Faulhuber and Steinerberger conjectured that the logarithmic derivative of has the property that is strictly decreasing and strictly convex. In this small note, we prove this conjecture.
Faulhuber et Steinerberger ont conjecturé que la dérivée logarithmique de possède la propriété selon laquelle est strictement décroissant et strictement convexe. Dans cette courte note, nous démontrons cette conjecture.
Accepted:
Published online:
Anne-Maria Ernvall-Hytönen 1; Esa V. Vesalainen 1
@article{CRMATH_2018__356_5_457_0, author = {Anne-Maria Ernvall-Hyt\"onen and Esa V. Vesalainen}, title = {On a conjecture of {Faulhuber} and {Steinerberger} on the logarithmic derivative of \protect\emph{\ensuremath{\vartheta}}\protect\textsubscript{4}}, journal = {Comptes Rendus. Math\'ematique}, pages = {457--462}, publisher = {Elsevier}, volume = {356}, number = {5}, year = {2018}, doi = {10.1016/j.crma.2018.04.006}, language = {en}, }
TY - JOUR AU - Anne-Maria Ernvall-Hytönen AU - Esa V. Vesalainen TI - On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ϑ4 JO - Comptes Rendus. Mathématique PY - 2018 SP - 457 EP - 462 VL - 356 IS - 5 PB - Elsevier DO - 10.1016/j.crma.2018.04.006 LA - en ID - CRMATH_2018__356_5_457_0 ER -
Anne-Maria Ernvall-Hytönen; Esa V. Vesalainen. On a conjecture of Faulhuber and Steinerberger on the logarithmic derivative of ϑ4. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 457-462. doi : 10.1016/j.crma.2018.04.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.006/
[1] On the log-concavity of a Jacobi theta function, Math. Comput., Volume 82 (2013), pp. 2265-2272
[2] Convexity of quotients of theta functions, J. Math. Anal. Appl., Volume 386 (2012), pp. 319-331
[3] On the secrecy gain of ℓ-modular lattices | arXiv
[4] Extremal Bounds of Gaussian Gabor Frames and Properties of Jacobi's Theta Functions, University of Vienna, 2016 (Doctoral dissertation)
[5] Properties of logarithmic derivatives of Jacobi's theta functions on a logarithmic scale | arXiv
[6] Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions, J. Math. Anal. Appl., Volume 445 (2017), pp. 407-422
[7] Minimal theta functions, Glasg. Math. J., Volume 30 (1988), pp. 75-85
[8] Some new properties of Jacobi theta functions, J. Comput. Appl. Math., Volume 178 (2005), pp. 419-424
[9] Harmonic measure of radial segments and symmetrization, Sb. Math., Volume 189 (1998), pp. 1701-1718
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☆ This work was supported by the Academy of Finland project 303820, and E. V. V. was supported by the Magnus Ehrnrooth Foundation.
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