The classical Bernoulli numbers $B_j$ are generated by \begin{equation}\label{Bernoulli-No-Generating} \frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty B_{2j}\frac{x^{2j}}{(2j)!}, \quad 0<|x|<2\pi. \end{equation} From the series expansion \begin{equation}\label{log-sin-series-expansion} \ln\sin x=\ln x+\sum_{j=1}^{\infty}(-1)^j\frac{2^{2j-1}}{j}B_{2j}\frac{x^{2j}}{(2j)!}, \quad 0<x<\pi \end{equation} on Page 55 in the handbook
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8
we can derive the Maclaurin power series expansion \begin{equation}\label{F(x)-ser-expan-rew} \ln\frac{\sinh x}{x}=\sum_{j=1}^\infty\frac{2^{2j-1}}{j}B_{2j}\frac{x^{2j}}{(2j)!}, \quad |x|<\pi. \end{equation} For $n\in\mathbb{N}$ and $x\in\mathbb{R}$, define \begin{equation}\label{Tail-F(x)frak-Dfn} \mathfrak{T}_n(x)= \begin{cases}\displaystyle \frac{n+1}{2^{2n+1}}\frac{1}{B_{2n+2}} \frac{(2n+2)!}{x^{2n+2}} \Biggl[\ln\frac{\sinh x}{x}-\sum_{j=1}^n\frac{2^{2j-1}}{j}B_{2j}\frac{x^{2j}}{(2j)!}\Biggr], & x\ne0;\\ 1, & x=0. \end{cases} \end{equation} We call the quantity $\mathfrak{T}_n(x)$ the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function \begin{equation*} \operatorname{sinhc} x= \begin{cases} \dfrac{\sinh x}{x}, & x\ne0;\\ 1, & x=0. \end{cases} \end{equation*}
Problem. Prove that the reciprocal $\dfrac{1}{\mathfrak{T}_n(x)}$ for $n\in\mathbb{N}$ is a convex function of $x\in\mathbb{R}$.
Motivations and Backgrounds. The idea proposing this problem is similar to the one in another problem at https://math.stackexchange.com/q/4946469. As for the concept of normalized tails, please refer to the following three papers.
Related References
- D.-W. Niu and F. Qi, Monotonicity results of ratios between normalized tails of Maclaurin power series expansions of sine and cosine, Mathematics 12 (2024), no. 12, Article 1781, 20 pages; available online at https://doi.org/10.3390/math12121781.
- G.-Z. Zhang, Z.-H. Yang, and F. Qi, On normalized tails of series expansion of generating function of Bernoulli numbers, Proc. Amer. Math. Soc. (2024), in press; available online at https://doi.org/10.1090/proc/16877.
- T. Zhang, Z.-H. Yang, F. Qi, and W.-S. Du, Some properties of normalized tails of Maclaurin power series expansions of sine and cosine, Fractal Fract. 8 (2024), no. 5, Art. 257, 17 pages; available online at https://doi.org/10.3390/fractalfract8050257.
