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arxiv: 2505.19160 · v1 · submitted 2025-05-25 · 🧮 math.CV

On certain subclasses of analytic and harmonic mappings

Raju Biswas This is my paper

Pith reviewed 2026-05-19 13:30 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C4531A05
keywords harmonic mappingsanalytic univalent functionscoefficient boundsHankel determinantstarlikenessgrowth estimatesunit disk
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The pith

Harmonic mappings in the class D_H^0(α, M) satisfy coefficient bounds, growth estimates and starlikeness criteria while functions in P(M) have a sharp second Hankel determinant bound on inverse logarithmic coefficients.

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

The paper defines the class D_H^0(α, M) consisting of harmonic functions f = h + conjugate(g) in the unit disk that obey the inequality |(1 - α)h'(z) + α z h''(z) - 1 + α| ≤ M + |(1 - α)g'(z) + α z g''(z)| with g'(0) = 0. It derives bounds on the coefficients of the series expansions for h and g, establishes growth estimates for |f(z)|, and supplies conditions under which these mappings are starlike. The paper further obtains the sharp upper bound for the second Hankel determinant formed from the inverse logarithmic coefficients of normalized analytic univalent functions f that belong to the auxiliary class P(M) and satisfy Re(z f''(z)) > -M, provided 0 < M ≤ 1/log 4. A sympathetic reader cares because these controls on size and shape extend classical results in geometric function theory to a broader family of mappings.

Core claim

Functions belonging to the newly defined class D_H^0(α, M) satisfy coefficient bounds, growth estimates and starlikeness criteria; moreover the sharp bound of the second Hankel determinant of the inverse logarithmic coefficients is obtained for normalized analytic univalent functions f in P(M) satisfying Re(z f''(z)) > -M when 0 < M ≤ 1/log 4.

What carries the argument

The class D_H^0(α, M) of harmonic mappings defined by the inequality |(1-α)h'(z) + α z h''(z) -1 +α| ≤ M + |(1-α)g'(z) + α z g''(z)| with g'(0)=0 holding throughout the unit disk.

If this is right

  • The coefficients of the analytic and co-analytic parts of functions in D_H^0(α, M) remain bounded by quantities depending only on α and M.
  • Growth estimates limit the modulus |f(z)| for |z| < 1 in terms of α and M.
  • Functions in D_H^0(α, M) satisfy explicit starlikeness criteria for suitable ranges of the parameters.
  • The second Hankel determinant of inverse logarithmic coefficients for functions in P(M) attains a sharp upper bound when M lies in (0, 1/log 4].

Where Pith is reading between the lines

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

  • Special choices of α and M recover earlier subclasses of harmonic mappings already studied in the literature.
  • The restriction M ≤ 1/log 4 may be tied to the radius of univalence or convexity for the associated analytic functions.
  • The same differential inequality could be combined with other operators such as convolution or integral transforms to generate further subclasses.

Load-bearing premise

The defining inequality is assumed to hold for every z inside the unit disk and the auxiliary class P(M) is restricted to the interval 0 < M ≤ 1/log 4.

What would settle it

A concrete harmonic function satisfying the defining inequality but whose coefficient magnitudes exceed the claimed bounds, or an analytic function in P(M) whose second Hankel determinant of inverse logarithmic coefficients exceeds the stated sharp value.

Figures

Figures reproduced from arXiv: 2505.19160 by Raju Biswas.

Figure 2
Figure 2. Figure 2: The graph of (−2 + 3p1)/(3p 2 1 ) for p1 ∈ (0, 1) Sub-case 3.2. Note that 4 (1 + |C|) 2 = 4  1 + (2 + p 2 1 ) 3p1 2 = 4(p 4 1 + 6p 3 1 + 13p 2 1 + 12p1 + 4) 9p 2 1 > 0 and −4AC C −2 − 1  = − 2p 2 1 9 · (39M2 − 12M + 2)(p 4 1 − 5p 2 1 + 4) (1 − p 2 1 )(2 + p 2 1 ) < 0. Therefore, min n 4 (1 + |C|) 2 , −4AC C −2 − 1 o = −4AC C −2 − 1  . Also from (5.7), we know that −4AC C −2 − 1  ≤ B 2 hold for M > 0 … view at source ↗
Figure 3
Figure 3. Figure 3: The graph of the polynomial Ψ(M) for 1/3 < M ≤ 1/ log 4 Sub-case 3.4. Corresponding to the values of M, we consider the following cases. Sub-case 3.4.1 For 0 < M < 1/3, the inequality |AB| − |C|(|B| − 4|A|) ≤ 0 is equivalent to the inequality (5.8). By using similar arguments to those of Sub-case 3.3, we get (5.9) with ∆(M) := 12(19 − 186M + 899M2 − 2172M3 + 2496M4 ) > 0, 117M3 − 153M2 + 48M − 8 < 0 and 78… view at source ↗
Figure 4
Figure 4. Figure 4: The graph of the polynomials Ψ1(M) and Ψ2(M) for 0 < M < 1/3 Thus, the inequality (5.8) is valid for √ t1 ≤ p1 < 1 whenever M ∈ (0, 1/3). In view of Lemma 4.2 and (5.6), we have [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: The graph of the polynomials Ψ3(M) and Ψ4(M) respectively within 1/3 < M ≤ 1/ log 4 Thus, the inequality (5.13) is valid for 0 < p1 ≤ √ t4 whenever M ∈ (1/3, 1/ log 4]. In view of Lemma 4.2 and (5.6), we have [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The graph of the polynomial ϕ3(M) for 4/13 < M < 1/3 Thus ξ1(x), x ∈ (0, t1) is decreasing for 4/13 < M < 1/3. As a result, both the functions ξ1(x) and ξ2(x) are decreasing for x ∈ (0, t1) and 0 < M < 1/3. Hence, we derive from (5.17) that [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The graph of the polynomials Ψj (M) (1 ≤ j ≤ 5) for 1/3 ≤ M ≤ 1/ log 4 From (5.17), we have [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The graph of the polynomial Φ5(M) for 0 < M ≤ 1/3 0.34 0.36 0.38 0.40 0.42 0.44 0 1000 2000 3000 4000 [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
read the original abstract

Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let $\mathcal{D}_{\mathcal{H}}^0(\alpha, M)$ denote the class of functions $f=h+ \overline{g}\in\mathcal{H}$ satisfying the conditions $\left|(1-\alpha)h'(z)+\alpha zh''(z)-1+\alpha\right|\leq M+\left|(1-\alpha)g'(z)+\alpha zg''(z)\right|$ with $g'(0)=0$ for $z\in\mathbb{D}$, $M>0$ and $\alpha\in(0,1]$. In this paper, we investigate fundamental properties for functions in the class $\mathcal{D}_{\mathcal{H}}^0(\alpha, M)$, such as the coefficient bounds, growth estimates, starlikeness and some other properties. Furthermore, we obtain the sharp bound of the second Hankel determinant of inverse logarithmic coefficients for normalized analytic univalent functions $f\in\mathcal{P}(M)$ in $\mathbb{D}$ satisfying the condition $\text{Re}\left(zf''(z)\right)>-M$ for $0<M\leq 1/\log4$ and $z\in\mathbb{D}$.

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

0 major / 3 minor

Summary. The paper introduces the class D_H^0(α, M) of normalized harmonic mappings f = h + conjugate(g) in the unit disk satisfying the inequality |(1-α)h'(z) + α z h''(z) - 1 + α| ≤ M + |(1-α)g'(z) + α z g''(z)| with g'(0) = 0. It derives coefficient bounds, growth estimates, and starlikeness criteria for functions in this class. Separately, for the class P(M) of normalized analytic univalent functions satisfying Re(z f''(z)) > -M, the paper obtains the sharp bound on the second Hankel determinant of the inverse logarithmic coefficients when 0 < M ≤ 1/log 4.

Significance. If the derivations hold, the results extend the literature on subclasses of harmonic mappings and univalent functions by supplying explicit estimates obtained via power-series manipulation and standard growth/subordination theorems. The sharp Hankel-determinant bound, obtained after verifying positivity of the auxiliary function for the extremal case under the stated restriction on M, is a concrete contribution that could serve as a reference point for related problems in geometric function theory.

minor comments (3)
  1. [Abstract] The abstract presents two distinct sets of results (one for the harmonic class D_H^0(α, M) and one for the analytic class P(M)); a brief sentence separating these threads would improve readability.
  2. [Section 2] In the statement of the defining inequality for D_H^0(α, M), the normalization g'(0) = 0 is given but its role in the subsequent coefficient extractions could be recalled explicitly when the first few coefficients are computed.
  3. [Section 4] The range 0 < M ≤ 1/log 4 for the Hankel-determinant result is justified by direct verification on the extremal function; a short remark on whether the bound remains valid (perhaps with a different constant) for M slightly larger than 1/log 4 would clarify the sharpness claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript on the class D_H^0(α, M) of harmonic mappings and the sharp second Hankel determinant bound for inverse logarithmic coefficients in P(M). The recommendation for minor revision is noted. No specific major comments appear in the report, so we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claims follow from direct manipulation of the defining inequality for class D_H^0(α, M) using power-series coefficients together with standard growth and subordination theorems for harmonic mappings. The separate Hankel-determinant bound for P(M) is obtained under the explicit restriction 0 < M ≤ 1/log 4, which is justified by verifying that an auxiliary function remains positive and subordinate via direct computation on the extremal function. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; all derivations remain independent of the paper's own inputs and rest on externally verifiable analytic techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the ad-hoc definition of the new subclass via the given differential inequality and on standard background facts from complex analysis; no numerical free parameters or newly invented entities are introduced.

axioms (1)
  • standard math Basic properties of analytic and harmonic functions in the unit disk (normalization, analyticity, conjugation)
    Invoked throughout the definition of H and the new subclass.

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