A look at generalized trigonometric functions as functions of their two parameters and further new properties

Dmitrii Karp; Elena Prilepkina

REVIEW 3 major objections 6 minor 24 references

A look at generalized trigonometric functions as functions of their two parameters and further new properties

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The two-parameter generalized sine is concave in each parameter at fixed argument, while the hyperbolic sine is log-convex in p.

desk verdict Real new content with a load-bearing proof error in Theorem 4; worth reviewing, but the log-convexity claim is unproven as written. read the letter →

arxiv 2411.13442 v1 pith:2PHOAKS3 submitted 2024-11-20 math.CA

Dmitrii Karp , Elena Prilepkina This is my paper · ORCID

classification math.CA MSC 33E3033E2026D0733C05

keywords generalizedtrigonometricfunctionshyperbolic(pq)-Laplacianlog-convexitylog-concavityintegralrepresentationhypergeometricfunctionparametermonotonicity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the two-parameter generalized sine, cosine, tangent, and their hyperbolic counterparts—functions that invert integrals such as $\arcsin_{p,q}(x)=\int_0^x (1-t^q)^{-1/p}\,dt$—as functions of the parameters $p$ and $q$ with the argument held fixed. Its central results are new parameter-dependent inequalities: for fixed $y\in[0,1]$ and $q>1$, the map $p\mapsto \sin_{p,q}(y)$ is concave on $(1,\infty)$, and the same holds for $q\mapsto \sin_{p,q}(y)$; for fixed $y$ and $q>1$, the map $p\mapsto \sinh_{p,q}(y)$ is logarithmically convex on $(1,\infty)$. The paper also proves logarithmic concavity for $p\mapsto \operatorname{tamh}_{p',p}(y)$ and gives necessary and sufficient conditions for concavity or log-concavity in the cases where only monotonicity is established. These results matter because the two-parameter functions arise in eigenvalue problems for the $(p,q)$-Laplacian and in explicit solutions of nonlinear Schr\"odinger-type equations, where varying $p$ and $q$ changes the shape of the solutions. The paper also records two new hypergeometric representations for the inverse generalized cosine and hyperbolic cosine, and evaluates four integrals of inverse and four of direct generalized trigonometric and hyperbolic functions in terms of generalized hypergeometric functions.

What carries the argument

The machinery is the integral representation of each inverse function as $y=f(x,p)=\int_0^x \varphi(t,p)\,dt$ with a positive kernel $\varphi$, combined with the inverse-function differentiation formulas (Lemma 2) that express $\partial_p g$ and $\partial_p^2 g$ in terms of $\varphi$ and its $p$-derivatives. For each target function, the desired convexity or log-convexity inequality becomes the statement that a quadratic form in $z=\int_0^x \varphi'_p(t,p)\,dt$ has a fixed sign; the proofs then control these forms using the monotone L'H\^opital-type rule of Lemma 4. In the concrete cases the kernels are explicit: for $\sin_{p,q}$ the kernel is $\varphi(t,p)=(1-t^q)^{-1/p}$, and for $\operatorname{tamh}_{p',p}$ it is $\varphi(x,p)=(1-x^p)^{-2/p}$.

What would settle it

Take $q=2$, so $\pi_q=\pi$, and choose $y=2.5$, which lies in $(\pi/2,\pi)$. Since $\hat\pi_{p,q}/2\downarrow \pi/2$ as $p\downarrow 1$, for $p$ sufficiently close to $1$ the value $\sinh_{p,q}(2.5)$ is undefined, directly falsifying the stated domain $[0,\pi_q]$ of Theorem 4; the theorem can only hold on $[0,\pi_q/2]$. A separate direct check of the log-convexity inequality at a few pairs $p_1,p_2$ and $\lambda$ using high-precision quadrature of the defining integrals would test the inequality itself on the corrected domain.

Watch

Extended reading notes

Core claim

The paper's central claim is that the generalized sine and hyperbolic sine, viewed as functions of their parameters at a fixed argument, obey parameter versions of concavity and log-convexity. Theorem 3 states that for each fixed $y\in[0,1]$ and $q>1$, the function $p\mapsto \sin_{p,q}(y)$ is concave on $(1,\infty)$, and symmetrically $q\mapsto \sin_{p,q}(y)$ is concave for fixed $p>1$. Theorem 4 states that for fixed $y\in[0,\pi_q]$ and $q>1$, the function $p\mapsto \sinh_{p,q}(y)$ is log-convex on $(1,\infty)$, where $\pi_q=2\pi/(q\sin(\pi/q))$ is the limiting half-period as $p\downarrow 1$. Theorem 2 adds that $p\mapsto \operatorname{tamh}_{p',p}(y)$, the inverse of $\int_0^x (1-t^p)^{-2/p}\,dt$, is logarithmically concave. The proof route is: write the inverse function as an integral, differentiate with respect to the parameter using the inverse-function formulas of Lemma 2, reduce the target inequality to the nonnegativity or nonpositivity of a quadratic form in an auxiliary integral, and verify that form by monotone L'H\^opital-type ratio comparisons.

Load-bearing premise

The proofs assume the functions are defined and smoothly invertible on the full parameter and argument intervals they state; for the hyperbolic sine, the common domain over all $p>1$ is actually $[0,\pi_q/2]$, not the full $[0,\pi_q]$ used in Theorem 4.

Editorial extensions

If this is right

Concavity of $p\mapsto \sin_{p,q}(y)$ yields parameter-interpolation inequalities such as $\sin_{\lambda p_1+(1-\lambda)p_2,q}(y)\ge \lambda\sin_{p_1,q}(y)+(1-\lambda)\sin_{p_2,q}(y)$ for $\lambda\in[0,1]$, and similarly for the second parameter.
Log-convexity of $p\mapsto \sinh_{p,q}(y)$ yields the geometric-mean bound $\sinh_{\lambda p_1+(1-\lambda)p_2,q}(y)\le [\sinh_{p_1,q}(y)]^\lambda[\sinh_{p_2,q}(y)]^{1-\lambda}$.
The necessary and sufficient conditions in Corollaries 1--4 give explicit sign criteria that determine when strict concavity or strict log-convexity holds in parameters, not just convexity.
The new hypergeometric representations for $\arccos_{p,q}$ and $\arccosh_{p,q}$ provide computable closed forms for these inverse functions, which can make numerical evaluation of the direct functions more efficient.
The four integral evaluations give closed forms for weighted averages of the inverse generalized trigonometric and hyperbolic functions, extending the single-parameter evaluations previously known.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The stated domain in Theorem 4 is wider than the common domain of definition: for $y\in(\pi_q/2,\pi_q)$, the value $\sinh_{p,q}(y)$ is undefined for $p$ sufficiently close to $1$, so the theorem as stated needs the restriction $y\in[0,\pi_q/2]$; the log-convexity claim itself should survive on the corrected interval.
The same inverse-function quadratic-form machinery could plausibly be applied to kernels of the form $(1\pm t^q)^{-a}$ for other exponents $a$, yielding parameter convexity for a wider family of generalized inverse functions; the paper does not pursue this extension.
The explicit integral evaluations provide closed forms for weighted moments of the inverse functions over the $q$-circle, which could be used in numerical quadrature for $(p,q)$-Laplacian eigenvalue problems or in error estimates for basis approximations by $\sin_{p,q}$ functions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Short take: this paper is worth reviewing, but the central log-convexity result, Theorem 4, is not proven as written. The stress-test note is right: the derivative of f1 - f2 in (35) is miscomputed. Differentiating f1 = q x^q/[p(1+x^q)] μ + x log(1+x^q) + 2px/(1+x^q)^{1/p} and f2 = μ leaves extra terms: (q x^q/[p(1+x^q)] - 1) log(1+x^q)/(1+x^q)^{1/p} plus log(1+x^q) + q x^q/(1+x^q) - q x^q/(1+x^q)^{1+1/p}. They don't cancel. So the positivity argument collapses; the log-convexity claim needs a new proof or a modified statement. The reader also caught the domain error: Theorem 4 and Corollary 3 claim y ∈ [0, π_q], but the common domain over p > 1 is [0, π_q/2], since sinh_{p,q} lives on [0, \hatπ_{p,q}/2] and \hatπ_{p,q} ↓ π_q as p ↓ 1. Both issues are real.

What is genuinely new and works: Lemma 1's hypergeometric representations for arccos_{p,q} and arccosh_{p,q} look correct (the p=q=2 case checks out). Theorem 3, on concavity of p→sin_{p,q} and q→sin_{p,q} on [0,1], has an explicit, checkable proof; the monotone L'Hôpital step is sound. The four integral evaluations in Theorem 5 are ambitious and the derivations are written out in enough detail to verify. The paper is also honest: the abstract admits the results are incomplete, and the citations match the claims.

Softer points, in proportion: Theorem 2 states log-concavity of p→tamh_{p',p} but omits the y-range; that's a presentation fix. The non-monotonicity claims in Theorem 1 rest on numerical evidence and a figure, with a promise of rigorous guaranteed-precision checks that isn't delivered; a referee should ask for the actual computations.

Bottom line: there is real content here, but the paper currently has a load-bearing gap in its flagship theorem. I'd send it to peer review, not desk reject: the flaws are identifiable and fixable, and the surrounding results are solid. Reading group: maybe, if you want a concrete example of why derivative identities in parameter-monotonicity arguments need careful checking.

Referee Report

3 major / 6 minor

Summary. The paper studies monotonicity and convexity/log-concavity in the parameters p and q of the two-parameter generalized trigonometric and hyperbolic functions. It states two new hypergeometric representations for the inverse generalized cosine and hyperbolic cosine (Lemma 1), derives necessary and sufficient conditions for concavity/log-concavity in parameters (Corollaries 1–4), proves concavity of sin_{p,q}(y) in each parameter for y in [0,1] (Theorem 3), claims log-convexity of sinh_{p,q}(y) in p for y in [0,pi_q] (Theorem 4) and log-concavity of tamh_{p',p}(y) in p (Theorem 2), and evaluates four parameter-dependent integrals in terms of generalized hypergeometric functions (Theorem 5 and Corollary 5). The paper is written in the conventional style, with explicit formulas and references to standard identities.

Significance. If correct, the convexity results would provide new parameter-dependent inequalities for these functions, a direction that the paper correctly identifies as underdeveloped. The hypergeometric representations in Lemma 1 and the integral evaluations in Section 3 are of independent interest and appear to be derived by standard, parameter-free methods from the integral definitions, with no fitted parameters or target-dependent assumptions. However, the proofs of Theorem 4 and Theorem 2 contain serious technical gaps and domain errors, so the central convexity claims for sinh_{p,q} and tamh_{p',p} are not established as written.

major comments (3)

[Theorem 4, proof around Eq. (35)] The derivative f_1'(x)-f_2'(x) is computed incorrectly. Writing μ(x)=∫_0^x log(1+t^q)/(1+t^q)^{1/p} dt and a(x)=q x^q/[p(1+x^q)], the functions in (35) satisfy f_1-f_2=(a-1)μ+x log(1+x^q)+2px/(1+x^q)^{1/p}. Differentiating gives (f_1-f_2)' = a'μ + (a-1)μ' + log(1+x^q) + qx^q/(1+x^q) + 2p/(1+x^q)^{1/p} - 2q x^q/(1+x^q)^{1+1/p}. The displayed formula in the proof retains only a'μ and replaces the remaining terms by [2p(1+x^q)-q x^q]/(1+x^q)^{1+1/p}, omitting (a-1)μ' + log(1+x^q) + qx^q/(1+x^q) - q x^q/(1+x^q)^{1+1/p}. These omitted terms do not cancel, so the conclusion f_1'>f_2' is not justified. The proof of log-convexity of p→sinh_{p,q}(y) therefore fails.
[Theorem 4 and Corollary 3] The common y-domain over p>1 is misstated. Since sinh_{p,q}(y) is defined on [0, π̂_{p,q}/2], the intersection ∩_{p>1}[0, π̂_{p,q}/2] is [0, π_q/2], not [0, π_q]. For y in (π_q/2, π_q], sinh_{p,q}(y) is undefined for p sufficiently close to 1 because π̂_{p,q}/2 converges to π_q/2 as p↓1. Both Theorem 4 and Corollary 3 must be restricted to y∈[0, π_q/2], and the proof must be reworked for that domain.
[Theorem 2] The proof does not support the stated logarithmic concavity. Lemma 3 requires checking D_2<0 for log-concavity, but the proof verifies D_1<0, which is the sufficient condition for plain concavity, not for log-concavity. Moreover, the statement omits the y-range: for a fixed y, the function p→tamh_{p',p}(y) is not defined for all p>1 unless y belongs to an appropriate intersection of domains, and that intersection needs to be specified. As written, Theorem 2 is unproven.

minor comments (6)

[Theorem 1 proof] The line 'inf_{p,q>1} π_{p,q} = inf_{p,q>1} π̂_{p,q} = 1' is incorrect; the infimum of the full periods is 2, and the relevant statement is that the half-periods are bounded below by 1. This appears to be a typo but should be corrected.
[Theorem 1] The claims that certain functions are 'generally speaking, not monotonic' rely on numerical examples and a promise of guaranteed-precision computation; no such rigorous computations are included. Either provide the computations or reformulate these claims as observations supported by numerical evidence.
[Theorem 2 proof] The text alternates between 'logarithmic convexity' and 'logarithmic concavity,' and the theorem statement says 'logarithmically concave' while the proof begins with 'To establish logarithmic convexity.' Please make the terminology consistent and aligned with the statement.
[Equation (18)] The notation 'sinr,q(y)' should read 'sin_{r,q}(y)' or otherwise be clarified, and the condition for the validity of the identity (which requires the parameter r>1) should be stated explicitly.
[Corollary 5] The parameter restrictions inherited from Theorem 5 are not repeated in the statements of (41)–(44); adding a sentence specifying the ranges of α and β would improve readability.
[Abstract and Introduction] There are several typos: 'managed the prove' should be 'managed to prove' in the Abstract; 'the forgoing investigation' should be 'the following investigation' or 'the forthcoming investigation' in Section 2; and 'L'Hôpital' should be spelled consistently. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the parameter convexity theorems are derived from integral representations and standard calculus and hypergeometric tools; the self-citations are to elementary lemmas and are not load-bearing.

full rationale

The derivation chain is self-contained in the relevant sense. The monotonicity and (log-)convexity claims in Theorems 1-4 are obtained by differentiating the defining integral representations (1), (6), (16)-(20) and applying Lemma 2 or Lemma 3, which are standard inverse-function differentiation identities. Although Lemma 2 and Lemma 4 are cited to the authors' earlier paper [10] for details, they are general parameter-free calculus facts whose assumptions do not include the target inequalities, so the self-citation is not load-bearing. Theorem 3 uses Corollaries 1-2, which restate the sign conditions derived from Lemma 3; neither the corollaries nor the theorem assume the concavity they prove. Theorem 4 is proved from Corollary 3's equivalent inequality, with no fitted parameter or target-dependent input. The hypergeometric representations in Lemma 1 and the integral evaluations in Theorem 5 are proven from (5), (10) and standard Euler and connection formulas; they are not used as disguised assumptions. There is no fitted-input-called-prediction pattern and no uniqueness claim imported from the authors' prior work. The reader's noted issues are correctness concerns, not circularity: the domain declaration [0, pi_q] in Theorem 4 and Corollary 3 should be [0, pi_q/2] for the common domain over p > 1, and the displayed derivative in the proof of Theorem 4 appears algebraically incomplete. Even if those defects invalidate the proof as written, the argument does not reduce to its conclusion by construction, so the circularity score remains at the level of minor, non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims are pure mathematics; no free parameters are fitted to data and no new entities are postulated. The paper builds on standard hypergeometric identities and the authors' earlier inverse-function lemma, which is a general tool rather than a restatement of the results.

assumptions (4)

standard math Inverse-function second-derivative formulas, Lemma 2, equations (21)-(23).
Used in all parameter-convexity proofs; proven in the authors' earlier paper [10], not re-proven here.
standard math Gauss hypergeometric connection formula, [1], 2.3.13.
Used in Lemma 1 and Theorem 5 to transform hypergeometric functions; this is a standard identity.
standard math Monotone L'Hopital rule, Lemma 4 from Pinelis [20].
Used in Theorems 3 and 4 to compare integral ratios; this is a standard lemma with proof cited.
standard math Differentiation under the integral sign, Feynman's trick, in Theorem 5.
Used to evaluate the four integrals; convergence conditions are stated but full justification of interchange is not shown.

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Pith. "Pith review of A look at generalized trigonometric functions as functions of their two parameters and further new properties." pith.science (2026). https://pith.science/paper/2PHOAKS3

@misc{pith2026241113442,
  author       = {Pith},
  title        = {Pith review of: A look at generalized trigonometric functions as functions of their two parameters and further new properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2PHOAKS3}},
  note         = {Machine review of arXiv:2411.13442}
}

read the original abstract

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Special Functions; Cambridge University Press: Cambridge, UK, 1999

Andrews, G.E.; Askey, R.; Roy, R. Special Functions; Cambridge University Press: Cambridge, UK, 1999

work page 1999

[2] [2]

Convexity properties of generalized trigonometric and hyperbolic functions

Baricz, A.; Bhayo, B.A.; Klen, R. Convexity properties of generalized trigonometric and hyperbolic functions. Aequat. Math. 2015, 89, 473–484

work page 2015

[3] [3]

Tur´ an type inequalities for generalized inverse trigonometric functions

Baricz, A.; Bhayo, B.A.; Vuorinen, M.M. Tur´ an type inequalities for generalized inverse trigonometric functions. Filomat 2015, 29, 303–313

work page 2015

[4] [4]

On generalized trigonometric functions with two parame- ters

Bhayo, B.A.; Vuorinen, M. On generalized trigonometric functions with two parame- ters. J. Approx. Theory. 2012, 164, 1415–1426

work page 2012

[5] [5]

Basis properties of the p, q-sine functions

Boulton, L.; Lord, G.J. Basis properties of the p, q-sine functions. Proc. R. Soc. A 2015, 471, 20140642

work page 2015

[6] [6]

On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian

Dr´ abek, P.; Man´ asevich, R. On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian. Differ. Integral Equ. 1999, 12, 773–788

work page 1999

[7] [7]

Activation functions in deep learning: A comprehensive survey and benchmark

Dubey, S.R.; Singh S.K.; Chaudhuri B.B. Activation functions in deep learning: A comprehensive survey and benchmark. Neurocomputing 2022, 503, 92–108

work page 2022

[8] [8]

Properties of generalized trigonometric functions

Edmunds, D.E.; Gurka, P.; Lang, J. Properties of generalized trigonometric functions. Approx. Theory. 2012, 164, 47–56

work page 2012