Elementary magnitude bounds for Jacobian elliptic functions with complex modulus beyond the regime $|m|\le1$, $|z|<\pi/2$

Schiefermayr (2012) proved the only known elementary upper bounds for the magnitudes of the Jacobian elliptic functions with complex parameter: for $|m|\le1$ and $|z|\le R<\pi/2$, $|\operatorname{sn}(z,m)|\le\tan|z|$, $|\operatorname{cn}(z,m)|\le1/\cos|z|$, $|\operatorname{dn}(z,m)|\le1/\cos|z|$, together with sharper intermediate bounds in terms of Jacobian functions with real parameter. The complementary regimes — modulus with $|m|>1$, and arguments beyond $|z|=\pi/2$ — remain open. Because these functions have poles in the argument, no literal global-in-$z$ upper bound exists, and for $|m|>1$ poles can enter the disk $|z|<\pi/2$ (e.g. $m=2$), so the requested extensions must be stated on domains consistent with the pole lattice — the natural candidate being the analyticity disk $|z|<|K(1-m)|$ — with sharpness analysis.

Background

The Jacobian elliptic functions $\operatorname{sn}(z,m)$, $\operatorname{cn}(z,m)$ and $\operatorname{dn}(z,m)$ are the classical doubly periodic generalizations of the sine and cosine: they solve $\frac{d}{dz}\operatorname{sn}=\operatorname{cn}\operatorname{dn}$, $\frac{d}{dz}\operatorname{cn}=-\operatorname{sn}\operatorname{dn}$, $\frac{d}{dz}\operatorname{dn}=-m\,\operatorname{sn}\operatorname{cn}$ with $\operatorname{sn}(0)=0$ and $\operatorname{cn}(0)=\operatorname{dn}(0)=1$, so that $\operatorname{sn}(z,0)=\sin z$, $\operatorname{cn}(z,0)=\cos z$, $\operatorname{dn}(z,0)=1$, while $\operatorname{sn}(z,1)=\tanh z$ and $\operatorname{cn}(z,1)=\operatorname{dn}(z,1)=1/\cosh z$. Here $m=k^2$ is the parameter (the modulus is $k$; the sources use both notations). As functions of the argument $z$ the three functions are meromorphic: $\operatorname{sn}$ has periods $4K(m)$ and $2iK(1-m)$, while $\operatorname{cn}$ and $\operatorname{dn}$ have the period pairs $4K(m),\,2K(m)+2iK(1-m)$ and $2K(m),\,4iK(1-m)$ respectively, where

$K(m)=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-mt^2)}}$

is the complete elliptic integral of the first kind, and each function has simple poles at the points congruent to $iK(1-m)$ modulo its period lattice; for $\operatorname{sn}$ these are exactly the points $2jK(m)+(2\ell+1)iK(1-m)$, $j,\ell\in\mathbb{Z}$. For real $m\in(0,1)$, where $K(m)$ and $K(1-m)$ are real and positive, the pole nearest the origin is $iK(1-m)$, so the largest disk $|z|<\rho$ free of poles has radius $\rho=|K(1-m)|$; for general complex $m$ the pole-free radius is the distance from the origin to the nearest point of the pole lattice, and $|K(1-m)|$ is the natural candidate radius that recurs throughout this record. When $1-m$ lies on the branch cut $[1,\infty)$ of $K$ (that is, for real $m\le0$), $|K(1-m)|$ is understood via either boundary value of $K$ there; the two boundary values have imaginary parts differing only in sign and hence equal absolute values. The pole-free radius tends to $\pi/2$ as $m\to1$ and equals $\pi/2$ exactly in the degenerate case $m=1$, where $\operatorname{sn}=\tanh$ has a pole at $i\pi/2$.

Almost the entire literature on these functions — in particular the literature on inequalities for them (Mercer; Wang–Zhang–Jiang; Neuman) — restricts the parameter to the real interval $m\in(0,1)$. The exceptions are the work of Walker, who proved that the Jacobian functions are analytic in the parameter on the cut plane $\mathbb{C}\setminus[1,\infty)$ and studied the distribution of their zeros as functions of the parameter, and the short note of Schiefermayr that is the source of the present problem. Schiefermayr's theorem, the only known elementary magnitude bounds for these functions with complex parameter, reads as follows. Let $m\in\mathbb{C}$ with $|m|\le1$, let $z\in\mathbb{C}$ with $|z|\le R<\pi/2$, and put $m_1:=1-|m|\in[0,1]$. Then

$|\operatorname{sn}(z,m)|\le\frac{\operatorname{sn}(|z|,m_1)}{\operatorname{cn}(|z|,m_1)}\le\tan|z|,$

$|\operatorname{cn}(z,m)|\le\frac{1}{\operatorname{cn}(|z|,m_1)}\le\frac{1}{\cos|z|},\qquad |\operatorname{dn}(z,m)|\le\frac{\operatorname{dn}(|z|,m_1)}{\operatorname{cn}(|z|,m_1)}\le\frac{1}{\cos|z|}.$

Equality holds throughout at $z=0$ and on the imaginary segment $z=iy$, $|y|\le R$, when $m=1$; the first inequality of each line is additionally sharp for $z=iy$ and real $m=|m|\ge0$. The proof compares the Taylor coefficients of $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ in $z$ (Walker's series, whose coefficients are polynomials with positive integer coefficients) and uses Walker's convergence theorem, which is available precisely for $|m|\le1$ and $|z|<\pi/2$; the two restrictions are inherited from the method.

Both restrictions leave genuine room. For a fixed $m$ with $|m|\le1$, $m\ne1$, the functions are analytic on the strictly larger disk $|z|<|K(1-m)|$ (for example $|K(0.1)|\approx1.612>\pi/2$), so $\tan|z|$, which blows up at $|z|=\pi/2$, is not the natural boundary of the phenomenon. For $|m|>1$ the situation is worse than merely unrestricted: poles can enter the disk $|z|<\pi/2$ — for $m=2$ the function $\operatorname{sn}(z,2)$ has a pole at $iK(-1)$ with $K(-1)=\Gamma(1/4)^2/(4\sqrt{2\pi})\approx1.3110<\pi/2$ — so no bound of the above shape can hold on that disk for all $|m|>1$. The classical reciprocal-modulus transformation

$\operatorname{sn}(z\mid m)=m^{-1/2}\operatorname{sn}(m^{1/2}z\mid 1/m),\quad \operatorname{cn}(z\mid m)=\operatorname{dn}(m^{1/2}z\mid 1/m),\quad \operatorname{dn}(z\mid m)=\operatorname{cn}(m^{1/2}z\mid 1/m)$

exchanges $m$ with $1/m$ and therefore maps the region $|m|>1$ into $|m|<1$ at the cost of shrinking the usable $z$-disk by the factor $|m|^{-1/2}$; the transformation carries a branch choice since $m^{1/2}$ is two-valued.

The only substantial later work on Jacobian functions of complex parameter is by Siegl and Šťampach, who analyzed the normalized function $m\mapsto\operatorname{sn}(K(m)u\mid m)$ for real $u\in(0,1)$ over the whole complex plane: they proved that $|\operatorname{sn}(K(m)u\mid m)|<1$ outside the lens-shaped region $\{w:\ |w-1|<1\}\cap\{|w|>1\}$, that for $u\le1/2$ the global maximum equals $1$ and is attained at $m=1$, and that for $u>1/2$ the global maximum lies in $(1,2)$ and slightly exceeds $1$ (numerically about $1.01038$). Carlson and Todd had earlier shown that for real $m\in(0,1)$ this normalized function is strictly increasing in $m$. These are extremal statements about a normalized function of the parameter; they do not provide bounds of Schiefermayr's type for the raw functions on argument disks, and they say nothing about $|\operatorname{cn}|$ or $|\operatorname{dn}|$ beyond asymptotics near $m\in\{0,1\}$.

By an elementary bound, following the source, we mean an upper bound whose right-hand side is built from elementary functions (trigonometric, hyperbolic, exponential) or from Jacobian functions with real parameter in $[0,1]$, as in $\operatorname{sn}(|z|,m_1)/\operatorname{cn}(|z|,m_1)$ above.

Problem Statement

Schiefermayr's theorem bounds $|\operatorname{sn}(z,m)|$, $|\operatorname{cn}(z,m)|$ and $|\operatorname{dn}(z,m)|$ only for $|m|\le1$ and $|z|<\pi/2$, and nothing comparable is known outside these regimes. The problem is to complete this bound map. Because the three functions have poles in the argument $z$, a literal upper bound valid for all $z$ cannot exist, and because for $|m|>1$ poles can lie inside the disk $|z|<\pi/2$ (for example $m=2$), any correct extension must be stated on domains consistent with the pole lattice; identifying those domains is part of the problem. Concretely:

(A) Parameter regime $|m|>1$. Establish explicit elementary upper bounds for $|\operatorname{sn}(z,m)|$, $|\operatorname{cn}(z,m)|$ and $|\operatorname{dn}(z,m)|$ for complex $m$ with $|m|>1$, valid on an explicitly described $z$-domain contained in the analyticity region — the natural candidate being the pole-free disk $|z|<|K(1-m)|$, or another pole-aware region — with the same two-level structure (a parameter-dependent bound plus a purely elementary one) and an equality/sharpness analysis as in the source theorem. The sub-regime $|m^{1/2}z|<\pi/2$ follows at once by composing the source bounds with the reciprocal-modulus transformation; the substantive content is the remaining range, in particular arguments with $\pi/(2|m|^{1/2})\le|z|<|K(1-m)|$.

(B) Argument regime beyond $|z|=\pi/2$ for $|m|\le1$. Establish comparable elementary bounds valid for $|z|$ beyond $\pi/2$ — for instance on the full pole-free disk $|z|<|K(1-m)|$, which strictly contains $|z|<\pi/2$ unless $m=1$, or pole-aware statements valid for all $z$ outside neighborhoods of the pole lattice, with the bound diverging at the poles.

(C) Sharpness. Determine whether the bounds of the source theorem and of (A)–(B) are sharp in their classes (best possible comparison functions and constants), extending the source's equality analysis, or exhibit tighter elementary bounds.

An answer should state precisely the $(m,z)$-region covered and treat the three functions coherently. The common objective of (A)–(C) is a complete, sharp, elementary magnitude-bound map for the Jacobian elliptic functions over the $(m,z)$ plane, of which Schiefermayr's theorem covers one corner.

References

1. Klaus Schiefermayr, "Inequalities for the Jacobian elliptic functions with complex modulus", Journal of Mathematical Inequalities 6 (2012), no. 1, 91–94. DOI: 10.7153/jmi-06-09. https://doi.org/10.7153/jmi-06-09 (preprint: arXiv:1306.5873, https://arxiv.org/abs/1306.5873)
2. Petr Siegl, František Šťampach, "On extremal properties of Jacobian elliptic functions with complex modulus", Journal of Mathematical Analysis and Applications 442 (2017), no. 2, 627–641. DOI: 10.1016/j.jmaa.2016.05.008. https://doi.org/10.1016/j.jmaa.2016.05.008 (preprint: arXiv:1512.06089, https://arxiv.org/abs/1512.06089)
3. Peter Walker, "The analyticity of Jacobian functions with respect to the parameter k", Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 459 (2003), 2569–2574. DOI: 10.1098/rspa.2003.1157. https://doi.org/10.1098/rspa.2003.1157
4. P. L. Walker, "The distribution of the zeros of Jacobian elliptic functions with respect to the parameter k", Computational Methods and Function Theory 9 (2009), no. 2, 579–591. DOI: 10.1007/bf03321746. https://doi.org/10.1007/bf03321746
5. B. C. Carlson, John Todd, "The Degenerating Behavior of Elliptic Functions", SIAM Journal on Numerical Analysis 20 (1983), no. 6, 1120–1129. DOI: 10.1137/0720081. https://doi.org/10.1137/0720081
6. Edward Neuman, "Product formulas and bounds for Jacobian elliptic functions with applications", Integral Transforms and Special Functions 23 (2012), no. 5, 347–354. DOI: 10.1080/10652469.2011.590807. https://doi.org/10.1080/10652469.2011.590807
7. Gendi Wang, Xiaohui Zhang, Yueping Jiang, "Hölder concavity and inequalities for Jacobian elliptic functions", Integral Transforms and Special Functions 23 (2012), no. 5, 337–345. DOI: 10.1080/10652469.2011.590482. https://doi.org/10.1080/10652469.2011.590482
8. A. McD. Mercer, "Shafer–Fink type inequalities for the elliptic function sn(u|k)", Journal of Mathematical Inequalities 2 (2008), no. 3, 377–382. DOI: 10.7153/jmi-02-33. https://doi.org/10.7153/jmi-02-33
9. Gendi Wang, "The inverse hyperbolic tangent function and Jacobian sine function", Journal of Mathematical Analysis and Applications 448 (2017), no. 1, 498–505. DOI: 10.1016/j.jmaa.2016.11.002. https://doi.org/10.1016/j.jmaa.2016.11.002
10. R. Cartas-Fuentevilla, "Analyticity of the elliptic functions on the hyperbolic complex plane: exact and duality symmetries for non-linear pendulum", Physics Letters A 590 (2026), 131853. DOI: 10.1016/j.physleta.2026.131853. https://doi.org/10.1016/j.physleta.2026.131853
11. M. Lopez-Fernandez, S. Sauter, "Fast and stable contour integration for high order divided differences via elliptic functions", Mathematics of Computation 84 (2015), no. 293, 1291–1315. DOI: 10.1090/S0025-5718-2014-02890-1. https://doi.org/10.1090/S0025-5718-2014-02890-1
12. Takashi Takebe, "Complex Jacobian Elliptic Functions", in: Elliptic Integrals and Elliptic Functions, Moscow Lectures 8, Springer, 2023, pp. 289–311. DOI: 10.1007/978-3-031-30265-7_18. https://doi.org/10.1007/978-3-031-30265-7_18
13. Avinash Khare, Uday Sukhatme, "Connecting Jacobi elliptic functions with different modulus parameters", Pramana 63 (2004), 921–936. DOI: 10.1007/bf02704331. https://doi.org/10.1007/bf02704331
14. NIST Digital Library of Mathematical Functions, Release 1.2.8, §19.7 (Connection Formulas) and §22.4 (Periods, Poles, and Zeros). https://dlmf.nist.gov/19.7, https://dlmf.nist.gov/22.4