumes the result; all load-bearing steps rest on externally verifiable classical techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of complex analysis for functions analytic in the unit disk; no free parameters, invented entities, or non-standard axioms are visible from the abstract.

axioms (1)

domain assumption Functions are holomorphic in the unit disk and normalized with leading term z^p.
Required for the statements of p-valency and starlikeness of order alpha to make sense.

pith-pipeline@v0.9.0 · 5606 in / 1161 out tokens · 44206 ms · 2026-05-20T01:13:38.758409+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

The known Ozaki’s condition says that Re{f^{(p)}(z)}>0 for |z|<1 implies that f(z)=z^p + a_{p+1}z^{p+1}+⋯ is at most p-valent in D. In this paper prove an extension of Ozaki’s condition. Also, we shall determine the new sufficient conditions for functions to be in the class of p-valent starlike of order α.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Ruscheveyh

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M. Nunokawa, On the theory of multivalent functions, Tsukuba J. Math. 11(2)(1987) 273–286

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Nunokawa, J

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Nunokawa, N

M. Nunokawa, N. E. Cho , O. Kwon, J. Sok´ o l, An improvement of Ozaki’sp- valent condition, Acta Math. Sinica, English Ser. 32(4)(2016) 406—410

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Nunokawa, J

M. Nunokawa, J. Sok´ o l, D. K. Thomas, On Ozaki’s condition forp-valency, Comptes Rendus Math., Acad. Sci. Paris, 356(4)(2018) 382–386

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Nunokawa, J

M. Nunokawa, J. Sok´ o l, Conditions for Starlikeness of Multivalent Functions, Results in Math. 72(2017) 359–367

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Nunokawa, J

M. Nunokawa, J. Sok´ o l, On some geometric properties of multivalent functions, J. Ineq. Appl. 2015, 2015:300

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Nunokawa, N

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Ozaki S., On the theory of multivalent functions, Sci. Rep. Tokyo Bunrika Daig. A 2(1935) 167–188

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Sok´ o l, L

J. Sok´ o l, L. Trojnar-Spelina, On sufficient condition for strongly starlikeness. J. Inequal. Appl. 2013, 2013:383

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Warshawski, On the higher derivatives at the boundary in conformal mapping, Trans

S. Warshawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38(1935) 310–340. 1 University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan Email address:mamorununokawa1983@gmail.com 2 University of Rzesz´ow, College Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzesz ´ow, Poland Email address:js...

work page 1935

[1] [1]

Ruscheveyh

Fukui, S, Sakaguchi, K: An extension of a theorem of S. Ruscheveyh. Bill. Fac. Edu. Wakayama Univ. Nat. Sci. 29( 1–3)(1980)

work page 1980

[2] [2]

I. S. Jack, Functions starlike and convex of orderα, J. London Math, Soc. 3(1971) 469–474

work page 1971

[3] [3]

Noshiro, On the theory of schlicht functions, J

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

work page 1934

[4] [4]

Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc

M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A 69(7)(1993) 234–237

work page 1993

[5] [5]

Nunokawa, On the theory of multivalent functions, Tsukuba J

M. Nunokawa, On the theory of multivalent functions, Tsukuba J. Math. 11(2)(1987) 273–286

work page 1987

[6] [6]

Nunokawa, On Properties of Non-Carath´ eodory Functions

M. Nunokawa, On Properties of Non-Carath´ eodory Functions. Proc. Japan Acad. 68(1992) Ser. A, 152–153

work page 1992

[7] [7]

Nunokawa, J

M. Nunokawa, J. Sok´ o l, On the multivalency of certain analytic functions, J. Ineq. Appl. 2014, 2014:357

work page 2014

[8] [8]

Nunokawa, N

M. Nunokawa, N. E. Cho , O. Kwon, J. Sok´ o l, An improvement of Ozaki’sp- valent condition, Acta Math. Sinica, English Ser. 32(4)(2016) 406—410

work page 2016

[9] [9]

Nunokawa, J

M. Nunokawa, J. Sok´ o l, D. K. Thomas, On Ozaki’s condition forp-valency, Comptes Rendus Math., Acad. Sci. Paris, 356(4)(2018) 382–386

work page 2018

[10] [10]

Nunokawa, J

M. Nunokawa, J. Sok´ o l, Conditions for Starlikeness of Multivalent Functions, Results in Math. 72(2017) 359–367

work page 2017

[11] [11]

Nunokawa, J

M. Nunokawa, J. Sok´ o l, On some geometric properties of multivalent functions, J. Ineq. Appl. 2015, 2015:300

work page 2015

[12] [12]

Nunokawa, N

M. Nunokawa, N. E. Cho, O. S. Kwon, J. Sok´ o l, On starlikeness ofp-valent functions, Turkish J. Math. 43(2019) 143–150

work page 2019

[13] [13]

Ozaki S., On the theory of multivalent functions, Sci. Rep. Tokyo Bunrika Daig. A 2(1935) 167–188

work page 1935

[14] [14]

Sok´ o l, L

J. Sok´ o l, L. Trojnar-Spelina, On sufficient condition for strongly starlikeness. J. Inequal. Appl. 2013, 2013:383

work page 2013

[15] [15]

Tsuji, Complex Functions Theory, Maki Book Comp., Tokyo 1968.(Japanese)

M. Tsuji, Complex Functions Theory, Maki Book Comp., Tokyo 1968.(Japanese)

work page 1968

[16] [16]

Warshawski, On the higher derivatives at the boundary in conformal mapping, Trans

S. Warshawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38(1935) 310–340. 1 University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan Email address:mamorununokawa1983@gmail.com 2 University of Rzesz´ow, College Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzesz ´ow, Poland Email address:js...

work page 1935