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arxiv: 2504.19132 · v1 · submitted 2025-04-27 · 🧮 math.CV

An estimation of the pre-Schwarzian norm for certain classes of analytic functions

Vasudevarao Allu , Raju Biswas , Rajib Mandal This is my paper

Pith reviewed 2026-05-22 19:09 UTC · model grok-4.3

classification 🧮 math.CV
keywords pre-Schwarzian normMa-Minda starlike classMa-Minda convex classsubordinationanalytic univalent functionssharp estimatesstarlike functionsconvex functions
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The pith

The pre-Schwarzian norm admits sharp upper bounds for Ma-Minda starlike and convex functions tied to two specific subordinating functions.

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

The paper derives the largest possible value of the pre-Schwarzian norm for functions belonging to the Ma-Minda starlike class S*(φ) and the Ma-Minda convex class C(φ). The bounds are obtained when the subordinating function φ is the power map 1/(1-z)^s for 0 < s ≤ 1 or the quadratic map (1 + s z)^2 for 0 < s ≤ 1/√2. These classes consist of normalized analytic functions whose image or derivative image is contained in a region controlled by φ. The estimates supply uniform control on how far such functions can deviate from being linear inside the unit disk. A reader would care because the pre-Schwarzian norm governs distortion, growth, and univalence criteria for the functions in these families.

Core claim

For the classes S*(φ) and C(φ) with φ(z) = 1/(1 - z)^s where 0 < s ≤ 1 and with φ(z) = (1 + s z)^2 where 0 < s ≤ 1/√2, the paper proves that the pre-Schwarzian norm attains a sharp maximum value attained by an explicit extremal function in each case.

What carries the argument

The pre-Schwarzian derivative norm, given by the supremum of (1 - |z|^2) |f''(z)/f'(z)| over the unit disk, which quantifies the local distortion of f.

If this is right

  • The growth and distortion theorems for these classes follow directly from the norm bound.
  • Coefficient bounds and radius problems for the same families become accessible via the norm estimate.
  • The extremal functions are the same ones that maximize the norm, confirming the sharpness.

Where Pith is reading between the lines

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

  • The same technique may yield bounds when φ is replaced by other convex functions with known Taylor coefficients.
  • The estimates could be compared with the corresponding bounds for the Schwarzian derivative to see which norm gives tighter control on univalence.
  • Numerical verification of the bound for a random sample of functions generated by the subordination condition would test consistency with the analytic proof.

Load-bearing premise

The functions must satisfy the subordination relation that defines membership in S*(φ) or C(φ) for the chosen φ.

What would settle it

Compute the pre-Schwarzian norm of the explicit extremal function for one of the given φ and check whether any other function in the same class exceeds that value.

Figures

Figures reproduced from arXiv: 2504.19132 by Rajib Mandal, Raju Biswas, Vasudevarao Allu.

Figure 2
Figure 2. Figure 2: Image of D un￾der the mapping (1 + sz) 2 for s = 1/ √ 2 The function φ(z) = (1 + sz) 2 maps the unit disk D onto a domain bounded by a lima¸con given by n u + iv ∈ C [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Thus, F2(t) attains its maximum value at t = ts. From (3.1), we have ∥Pf ∥ ≤ F2(ts) = sts(1 + ts) + (1 + ts)(1 − ts) 1−s − (1 − t 2 s ) ts , where ts ∈ (0, 1) is the unique positive root of the equation Fs(t) := (1 − t) −s [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graph of Fs(t) for different values of s in (0, 1) In the following result, we obtain the estimate of the pre-Schwarzian norm for func￾tions in the class S ∗ L . Theorem 3.2. Let f ∈ S∗ L . Then the pre-Schwarzian norm satisfies the following inequality ∥Pf ∥ ≤ 2s(1 − t 2 s ) 1 − sts + (1 − t 2 s ) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graph of Gs(t) for different values of s in (0, 1/ √ 2] In the following result, we establish the sharp estimate of the pre-Schwarzian norm for the functions in the class Chyp. Theorem 3.3. For any g ∈ Chyp, the pre-Schwarzian norm satisfies the following sharp inequality ∥Pg∥ ≤    (1 + rs)(1 − rs) 1−s − (1 − r 2 s ) rs for 0 < s < 1 2 for s = 1, where rs ∈ (0, 1) is the unique root of the equation (1 −… view at source ↗
Figure 5
Figure 5. Figure 5: Graph of hs(r) for different values of s in (0, 1) In the following result, we establish the sharp estimate of the pre-Schwarzian norm for the functions in the class CL. Theorem 3.4. For any g ∈ CL, the pre-Schwarzian norm satisfies the following sharp inequality ∥Pg∥ ≤ 2 √ 3s 2 + 4 + 4 3s 2 + 2√ 3s 2 + 4 − 4  27s . Proof. Let g ∈ CL, then by the definition of the class CL, we have 1 + zg′′(z) g ′(z) ≺… view at source ↗
read the original abstract

The primary objective of this paper is to establish the sharp estimates of the pre-Schwarzian norm for functions $f$ in the class $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$ when $\varphi(z)=1/(1-z)^s$ with $0<s\leq 1$ and $\varphi(z)=(1+sz)^2$ with $0<s\leq 1/\sqrt{2}$, where $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$ are the Ma-Minda type starlike and Ma-Minda type convex classes associated with $\varphi$, respectively.

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.

Referee Report

2 major / 2 minor

Summary. The paper establishes sharp estimates for the pre-Schwarzian norm ||f|| = sup_{|z|<1} (1-|z|^2) |f''(z)/f'(z)| for functions f in the Ma-Minda starlike class S^*(φ) and convex class C(φ), specifically when φ(z) = 1/(1-z)^s for 0 < s ≤ 1 and when φ(z) = (1 + s z)^2 for 0 < s ≤ 1/√2. The classes are defined by the subordination conditions z f'/f ≺ φ and f' ≺ φ, respectively. The proofs rely on standard subordination arguments and relations such as f''/f' = p'/p + (p-1)/z for starlike and f''/f' = (p-1)/z for convex cases.

Significance. If the claimed sharp bounds are verified to be attained by admissible extremal functions in each class, the results would extend classical pre-Schwarzian estimates from the standard starlike and convex classes to these parameterized Ma-Minda families. This could be useful for distortion theorems and coefficient problems in geometric function theory, particularly since the parameter ranges on s are explicitly restricted to ensure the functions remain in the unit disk and satisfy the required analyticity and univalence.

major comments (2)
  1. [§3] §3 (main results for S^*(φ)): The upper bound for ||f|| is derived by maximizing over p ≺ φ, but the manuscript does not explicitly construct or evaluate the candidate extremal function f(z) = z exp(∫_0^z (φ(t)-1)/t dt) to confirm that the supremum is attained at an interior point of D for the full range 0 < s ≤ 1. Without this verification, the sharpness claim rests on the assumption that the bound is achieved rather than merely approached as |z| → 1.
  2. [§4] §4 (results for C(φ)): For φ(z) = (1 + s z)^2 with 0 < s ≤ 1/√2, the bound obtained from f''/f' = (p-1)/z requires explicit confirmation that the extremal function remains analytic and univalent in D and that the pre-Schwarzian norm equals the stated constant; the parameter restriction on s appears chosen to guarantee this, but the manuscript does not include a direct computation showing attainment inside the disk.
minor comments (2)
  1. [Introduction] The notation for the pre-Schwarzian norm is introduced without a numbered equation; adding ||f|| := sup ... would improve readability when the quantity is referenced in later theorems.
  2. [References] A few references to classical results on pre-Schwarzian norms (e.g., Becker's criterion or related works on Ma-Minda classes) are cited but could be expanded with one or two more recent papers on subordination techniques for sharpness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for providing constructive comments that help improve the presentation of our results. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (main results for S^*(φ)): The upper bound for ||f|| is derived by maximizing over p ≺ φ, but the manuscript does not explicitly construct or evaluate the candidate extremal function f(z) = z exp(∫_0^z (φ(t)-1)/t dt) to confirm that the supremum is attained at an interior point of D for the full range 0 < s ≤ 1. Without this verification, the sharpness claim rests on the assumption that the bound is achieved rather than merely approached as |z| → 1.

    Authors: We acknowledge the referee's observation. To establish sharpness rigorously, we will revise the manuscript to explicitly introduce the extremal function f(z) = z exp(∫_0^z (φ(t)-1)/t dt) for the case p(z) = φ(z). For φ(z) = 1/(1-z)^s with 0 < s ≤ 1, this function belongs to S^*(φ) by construction. We will provide a direct computation of (1 - |z|^2) |f''(z)/f'(z)| and verify that its supremum over the unit disk equals the bound derived in the paper. This verification will confirm attainment of the bound, either at an interior point for certain s or approached as |z| → 1, thereby justifying the sharpness claim for the entire range. revision: yes

  2. Referee: [§4] §4 (results for C(φ)): For φ(z) = (1 + s z)^2 with 0 < s ≤ 1/√2, the bound obtained from f''/f' = (p-1)/z requires explicit confirmation that the extremal function remains analytic and univalent in D and that the pre-Schwarzian norm equals the stated constant; the parameter restriction on s appears chosen to guarantee this, but the manuscript does not include a direct computation showing attainment inside the disk.

    Authors: We agree that a direct verification is beneficial. In the revised manuscript, we will explicitly construct the extremal function for the convex case, which satisfies f'(z) = φ(z), and confirm its analyticity and univalence in the unit disk under the given restriction 0 < s ≤ 1/√2. We will compute the pre-Schwarzian norm for this function and show that it attains the stated bound. The restriction on s is indeed to ensure that φ maps the unit disk into a region that guarantees univalence of the integrated function, and we will include this justification along with the attainment computation. revision: yes

Circularity Check

0 steps flagged

No circularity: estimates derived from subordination without reduction to inputs

full rationale

The paper derives pre-Schwarzian norm bounds for S*(φ) and C(φ) via standard subordination p ≺ φ and the relations f''/f' = p'/p + (p-1)/z or (p-1)/z. These steps use the given φ(z) = 1/(1-z)^s or (1+sz)^2 directly to maximize the expression over |z|<1; no parameter is fitted to data and then renamed as a prediction, no self-definition of the norm via the bound itself, and no load-bearing self-citation chain. The derivation remains self-contained against the subordination definition and the explicit φ forms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard theory of analytic functions in the unit disk and the definition of Ma-Minda classes via subordination; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard results from univalent function theory and subordination principle hold in the unit disk.
    The classes S*(φ) and C(φ) are defined using subordination, which presupposes the usual analytic-function machinery.

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Reference graph

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