Sharp Bounds for the Arc Lemniscate Sine Function

TL;DR

This paper refines bounds for the arc lemniscate sine function by replacing a constant factor with a sharper, optimal value derived using monotone l'Hopital's rule, improving previous inequalities.

Contribution

It introduces a tighter bound for arcsl(x) in terms of the Lerch zeta function, establishing the best possible constant factor.

Findings

The constant factor 1/4 is replaced by approximately 0.12836.

The new bound is proven to be optimal.

The method uses monotone l'Hopital's rule for the proof.

Abstract

The arc lemniscate sine function is given by $$ \mbox{arcsl}(x)=\int_0^x \frac{1}{\sqrt{1-t^4}}dt. $$ In 2017, Mahmoud and Agarwal presented bounds for $\mbox{arcsl}$ in terms of the Lerch zeta function $$ \Phi(z,s,a)=\sum_{k=0}^\infty \frac {z^k}{(k+a)^s}. $$ They proved $$ \frac{1}{8} \, x \, \Phi(x^4, 3/2, 1/4) < \mbox{arcsl}(x)< \frac{1}{4} \, x \, \Phi(x^4,3/2,1/4)\qquad{(0<x<1)}. $$ We %use the monotone form of l'Hopital's rule to show that the factor $1/4$ can be replaced by $\mbox{arcsl}(1)/\Phi(1,3/2,1/4)=0.12836...$. This constant is best possible.

Equations

$$\text{mbox}arcsl(x)={\int }_0^x\frac{1}{\sqrt{1-t^4}}dt.$$

$$\Phi (z,s,a)=\sum _{k=0}^{\infty }\frac{z^k}{(k+a{)}^s}.$$

$$\frac{1}{8}x\Phi (x^4,3/2,1/4)<\text{mbox}arcsl(x)<\frac{1}{4}x\Phi (x^4,3/2,1/4)(0<x<1).$$

$$L=4\sqrt{2}c\text{mbox}arcsl(1),$$

$$\text{mbox}arcsl(x)={\int }_0^x\frac{1}{\sqrt{1-t^4}}dt(-1\le x\le 1).$$

$$\Phi (z,s,a)=\sum _{k=0}^{\infty }\frac{z^k}{(k+a{)}^s}.$$

$$\frac{1}{8}x\Phi (x^4,3/2,1/4)<\text{mbox}arcsl(x)<\frac{1}{4}x\Phi (x^4,3/2,1/4)(0<x<1).$$

$$\alpha x\Phi (x^4,3/2,1/4)<\text{mbox}arcsl(x)<\beta x\Phi (x^4,3/2,1/4)$$

$$\alpha =\frac{1}{8}\text{mbox}and\beta =\frac{\text{mbox}arcsl(1)}{\Phi (1,3/2,1/4)}=\mathrm{0.12836....}$$

$$B(x,y)={\int }_0^1{t}^{x-1}(1-t{)}^{y-1}dt\text{mbox}and\zeta (s,a)=\Phi (1,s,a)=\sum _{k=0}^{\infty }\frac{1}{(k+a{)}^s}.$$

$$\text{mbox}arcsl(1)={\int }_0^1\frac{1}{\sqrt{1-t^4}}dt=\frac{1}{4}{\int }_0^1{s}^{-3/4}(1-s{)}^{-1/2}ds=\frac{1}{4}B(1/4,1/2).$$

$$\beta =\frac{B(1/4,1/2)}{4\zeta (3/2,1/4)}=\frac{\Gamma (1/4{)}^2}{4\sqrt{2\pi }\zeta (3/2,1/4)},$$

$$x\mapsto \frac{u(x)-u(a)}{v(x)-v(a)}$$

$$F(x)=\frac{\text{mbox}arcsl(x)}{x\Phi (x^4,3/2,1/4)}.$$

$$u(x)=\text{mbox}arcsl(x)\text{mbox}andv(x)=x\Phi (x^4,3/2,1/4).$$

$$u(0)=v(0)=0\text{mbox}and{u}^{\prime }(x)=\frac{1}{\sqrt{1-x^4}},v^{\prime }(x)=8\sum _{k=0}^{\infty }\frac{x^{4k}}{\sqrt{4k+1}}.$$

$$\frac{u^{\prime }(x)}{v^{\prime }(x)}=\frac{1}{8h(x^4)}$$

$$h(s)=\sqrt{1-s}\sum _{k=0}^{\infty }\frac{s^k}{\sqrt{4k+1}}.$$

$$2\sqrt{1-s}{h}^{\prime }(s)=2(1-s)\sum _{k=1}^{\infty }\frac{k{s}^{k-1}}{\sqrt{4k+1}}-\sum _{k=0}^{\infty }\frac{s^k}{\sqrt{4k+1}}=\sum _{k=0}^{\infty }{a}_k{s}^k,$$

$$a_k=\frac{2k+2}{\sqrt{4k+5}}-\frac{2k+1}{\sqrt{4k+1}}=\frac{-1}{(2k+2)(4k+1)\sqrt{4k+5}+(2k+1)(4k+5)\sqrt{4k+1}}.$$

$$F(0)<F(x)<F(1)\text{mbox}forx\in (0,1).$$

$$F(1)=\frac{\text{mbox}arcsl(1)}{\Phi (1,3/2,1/4)}.$$

$$\underset{x\to 0}{\lim }\frac{\text{mbox}arcsl(x)}{x}={\frac{d}{dx}\text{mbox}arcsl(x)|}_{x=0}=1\text{mbox}and\Phi (0,3/2,1/4)=8,$$

$$F(0)=\frac{1}{8}.$$

Full text

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In 2017, Mahmoud and Agarwal presented bounds for arcsl in terms of the Lerch zeta function

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They proved

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We show that the factor $1/4$ can be replaced by $\mbox{arcsl}(1)/\Phi(1,3/2,1/4)=0.12836...$. This constant is best possible.

Keywords. Arc lemniscate sine function, Lerch zeta function, beta function, Hurwitz zeta function, inequalities.

1 Introduction and statement of result

Let $F_{1}$ and $F_{2}$ be two points in the plane, with distance $F_{1}F_{2}=2c$. The lemniscate of Bernoulli is the locus of all points $P$ such that $PF_{1}\cdot PF_{2}=c^{2}$. It is named after the Swiss mathematician Jakob Bernoulli (1655-1705) who was the first who studied the lemniscate in detail. The arc length of the lemniscate curve $L$ is given by the formula

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where arcsl is the so-called arc lemniscate sine function, defined by

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Many interesting information on this subject including historical comments can be found in Ayoub [1] and Langer & Singer [4].

This note is inspired by a remarkable paper published by Mahmoud and Agarwal [1] in 2017. Among others, the authors offered upper and lower bounds for arcsl in terms of the Lerch zeta function

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They proved the elegant double-inequality

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It is natural to ask whether the constant factors $1/8$ and $1/4$ are sharp. In this note, we refine the upper bound given in (1.1). Indeed, the constant $1/4$ can be replaced by a smaller number as the following theorem reveals.

THEOREM. For all $x\in(0,1)$ we have

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with the best possible constant factors

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In particular, we obtain that for all $x\in(0,1)$ the ratio \mbox{arcsl}(x)/\bigl{(}x\Phi(x^{4},3/2,1/4)\bigr{)} lies between $1/8$ and $1/7$. The constant $\beta$ can be expressed in terms of the Euler beta function and the Hurwitz zeta function, respectively, which are given by

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The substitution $t=s^{1/4}$ gives

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Thus,

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where $\Gamma$ denotes the classical gamma function.

Schneider [7] proved in 1937 that the lemniscate constant $\mbox{arcsl}(1)$ is a transcendental number; see also Todd [8].

2 Proof of Theorem

The following lemma plays an important role in the proof of our theorem. It is known in the literature as the monotone form of l’Hopital’s rule; see Hardy et al. [2, p. 106], Kwong [3] and Pinelis [6].

LEMMA. Let $u,v:[a,b]\rightarrow\mathbb{R}$ be continuous functions. Moreover, let $u,v$ be differentiable on $(a,b)$ and $v^{\prime}\neq 0$ on $(a,b)$. If $u^{\prime}/v^{\prime}$ is strictly increasing on $(a,b)$, then

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is strictly increasing on $(a,b)$.

PROOF OF THEOREM. Let

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In order to prove that $F$ is strictly increasing on $(0,1)$ we apply the lemma with

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Let $x\in(0,1)$. We have

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It follows that

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with

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Then,

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where

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Since $a_{k}<0$ for $k=0,1,2,...$, we conclude that $h^{\prime}(s)<0$ for $s\in(0,1)$. Thus, $h$ is strictly decreasing on $(0,1)$. Using (2.1) yields that $u^{\prime}/v^{\prime}$ is strictly increasing on $(0,1)$, so that the lemma reveals that $F=u/v$ is strictly increasing on $(0,1)$. It follows that

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We have

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Since

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we obtain

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From (2.2), (2.3) and (2.4) we conclude that (1.2) is valid and that the constant factors $\alpha$ and $\beta$ as given in (1.3) are best possible. This completes the proof of the theorem.