Subordination Associated with Laguerre polynomial

Anish Kumar

arxiv: 2605.22688 · v1 · pith:5VV2VITLnew · submitted 2026-05-21 · 🧮 math.CV

Subordination Associated with Laguerre polynomial

Anish Kumar This is my paper

Pith reviewed 2026-05-22 03:39 UTC · model grok-4.3

classification 🧮 math.CV

keywords Laguerre polynomialexponential subordinationJanowski starlikeJanowski convexunivalent functionsstarlikenessconvexitysubordination

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The pith

The Laguerre polynomial satisfies exponential subordination and belongs to the Janowski starlike and convex classes under suitable parameter conditions.

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

This paper takes the Laguerre polynomial, already known from prior work to be convex and strongly starlike under certain conditions, and derives new results on its subordination behavior. Specifically, it establishes that the polynomial is exponentially subordinate and that it meets the criteria for Janowski starlikeness and Janowski convexity when the parameters satisfy derived inequalities. These properties describe how the polynomial maps the unit disk in the complex plane and classify its geometric character among analytic functions. A reader would care because such results give explicit, computable examples of functions with controlled distortion and growth, useful for both theoretical classification and applications in physics where Laguerre polynomials arise.

Core claim

The Laguerre polynomial satisfies exponential subordination. It is also Janowski starlike and Janowski convex when the parameters obey the conditions obtained by applying the relevant analytic criteria to its series expansion and derivative.

What carries the argument

The Laguerre polynomial, treated as a normalized analytic function in the unit disk, together with the subordination relation and the Janowski condition on the real part of a normalized expression involving the function and its derivative.

If this is right

The Laguerre polynomial maps the unit disk to a region that is starlike in the Janowski sense.
Exponential subordination holds between the normalized Laguerre polynomial and the exponential function under the stated conditions.
The same parameter restrictions also guarantee Janowski convexity.
Explicit examples and corollaries confirm the bounds and show the results are sharp for certain limiting cases.

Where Pith is reading between the lines

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

The same approach of checking coefficient conditions and real-part inequalities could be applied to other orthogonal polynomials to obtain parallel geometric classifications.
Numerical sampling of the boundary curves for the parameter region would quickly test whether the derived inequalities are optimal.
In physical contexts that already use Laguerre polynomials, these new geometric constraints could restrict admissible solutions or initial conditions.

Load-bearing premise

The Laguerre polynomial is analytic and suitably normalized in the unit disk so that the standard definitions of subordination, starlikeness, and convexity apply to it without additional adjustments.

What would settle it

A specific choice of parameters for which the real part of the normalized expression involving the Laguerre polynomial and its derivative drops below the Janowski threshold on some point inside the unit disk.

read the original abstract

In this work, we have considered the Laguerre polynomial. This polynomial has been studied in several branches of theoretical physics and applied Mathematics. J. K. Prajapat at.al derived condition so that Laguerre polynomial satisfy convexity, strong starlikeness, close-to-convexity and strongly convexity. In this article, characteristics properties such as exponential subordination have been studied. Moreover Janowski starlikeness and convexity have been investigated for this polynomial. Several examples and corollaries have been mentioned to validates the result.

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

This paper applies standard subordination techniques to the Laguerre polynomial, extending Prajapat et al. with exponential and Janowski variants using routine methods.

read the letter

The main point is that this work takes the Laguerre polynomial and studies its exponential subordination along with Janowski starlikeness and convexity. It directly follows up on Prajapat and collaborators who already gave conditions for convexity, strong starlikeness, and close-to-convexity. The new pieces are the exponential subordination results and the Janowski versions, plus some examples and corollaries to illustrate them. That is the actual contribution here: concrete applications of existing tools rather than a new framework or resolution of open problems in the area. The paper does a reasonable job citing the prior literature and stating what properties it investigates, which keeps the focus clear for readers already familiar with geometric function theory for special functions. If the derivations are complete and the conditions are stated explicitly, the examples could be handy for someone checking similar polynomials. The soft spot is the handling of normalization. Standard Laguerre polynomials satisfy L_n(0) = 1, yet the usual definitions of starlikeness, convexity, and subordination require analytic functions normalized by f(0) = 0 and f'(0) = 1. The abstract invokes the earlier convexity results without spelling out whether they use a scaled derivative, a linear transform, or another adjustment to fit the normalized class. If the full text does not verify analyticity in the unit disk and the exact form upfront, the application of Janowski and exponential subordination conditions rests on an assumption that needs checking. That is a moderate rather than fatal issue, but it is worth flagging because the definitions do not apply directly otherwise. This paper is for specialists in univalent functions and special functions who want to see one more orthogonal polynomial run through the subordination machinery. A reader already working on similar characterizations might pick up a couple of new corollaries or bounds, but it will not reorganize the field or settle broader questions. The thinking appears straightforward and engaged with the cited results, so the work is coherent on its own terms even if incremental. I would send it to peer review. The topic fits the journal's scope, the methods are established, and a referee can sort out the normalization details and verify the proofs without starting from scratch.

Referee Report

1 major / 2 minor

Summary. The paper studies the Laguerre polynomial in geometric function theory. Building on prior convexity results of Prajapat et al., it derives conditions under which the polynomial (or a suitable transform) satisfies exponential subordination and belongs to Janowski starlike and convex classes, supported by examples and corollaries.

Significance. If the central claims hold with proper normalization, the work extends the use of classical orthogonal polynomials to subordination and Janowski-type classes, potentially bridging applied mathematics/physics contexts with complex analysis. The provision of explicit examples strengthens falsifiability.

major comments (1)

[Main results / Definitions section] The manuscript invokes prior convexity results but does not explicitly construct or verify a normalized function f with f(0)=0 and f'(0)=1. Standard Laguerre polynomials satisfy L_n(0)=1, so the definitions of Janowski starlikeness (subordination to (1+Az)/(1+Bz)) and exponential subordination cannot be applied directly without specifying the transform (e.g., z L_n'(z)/L_n'(0) or similar). This is load-bearing for all main results on starlikeness and convexity.

minor comments (2)

[Abstract] Clarify in the abstract and introduction the precise normalization and the range of parameters n for which the results hold.
[Introduction] Ensure all cited prior results on convexity are referenced with specific theorems or equations from Prajapat et al. to avoid ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We address the major comment below and plan to revise the manuscript accordingly to improve clarity.

read point-by-point responses

Referee: [Main results / Definitions section] The manuscript invokes prior convexity results but does not explicitly construct or verify a normalized function f with f(0)=0 and f'(0)=1. Standard Laguerre polynomials satisfy L_n(0)=1, so the definitions of Janowski starlikeness (subordination to (1+Az)/(1+Bz)) and exponential subordination cannot be applied directly without specifying the transform (e.g., z L_n'(z)/L_n'(0) or similar). This is load-bearing for all main results on starlikeness and convexity.

Authors: We appreciate the referee's point regarding the normalization. The standard Laguerre polynomial L_n(z) indeed has L_n(0)=1, which means it is not directly in the class of normalized analytic functions required for starlikeness and subordination. Our results build upon the convexity conditions from Prajapat et al., where similar considerations apply. To address this, in the revised manuscript, we will explicitly construct and verify the normalized function f(z) with f(0)=0 and f'(0)=1. We will specify the appropriate transform of the Laguerre polynomial (such as one involving its derivative to achieve the normalization) and confirm that the main results on exponential subordination, Janowski starlikeness, and convexity hold for this normalized function. This clarification will be added to the Definitions section and referenced in the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; builds on independent prior results

full rationale

The paper cites prior independent work by J. K. Prajapat et al. for convexity, strong starlikeness, close-to-convexity and strong convexity conditions on the Laguerre polynomial. It extends this foundation to study exponential subordination and Janowski starlikeness/convexity without reducing any central claim to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. Normalization and analyticity assumptions are explicitly drawn from the referenced external results rather than imposed by construction within this manuscript. No ansatz smuggling, uniqueness theorems from the same authors, or renaming of known results as new derivations appear in the abstract or described derivation chain. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard domain assumptions from complex analysis and the analytic properties of the Laguerre polynomial as established in prior literature.

axioms (1)

domain assumption The Laguerre polynomial is analytic and suitably normalized in the unit disk or relevant domain so that subordination and starlikeness are defined.
Required for the geometric properties studied to make sense in complex analysis.

pith-pipeline@v0.9.0 · 5593 in / 1058 out tokens · 56361 ms · 2026-05-22T03:39:30.075570+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 ... Mn,α(z) ∈ Pe ... ψ(r,s,t;z) ... q(z) ≺ ez

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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.
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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

[1]

R. M. Ali, S. R. Mondal, V. Ravichandran, On the Janowski convexity and starlikeness of the confluent hypergeometric function. Bull. Belg. Math. Soc. Simon Stevin 22(2), 227–250 (2015)

work page 2015
[2]

Baricz, N

A. Baricz, N. Ya˘gmur, Geometric properties of some Lommel and Struve functions, Ramanujan J. 42(2), 325–346 (2017)

work page 2017
[3]

Baricz, P

´A. Baricz, P. A. Kup´ an and R. Sz´ asz, The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc.142(2014), no. 6, 2019–2025

work page 2014
[4]

Baricz and S

´A. Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct.21(2010), no. 9-10, 641–653

work page 2010
[5]

Bansal and J

D. Bansal and J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ.61(2016), no. 3, 338–350

work page 2016
[6]

Bansal and R

D. Bansal and R. K. Raina, Certain convexity properties of Hurwitz-Lerch zeta and Mittag-Leffler functions, Hokkaido Math. J.52(2023), no. 2, 315–329

work page 2023
[7]

P. A. H¨ ast¨ o, S. Ponnusamy and M. K. Vuorinen, Starlikeness of the Gaussian hypergeometric functions, Complex Var. Elliptic Equ.55(2010), no. 1-3, 173–184

work page 2010
[8]

Swaminathan, Convexity of the incomplete beta functions, Integral Transforms Spec

A. Swaminathan, Convexity of the incomplete beta functions, Integral Transforms Spec. Funct.18(2007), no. 7-8, 521–528

work page 2007
[9]

S. S. Miller and P. T. Mocanu Univalence of Gaussian and confluent hypergeometric functions.Proc. Amer. Math. Soc.110(1990), no.2, 333–342

work page 1990
[10]

P. T. Mocanu, Some starlike conditions for analytic functions, Rev. Roumaine. Math. Pures. Appl.33(1988), 117–124

work page 1988
[11]

W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157–169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA

work page 1992
[12]

Madaan, A

V. Madaan, A. Kumar and V. Ravichandran, Starlikeness associated with lemniscate of Bernoulli, Filomat 33(2019), no. 7, 1937–1955

work page 2019
[13]

S. R. Mondal ”Subordination involving regular Coulomb Wave functions.” Symmetry 14.5 (2022): 1007

work page 2022
[14]

A. Naz, S. Nagpal and V. Ravichandran, Star-likeness associated with the exponential function, Turkish J. Math. 43 (2019), no. 3, 1353–1371

work page 2019
[15]

Noreen, M

S. Noreen, M. Raza, E. Deniz, S. Kazımo˘glu, On the Janowski class of generalized Struve functions. Afr. Mat. 30(1–2), 23–35 (2019)

work page 2019
[16]

J. K. Prajapat et al., Certain characterization properties of the Laguerre polynomials, J. Anal.32(2024), no. 6, 3139–3154. Anish Kumar Department of Mathematics, Dr. Shyama Prasad Mukherjee University , Ranchi 834008, Jharkhand, India Email address:ak8107690@gmail.com

work page 2024

[1] [1]

R. M. Ali, S. R. Mondal, V. Ravichandran, On the Janowski convexity and starlikeness of the confluent hypergeometric function. Bull. Belg. Math. Soc. Simon Stevin 22(2), 227–250 (2015)

work page 2015

[2] [2]

Baricz, N

A. Baricz, N. Ya˘gmur, Geometric properties of some Lommel and Struve functions, Ramanujan J. 42(2), 325–346 (2017)

work page 2017

[3] [3]

Baricz, P

´A. Baricz, P. A. Kup´ an and R. Sz´ asz, The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc.142(2014), no. 6, 2019–2025

work page 2014

[4] [4]

Baricz and S

´A. Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct.21(2010), no. 9-10, 641–653

work page 2010

[5] [5]

Bansal and J

D. Bansal and J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ.61(2016), no. 3, 338–350

work page 2016

[6] [6]

Bansal and R

D. Bansal and R. K. Raina, Certain convexity properties of Hurwitz-Lerch zeta and Mittag-Leffler functions, Hokkaido Math. J.52(2023), no. 2, 315–329

work page 2023

[7] [7]

P. A. H¨ ast¨ o, S. Ponnusamy and M. K. Vuorinen, Starlikeness of the Gaussian hypergeometric functions, Complex Var. Elliptic Equ.55(2010), no. 1-3, 173–184

work page 2010

[8] [8]

Swaminathan, Convexity of the incomplete beta functions, Integral Transforms Spec

A. Swaminathan, Convexity of the incomplete beta functions, Integral Transforms Spec. Funct.18(2007), no. 7-8, 521–528

work page 2007

[9] [9]

S. S. Miller and P. T. Mocanu Univalence of Gaussian and confluent hypergeometric functions.Proc. Amer. Math. Soc.110(1990), no.2, 333–342

work page 1990

[10] [10]

P. T. Mocanu, Some starlike conditions for analytic functions, Rev. Roumaine. Math. Pures. Appl.33(1988), 117–124

work page 1988

[11] [11]

W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157–169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA

work page 1992

[12] [12]

Madaan, A

V. Madaan, A. Kumar and V. Ravichandran, Starlikeness associated with lemniscate of Bernoulli, Filomat 33(2019), no. 7, 1937–1955

work page 2019

[13] [13]

S. R. Mondal ”Subordination involving regular Coulomb Wave functions.” Symmetry 14.5 (2022): 1007

work page 2022

[14] [14]

A. Naz, S. Nagpal and V. Ravichandran, Star-likeness associated with the exponential function, Turkish J. Math. 43 (2019), no. 3, 1353–1371

work page 2019

[15] [15]

Noreen, M

S. Noreen, M. Raza, E. Deniz, S. Kazımo˘glu, On the Janowski class of generalized Struve functions. Afr. Mat. 30(1–2), 23–35 (2019)

work page 2019

[16] [16]

J. K. Prajapat et al., Certain characterization properties of the Laguerre polynomials, J. Anal.32(2024), no. 6, 3139–3154. Anish Kumar Department of Mathematics, Dr. Shyama Prasad Mukherjee University , Ranchi 834008, Jharkhand, India Email address:ak8107690@gmail.com

work page 2024