lajos/Estimating the Q-array

Introduction

Let \(p\ge 2\) be an integer. If \({\mathcal{T}}_p\) denotes the Taylor polynomial of degree \(p\) of the exponential function around \(0\), then we let \[ {\mathcal{S}}_p:=\{ z\in\mathbb{C} : |{\mathcal{T}}_p(z)|\le 1 \}. \]

Some properties of the set \({\mathcal{S}}_p\) will be/are given here, but only numerically or asymptotically, together with an integral representation. Unfortunately, no simple analytical description/bounds are available to us. In the first part of this document, we are going to use only that \(0\in {\mathcal{S}}_p\).

By introducing the notation \(Q_{p,m,\ell}\), we have seen earlier that \[ \left|Q_{p,m,\ell}(z)\right|=\left|\alpha_m\left(1+\frac{z}{m}\right)^{\ell-1}\right|\equiv \frac{m^{p-\ell} }{(p-m)! (m-1)!}|z+m|^{\ell-1}, \] for \(z\in\mathbb{C}\) with \((p,m,\ell)\in\mathbb{N}^3\), \(p\ge 2\), \(2\le m\le p\) and \(2\le\ell\le m\).

The ideal goal would be to have information on

\[ \min_{z\in {\mathcal{S}}_p} \left|Q_{p,m,\ell}(z)\right|\quad\mathrm{and}\quad \max_{z\in {\mathcal{S}}_p} \left|Q_{p,m,\ell}(z)\right| \] for any admissible triple \((p,m,\ell)\).

First, for any \(p\ge 2\), we are going to give a lower bound on \[ \max_{2\le m\le p, 2\le\ell\le m} \left|Q_{p,m,\ell}(0)\right|, \] then we will prove an interesting statement about the \(p\to \infty\) asymptotic location of the index \(m\) within the interval \([2,p]\) at which the above maximum occurs. Finally, we show some numerical results about \[ \max_{2\le m\le p, 2\le\ell\le m}\max_{z\in {\mathcal{S}}_p, \mathrm{re}(z)\le 0} \left|Q_{p,m,\ell}(z)\right| \] as \(p\to \infty\).

A lower bound on the maximum at \(z=0\)

Let us consider \(\left|Q_{p,m,\ell}(0)\right|=\frac{m^{p-1} }{(p-m)! (m-1)!}\). If \(p=2\), then \(m=2\) and the maximal value is 2. If \(p=3\), then \(2\le m\le 3\) and the maximal value is \(\frac{9}{2}\). So we can suppose \(p\ge 4\). We set \[ F_p(m):=\frac{m^{p-1} }{(p-m)! (m-1)!}. \]

Proposition. For any \(n\in\mathbb{N}^+\) we have \[ \left(\frac{n}{e}\right)^n \sqrt{2\pi n} < n! \le e \left(\frac{n}{e}\right)^n \sqrt{n}. \]

Remark. We will need only the upper estimate for the factorial.

First suppose that \(p\) is divisible by 4. Then evaluating \(F_p\) at \(m=3p/4\) and using the upper estimate of the above proposition to estimate both factorials in the denominator, we get \[ F_p(3p/4)\ge 4\cdot 3^{p-1} e^{p-3} p^{\frac{3 (p-2)}{4}} (3 p-4)^{\frac{1}{2}-\frac{3 p}{4}}. \] We claim that for any \(p\ge 4\), the right-hand side is decreased further, if we use the following simpler expression: \[ \frac{\sqrt{3}}{2 e^2}\cdot \frac{\left( \sqrt[4]{3} e \right)^p}{p}\approx 0.117204\cdot \frac{ 3.57746 ^p}{p}. \] (This last claim can be shown by taking the ratio of the two expressions, then analyzing the appropriate limits and the sign of successive derivatives, until the transcendental inequality is reduced to a polynomial one.)

Finally, if \(p\) is of the form \(p=4k_1-1\) (with a suitable integer \(k_1\)), then we choose \(m:=3(p+1)/4\); if \(p=4k_2-2\), then \(m:=3(p+2)/4\); finally, if \(p=4k_3+1\), then \(m:=3(p-1)/4\) is chosen. In all these cases, we verify in a similar manner that \[ \max_{2\le m\le p} F_p(m)\ge \frac{\sqrt{3}}{2 e^2}\cdot \frac{\left( \sqrt[4]{3} e \right)^p}{p} \] for any \(p\ge 4\). Note that all the above estimates are valid for all \(p\ge 4\) values (and not only asymptotically or approximately).

Remark. The choice of \(m\approx \frac{3p}{4}\) will be interpreted by the next subsection.

The \(p\to\infty\) asymptotic behavior of the maximum at \(z=0\)

Since mathwiki rendering at this point is now toooo slow, I open a new document here.

The \(p\to\infty\) asymptotic behavior of the maximum on the whole region in the left half-plane, based on numerical information

Can we say something, at least numerically, about \[ \max_{2\le m\le p, 2\le\ell\le m}\max_{z\in {\mathcal{S}}_p, \mathrm{re}(z)\le 0} \left|Q_{p,m,\ell}(z)\right|=\max_{2\le m\le p, 2\le\ell\le m}\max_{z\in {\mathcal{S}}_p, \mathrm{re}(z)\le 0} \frac{m^{p-\ell} }{(p-m)! (m-1)!}|z+m|^{\ell-1}? \] By accepting the approximate semi-disk shape of \({\mathcal{S}}_p\) and considering \(p\) to be large enough, the largest distance between the point \(-m\) in the complex plane and the set \({\mathcal{S}}_p\) is attained at the “topmost” point of the vertical slice \({\mathcal{S}}_p\cap (i\mathbb{R})\). (Due to symmetry, it is enough to consider the upper half-plane.) But as mentioned already on this page \[ \max \{ y>0 : |{\mathcal{T}}_p(i y)|\le 1 \}\approx \frac{p}{e}+\frac{\log (p)}{2 e}+\frac{3}{5}+\frac{1+\log (2 \pi )}{2 e}\le \frac{p}{e}+\frac{\log (p)}{2 e}+\frac{9}{8}. \] We remark in advance that the structure of the final estimate will not change if we use the estimate \[ \frac{p}{e}+\frac{\log (p)}{2 e}+\frac{9}{8}\le \frac{2p}{e}, \] valid for \(p\ge 4\).

First we observe the following. If \(z=i y\) with \(y>0\), and \(p\ge 2\) and \(2\le m\le p\) are fixed, then for \(2\le \ell \le m\) the expression \[ \left|Q_{p,m,\ell}(i y)\right|= \frac{m^{p-\ell} }{(p-m)! (m-1)!}\left(m^2+y^2\right)^{\frac{\ell-1}{2}} \] attains its maximum, if \(\ell=m\). Indeed, the above expression attains its maximum precisely when \[ m^{-2\ell+2}\left(m^2+y^2\right)^{\ell-1}=\left(\frac{m^2+y^2}{m^2}\right)^{\ell-1}=\left(1+\frac{y^2}{m^2}\right)^{\ell-1} \] is maximal, which happens if \(\ell=m\).

So we would like to see the asymptotic behavior of \[ \max_{2\le m\le p} F_p(m) \] as \(p\to +\infty\), where \[ F_p(m):=\frac{m^{p-m} }{(p-m)! (m-1)!}\left(m^2+\left(\frac{2p}{e}\right)^2\right)^{\frac{m-1}{2}}. \] We consider \(m\) again as a continuous variable (\(m\in [2,p]\)) and proceed as before, but now only numerically verifying most of the steps. Numerical tests indicate that \[ m\mapsto G_p(m):=\frac{F_p(m+1)}{F_p(m)} \] is a strictly decreasing continuous function, \(G_p(2)>1\) (if \(p\ge 3\)), and (after cancelling the factorials) \(G_p(p)=0\), so there is a unique \(m_c^*(p)\in (2,p)\) such that \(G_p(m_c^*(p))=1\), corresponding to the unique maximum of the unimodal function \(m\mapsto F_p(m)\). Numerical tests show that \[ L:=\lim_{p\to+\infty}\frac{m_c^*(p)}{p}\approx 0.76138057392. \] (The fraction seems to be monotone increasing w.r.t. \(p\). We have used \(p\) values as large as \(p\approx 10^{12}\) to get the first 10 digits of the limit.) Now by computing the (first term of the) asymptotic series (as \(p\to +\infty\)) of \(F_p(L p)\), we get that \[ \max_{2\le m\le p} F_p(m)\approx 0.268509\cdot \frac{4.60862^p}{p}. \] (We have kept \(p\) in the denominator, because this way we get a better estimate for smaller \(p\) values.)