Some Rational Approximations and Bounds for Bateman’s G-Function

Abstract

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper, we present the following rational approximations for Bateman’s G-function $G\left(w\right)=\frac{1}{w}+{\left[2w^2+{\displaystyle \sum _{j=1}^n}4{\alpha }_j{w}^{2-2j}\right]}^{-1}+O\left(\frac{1}{w^{2n+2}}\right),$ where ${\alpha }_1=\frac{1}{4},$ and ${\alpha }_j=\frac{(1-2^{2j+2})B_{2j+2}}{j+1}+{\sum }_{\nu =1}^{j-1}\frac{(1-2^{2j-2\nu +2})B_{2j-2\nu +2}{\alpha }_{\nu }}{j-\nu +1},j>1.$ As a consequence, we introduced some new bounds of $G\left(w\right)$ and a completely monotonic function involving it.

1. Introduction

Bateman’s G-function is defined by Harry Bateman (1882–1946) ([1], p. 20) as

$$G\left(w\right)=\psi \left((1+w)/2\right)-\psi \left(w/2\right),w\in R-\left\{0,-1,-2,...\right\}$$

(1)

where the Psi function $\psi \left(w\right)=\frac{d}{dw}ln\mathsf{\Gamma }\left(w\right)$ and $\mathsf{\Gamma }$ is the classical Euler Gamma function ([2], p. 3). The function $G\left(w\right)$ satisfies

$$G\left(w\right)=2{\int }_0^{\infty }\frac{e^{-wv}}{1+e^{-v}}dv,w>0$$

(2)

and

$$G\left(w\right)+G(w+1)=2w^{-1}.$$

(3)

For more details about the applications and inequalities of the function $G\left(w\right)$, please see [1,3,4,5,6,7,8,9,10,11] and the references therein.

A function $C\left(w\right)$ defined for $w>0$ is called completely monotonic if $C^{\left(r\right)}\left(w\right)$ exists ∀$r\in \mathbb{N}$ such that

$${(-1)}^r{C}^{\left(r\right)}\left(w\right)\ge 0,w>0;r\in \mathbb{N}.$$

There are several remarkable applications of completely monotonic functions in many scientific branches; for more information about this topic, we refer to [12,13,14]. The convergence of the integral ([15], p. 160):

$$C\left(w\right)={\int }_0^{\infty }{e}^{-vw}d\kappa \left(v\right),w\ge 0$$

(4)

where $\kappa \left(w\right)$ is bounded and non-decreasing for $w\ge 0$, gives the necessary and sufficient condition for $C\left(w\right)$ to be completely monotonic on $w\ge 0$.

In [11], Qiu and Vuorinen deduced the double inequality:

$$\frac{4(1.5-ln4)}{w^2}<G\left(w\right)-w^{-1}<\frac{1}{2w^2},w>0.5$$

(5)

and Mortici [10] improve it by

$$\psi \left(w\right)<\psi (w+r)\le \psi \left(w\right)+\psi \left(r\right)+\gamma -r+r^{-1},w\ge 1;0<r<1$$

(6)

where $\gamma$ is the Euler constant. In [5], Mahmoud and Agarwal improved the lower bound of Inequality (5) for $w>{\left(\frac{9-12ln2}{16ln2-11}\right)}^{1/2}\approx 2.74$ by

$$G\left(w\right)>\frac{1}{2w^2+\frac{3}{2}},w>0.$$

(7)

Furthermore, they presented the asymptotic formula:

$$G\left(w\right)-w^{-1}\sim {\displaystyle \sum _{n=1}^{\infty }}\frac{(2^{2n}-1)B_{2n}}{n{w}^{2n}},w\to \infty$$

(8)

where the $B_n$s are the Bernoulli numbers.

In [7], Mahmoud, Talat, and Moustafa studied the following approximation family:

$$\lambda (\xi ,w)=ln\left(\frac{w+\xi +1}{w+\xi }\right)+\frac{2}{w(w+1)},\xi \in [1,2];w>0$$

which is asymptotically equivalent to $G\left(w\right)$ for $w\to \infty$, and they proved the inequality:

$$ln\left(\frac{w+a+1}{w+a}\right)-\frac{w-1}{w(w+1)}<G\left(w\right)-w^{-1}<ln\left(\frac{w+b+1}{w+b}\right)-\frac{w-1}{w(w+1)},$$

(9)

where $a=\frac{4}{e^2-4}$ and $b=1$ are the best possible. In [3] Hegazi, Mahmoud, Talat, and Moustafa deduced that:

$$G\left(w\right)-\frac{1}{w}>ln\left(1+\frac{1}{w+\sqrt{\frac{2}{3}}}\right)+\frac{\sqrt{6}-3w}{3w(w+1)}+\frac{1}{6\sqrt{6}{w}^4},w\ge 2$$

(10)

$$G\left(w\right)-\frac{1}{w}<ln\left(1+\frac{1}{w-\sqrt{\frac{2}{3}}}\right)-\frac{\sqrt{6}+3w}{3w(w+1)}-\frac{1}{6\sqrt{6}{w}^4},w>\sqrt{\frac{2}{3}}$$

(11)

$$G\left(w\right)-\frac{1}{w}<\frac{1}{2}ln\left(1+\frac{2w+3}{w^2+2w+\frac{48}{e^4-16}}\right)-\frac{w-1}{w(w+1)},w>\frac{576}{1000}$$

(12)

$$G\left(w\right)-\frac{1}{w}>\frac{1}{2}ln\left(1+\frac{2w+c}{w^2+2w+\frac{4}{3}}\right)-\frac{w-1}{w(w+1)},w>0$$

(13)

and

$$G\left(w\right)-\frac{1}{w}<\frac{1}{2}ln\left(1+\frac{2w+d}{w^2+2w+\frac{4}{3}}\right)-\frac{w-1}{w(w+1)},w>0$$

(14)

where $c=3$ and $d=\frac{e^4-16}{12}$ are the best possible.

In [9], Mahmoud, Talat, Moustafa, and Agarwal proved

$$\frac{1}{2w^2+1}<G\left(w\right)-1/w<\frac{1}{2w^2},w>0$$

(15)

which presents improvements of the lower bounds of (7) for $w>0$, Inequality (9) for $w>2.48$, Inequality (10) for $w>1.96$, and Inequality (13) for $w>4.13$. Furthermore, it improves the upper bounds of (9) for $w>2.54$, Inequality (11) for $w>\sqrt{\frac{2}{3}}$, Inequality (12) for $w>5.72$, and Inequality (14) for $w>2.53$. They showed that

$${\displaystyle \sum _{n=1}^{2m}}\frac{(2^{2n}-1)B_{2n}}{n{w}^{2n}}<G\left(w\right)-w^{-1}<{\displaystyle \sum _{n=1}^{2l-1}}\frac{(2^{2n}-1)B_{2n}}{n{w}^{2n}},m,l=1,2,...$$

(16)

and deduced that

$$\frac{1}{2w^2+l}<G\left(w\right)-w^{-1}<\frac{1}{2w^2+k},w>0$$

with the best possible constants $l=1$ and $k=0$, which is a refinement of the lower bound of Inequality (7).

Recently, Mahmoud and Almuashi [8] studied the generalized Bateman’s G-function $G_{\mu }\left(w\right)$ defined by

$$G_{\mu }\left(w\right)=\psi \left(\frac{w+\mu }{2}\right)-\psi \left(\frac{w}{2}\right),w\ne -2r,-2r-\mu ;0<\mu <2;r=0,1,2,...$$

and presented some of its properties. Furthermore, they presented the following inequality:

$$ln\left(1+\frac{\mu }{w+\varrho }\right)<G_{\mu }\left(w\right)-\frac{2\mu }{(w+\mu )w}<ln\left(1+\frac{\mu }{w+\iota }\right),w>0;\mu \in (0,2)$$

where $\varrho =\frac{\mu }{e^{\gamma +\frac{2}{\mu }+\psi \left(\frac{\mu }{2}\right)}-1}$ and $\iota =1$ are the best possible.

The outline of the paper is as follows. Section 1 provides the definition, some relations, asymptotic expansions, and some inequalities of the Bateman G-function. The Padé approximant is defined in Section 2, and some rational approximations of $G\left(w\right)$ are calculated. In Section 3, some new bounds of $G\left(w\right)$ are presented based on Padé approximants, and we show that our new inequalities improve some recently published ones. We prove the complete monotonicity property of a function involving $G\left(w\right)$ in Section 4.

2. Some Padé Approximants of Bateman’s G-Function

In this section, we present some Padé approximants of the function $G\left(w\right)$, which present the best rational approximations with the given order of a function.

Consider the formal power series:

$$h\left(w\right)=c_0+c_{1}w+c_2{w}^2+...,$$

then the rational function:

$${[r,s]}_h\left(w\right)=\frac{{\displaystyle \sum _{i=0}^r}{a}_i{w}^i}{1+{\displaystyle \sum _{i=1}^s}{b}_i{w}^i},r\ge 0;s\ge 1$$

(17)

is called the Padé approximant of order $(r,s)$ of the function $h\left(w\right)$ ([16], Chapter 1; [17], p. 96) and [18] where

$${[r,s]}_h\left(w\right)-h\left(w\right)=O\left(w^{r+s+1}\right),w\to \infty$$

(18)

and the coefficients $b_i^{\prime }s$ are the solution of the system:

$$\left.\begin{array}{ccc}\hfill 0& =& c_{r+1}+c_r{b}_1+\cdots +c_{r-s+1}{b}_s\hfill \\ \hfill 0& =& c_{r+2}+c_{r+1}{b}_1+\cdots +c_{r-s+2}{b}_s\hfill \\ \hfill & ⋮& \\ \hfill 0& =& c_{r+s}+c_{r+s-1}{b}_1+\cdots +c_r{b}_s,\hfill \end{array}\right\}$$

(19)

with $c_i=0$ for $i<0$, and the coefficients $a_i^{\prime }s$ are given by

$$\left.\begin{array}{ccc}\hfill a_0& =& c_0\hfill \\ \hfill a_1& =& c_1+c_0{b}_1\hfill \\ \hfill & ⋮& \\ \hfill a_r& =& c_r+c_{r-1}{b}_1+\cdots +c_{r-s}{b}_s.\hfill \end{array}\right\}$$

(20)

Theorem 1.

The Padé approximant of order $(2,2n)$ of the function $f\left(w\right)={\sum }_{k=1}^{\infty }\frac{(2^{2k}-1)B_{2k}}{k{w}^{2k}}$ is given by

$${[2,2n]}_f\left(w\right)=\frac{a_0+a_1{w}^{-2}}{1+{\displaystyle \sum _{i=1}^n}{b}_i{w}^{-2i}},$$

(21)

where $a_0=0$, $a_1=\frac{1}{2}$, $b_1=\frac{1}{2}$, and

$$b_j=-2\left(\frac{(2^{2j+2}-1)B_{2j+2}}{j+1}+{\displaystyle \sum _{v=1}^{j-1}}\frac{(2^{2j-2v+2}-1)B_{2j-2v+2}}{j-v+1}{b}_v\right),j>1.$$

Proof. 

For the function:

$$f\left(\sqrt{w}\right)={\displaystyle \sum _{k=1}^{\infty }}\frac{(2^{2k}-1)B_{2k}}{k{w}^k}$$

the Padé approximant of order $(1,n)$ is given by

$${[1,n]}_f\left(\sqrt{w}\right)=\frac{{\displaystyle \sum _{\nu =0}^1}{a}_{\nu }{w}^{-\nu }}{1+{\displaystyle \sum _{\nu =1}^n}{b}_{\nu }{w}^{-\nu }},$$

where the coefficients $b_{\nu }^{\prime }s$ are the solution of the system

$$\left.\begin{array}{ccc}\hfill 0& =& \frac{-1}{4}+\frac{1}{2}{b}_1\hfill \\ \hfill 0& =& \frac{1}{2}+\frac{-1}{4}{b}_1+\frac{1}{2}{b}_2\hfill \\ & ⋮& \\ \hfill 0& =& \frac{(2^{2n+2}-1)B_{2n+2}}{n+1}+\frac{(2^{2n}-1)B_{2i}}{n}{b}_1+\cdots +\frac{15B_4}{2}{b}_{n-1}+\frac{1}{2}{b}_n,\hfill \end{array}\right\}$$

and

$$\left.\begin{array}{ccc}\hfill a_0& =& 0\hfill \\ \hfill a_1& =& \frac{1}{2}.\hfill \end{array}\right\}$$

Then,

$${[2,2n]}_f\left(w\right)=\frac{w^{-2}}{2(1+{\displaystyle \sum _{i=1}^n}{b}_i{w}^{-2i})}$$

(22)

with $b_1=\frac{1}{2}$, and $b_j=2\left(\frac{(1-2^{2j+2})B_{2j+2}}{j+1}+{\sum }_{v=1}^{j-1}\frac{(1-2^{2j-2v+2})B_{2j-2v+2}}{j-v+1}{b}_v\right)$ for $j=2,3,...,n.$ □

To obtain the Padé approximant of order $(2,16)$ of the function $f\left(w\right)$, we consider the system:

$$\begin{array}{ccc}\hfill b_2& =& -2\left(\frac{1}{2}-\frac{1}{4}{b}_1\right)\hfill \\ \hfill b_3& =& -2\left(\frac{-17}{8}+\frac{1}{2}{b}_1+\frac{-1}{4}{b}_2\right)\hfill \\ \hfill b_4& =& -2\left(\frac{31}{2}+\frac{-17}{8}{b}_1+\frac{1}{2}{b}_2-\frac{1}{4}b3\right)\hfill \\ \hfill b_5& =& -2\left(\frac{-691}{4}+\frac{31}{2}{b}_1-\frac{17}{8}{b}_2+\frac{1}{2}{b}_3-\frac{1}{4}{b}_4\right)\hfill \\ \hfill b_6& =& -2\left(\frac{5461}{2}-\frac{691}{4}{b}_1+\frac{31}{2}{b}_2-\frac{17}{8}{b}_3+\frac{1}{2}{b}_4-\frac{1}{4}{b}_5\right)\hfill \\ \hfill b_7& =& -2\left(\frac{-929,569}{16}+\frac{5461}{2}{b}_1-\frac{691}{4}{b}_2+\frac{31}{2}{b}_3-\frac{17}{8}{b}_4+\frac{1}{2}{b}_5-\frac{1}{4}{b}_6\right)\hfill \\ \hfill b_8& =& -2\left(\frac{3,202,291}{2}-\frac{929,569}{16}{b}_1+\frac{5461}{2}{b}_2-\frac{691}{4}{b}_3+\frac{31}{2}{b}_4-\frac{17}{8}{b}_5+\frac{1}{2}{b}_6-\frac{1}{4}{b}_7\right),\hfill \end{array}$$

and hence, we obtain $b_2=-\frac{3}{4},b_3=\frac{27}{8},b_4=-\frac{423}{16},b_5=\frac{9927}{32},b_6=-\frac{324,423}{64}$, $b_7=\frac{14,098,527}{128}$, and $b_8=-\frac{787,622,823}{256}$. Then,

$$\begin{array}{ccc}\hfill {[2,2]}_f\left(w\right)& =& \frac{w^{-2}/2}{1+\frac{1}{2}{w}^{-2}}\hfill \\ \hfill {[2,4]}_f\left(w\right)& =& \frac{w^{-2}/2}{1+\frac{1}{2}{w}^{-2}-\frac{3}{4}{w}^{-4}}\hfill \\ \hfill {[2,6]}_f\left(w\right)& =& \frac{w^{-2}/2}{1+\frac{1}{2}{w}^{-2}-\frac{3}{4}{w}^{-4}+\frac{27}{8}{w}^{-6}}\hfill \\ \hfill {[2,8]}_f\left(w\right)& =& \frac{w^{-2}/2}{1+\frac{1}{2}{w}^{-2}-\frac{3}{4}{w}^{-4}+\frac{27}{8}{w}^{-6}-\frac{423}{16}{w}^{-8}}\hfill \\ \hfill {[2,10]}_f\left(w\right)& =& \frac{w^{-2}/2}{1+\frac{1}{2}{w}^{-2}-\frac{3}{4}{w}^{-4}+\frac{27}{8}{w}^{-6}-\frac{423}{16}{w}^{-8}+\frac{9927}{32}{w}^{-10}}\hfill \\ \hfill {[2,12]}_f\left(w\right)& =& \frac{w^{-2}/2}{1+\frac{1}{2}{w}^{-2}-\frac{3}{4}{w}^{-4}+\frac{27}{8}{w}^{-6}-\frac{423}{16}{w}^{-8}+\frac{9927}{32}{w}^{-10}-\frac{324,423}{64}{w}^{-12}}\hfill \\ \hfill {[2,14]}_f\left(w\right)& =& \frac{w^{-2}}{l_1\left(w\right)}\hfill \\ \hfill {[2,16]}_f\left(w\right)& =& \frac{w^{-2}}{l_2\left(w\right)},\hfill \end{array}$$

where

$l_1\left(w\right)=2(1+\frac{1}{2}{w}^{-2}-\frac{3}{4}{w}^{-4}+\frac{27}{8}{w}^{-6}-\frac{423}{16}{w}^{-8}+\frac{9927}{32}{w}^{-10}-\frac{324,423}{64}{w}^{-12}+\frac{14,098,527}{128}{w}^{-14})$ and $l_2\left(w\right)=2(1+\frac{1}{2}{w}^{-2}-\frac{3}{4}{w}^{-4}+\frac{27}{8}{w}^{-6}-\frac{423}{16}{w}^{-8}+\frac{9927}{32}{w}^{-10}-\frac{324,423}{64}{w}^{-12}$

$+\frac{14,098,527}{128}{w}^{-14}-\frac{787,622,823}{256}{w}^{-16})$.

3. Some Rational Bounds of Bateman’s G-Function

In this section, we use Padé approximants to formulate new bounds of the function $G\left(w\right)$.

Recall that, if the real-valued function $\zeta \left(t\right)$ defined for $t>0$, ${lim}_{t\to \infty }\zeta \left(t\right)=0$ and $\zeta (t+r)<\zeta \left(t\right)$, $r\in \mathbb{N}$, then $\zeta \left(t\right)>0$ for $t>0$ (see [19]).

Theorem 2.

The following inequality holds:

$${[2,6]}_f\left(w\right)<G\left(w\right)-w^{-1}<{[2,4]}_f\left(w\right),$$

(23)

where the lower bound and the upper bound hold for $w>0$ and $w>\frac{\sqrt{\sqrt{13}-1}}{2}\simeq 0.80709$, respectively.

Proof. 

Let $g_4\left(w\right)=G\left(w\right)-\frac{1}{w}-{[2,4]}_f\left(w\right)$. Then, by using Relation (3), we have

$$g_4(w+2)-g_4\left(w\right)=\frac{18\left(24w^2+48w+23\right)}{w(w+1)(w+2)\left(4w^4+2w^2-3\right)\left(4w^4+32w^3+98w^2+136w+69\right)}.$$

Since the equation $4w^4+2w^2-3=0$ has only one positive root at $w_1=\frac{\sqrt{\sqrt{13}-1}}{2}$, then $g_4(w+2)>g_4\left(w\right)$, ∀$w>w_1$ with ${lim}_{w\to \infty }{g}_4\left(w\right)=0$. Then, we obtain $g_4\left(w\right)<0$, ∀$w>w_1$, which implies $G\left(w\right)-\frac{1}{w}<{[2,4]}_f\left(w\right)$, ∀$w>w_1$. Let $g_6\left(w\right)=G\left(w\right)-\frac{1}{w}-{[2,6]}_f\left(w\right)$, then by using Relation (3), we have

$$g_6(w+2)-g_6\left(w\right)=-\frac{18\left(940w^4+3760w^3+6316w^2+5112w+1737\right)}{l_3\left(w\right)},$$

where $l_3\left(w\right)=w(w+1)(w+2)(8w^6+4w^4-6w^2+27)(8w^6+96w^5+484w^4+1312w^3+2010w^2+1640w+579)$. Since the equation $8w^3+4w^2-6w+27=0$ has no positive real root, then $g_6(w+2)<g_6\left(w\right)$, ∀$w>0$ with ${lim}_{w\to \infty }{g}_6\left(w\right)=0$. Then, we obtain $g_6\left(w\right)>0$, ∀$w>0$, which implies ${[2,6]}_f\left(w\right)<G\left(w\right)-\frac{1}{w}$, ∀$w>0$. □

Theorem 3.

The following inequality holds:

$${[2,14]}_f\left(w\right)<G\left(w\right)-\frac{1}{w}<{[2,16]}_f\left(w\right),$$

(24)

where the lower bound and the upper bound hold for $w>0$ and $w>\frac{13}{5}$, respectively.

Proof. 

Let $g_{14}\left(w\right)=G\left(w\right)-\frac{1}{w}-{[2,14]}_f\left(w\right)$, then by using Relation (3):

$$g_{14}(w+2)-g_{14}\left(w\right)=-\frac{18r\left(w\right)}{w(w+1)(w+2)s\left(w\right)}<0,$$

where

$$\begin{array}{ccc}\hfill r\left(w\right)& =& 576,752w^{13}+17,879,312w^{12}+247,006,376w^{11}+1,951,966,872w^{10}\hfill \\ & +& 9,023,932,048w^9+29,349,843,952w^8+69,495,292,288w^7\hfill \\ & +& 123,190,424,736w^6+164,419,139,520w^5+163,742,349,820w^4\hfill \\ & +& 118,336,385,968w^3+57,293,959,996w^2+15,984,084,792w+157,65,707,337\hfill \\ & >& 0,w>0\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill s\left(w\right)& & =4096w^{24}+114,688w^{23}+1,495,040w^{22}+12,034,048w^{21}+66,882,560w^{20}\hfill \\ & & +271,851,520w^{19}+834,863,104w^{18}+1,972,912,128w^{17}+3,617,878,272w^{16}\hfill \\ & & +5,148,431,360w^{15}+5,640,385,280w^{14}+4,677,807,616w^{13}+2,830,319,744w^{12}\hfill \\ & & +1,716,108,544w^{11}+8,253,448,064w^{10}+46,222,128,512w^9+176,800,158,576w^8\hfill \\ & & +468,131,097,984w^7+899,802,659,232w^6+1,276,187,276,736w^5\hfill \\ & & +1,331,567,184,348w^4+997,589,356,848w^3+496,128,363,804w^2\hfill \\ & & +141,254,459,640w+141,891,366,033>0,w>0.\hfill \end{array}$$

Furthermore, ${lim}_{w\to \infty }{g}_{14}\left(w\right)=0$, and hence, $g_{14}\left(w\right)>0$, ∀$w>0$, which gives ${[2,14]}_f\left(w\right)<G\left(w\right)-\frac{1}{w}$, ∀$w>0$. Let $g_{16}\left(w\right)=G\left(w\right)-\frac{1}{w}-{[2,16]}_f\left(w\right)$, then by using Relation (3), we have

$$g_{16}(w+2)-g_{16}\left(w\right)=\frac{18r_1\left(w\right)}{w(w+1)(w+2)s_1\left(w\right)s_2\left(w\right)},$$

where

$$\begin{array}{ccc}\hfill r_1(w+1)& & =7,832,993,923,840w^{14}+219,323,829,867,520w^{13}\hfill \\ & & +2,951,459,430,932,288w^{12}+25,215,669,729,930,752w^{11}\hfill \\ & & +152,317,731,513,578,304w^{10}+686,224,197,180,084,480w^9\hfill \\ & & +2,371,783,010,371,491,504w^8+6,374,300,275,957,107,456w^7\hfill \\ & & +13,359,225,218,824,632,416w^6+21,688,648,087,649,664,128w^5\hfill \\ & & +26,807,655,086,077,277,460w^4+24,427,148,696,743,456,928w^3\hfill \\ & & +15,494,208,668,209,128,860w^2+6,126,480,357,977,215,344w\hfill \\ & & +1,081,166,851,750,059,759>0w>0,\hfill \end{array}$$

$$\begin{array}{ccc}& & s_1(w+1)=\hfill \\ & & 256w^{16}+12,288w^{15}+276,608w^{14}+3,876,096w^{13}+37,844,160w^{12}\hfill \\ & & +272,975,616w^{11}+1,504,750,176w^{10}+6,466,091,328w^9\hfill \\ & & +21,889,518,768w^8+58,570,846,848w^7+123,458,967,864w^6\hfill \\ & & +202,842,619,632w^5+254,654,582,220w^4+236,146,329,936w^3\hfill \\ & & +152,546,318,910w^2+61,418,471,508w+10,955,689,299>0,w>0\hfill \end{array}$$

and

$$\begin{array}{ccc}& & s_2\left(w+\frac{13}{5}\right)=\hfill \\ & & 256w^{16}+\frac{94,208}{5}{w}^{15}+\frac{3,250,816}{5}{w}^{14}+\frac{349,058,304}{25}{w}^{13}\hfill \\ & & +\frac{26,107,488,704}{125}{w}^{12}+\frac{7,211,309,234,944}{3125}{w}^{11}+\frac{30,437,288,8683,872}{15,625}{w}^{10}\hfill \\ & & +\frac{2,002,484,736,219,456}{15,625}{w}^9+\frac{51,883,806,254,460,144}{78,125}{w}^8\hfill \\ & & +\frac{1,062,345,679,901,482,624}{390,625}{w}^7+\frac{85,663,214,450,859,557,752}{9,765,625}{w}^6\hfill \\ & & +\frac{1,076,675,005,717,334,916,016}{48,828,125}{w}^5+\frac{2,067,776,254,875,229,344,924}{48,828,125}{w}^4\hfill \\ & & +\frac{14,665,131,542,761,625,849,296}{244,140,625}{w}^3+\frac{72,444,882,017,256,139,899,286}{1,220,703,125}{w}^2\hfill \\ & & +\frac{1,113,540,298,780,701,469,947,172}{30,517,578,125}w+\frac{1,604,764,820,494,673,221,424,691}{152,587,890,625}\hfill \\ & & >0w>0.\hfill \end{array}$$

Then, $g_{16}(w+2)>g_{16}\left(w\right)$, ∀$w>\frac{13}{5}$ with ${lim}_{w\to \infty }{g}_{16}\left(w\right)=0$. Then, we have $g_{16}\left(w\right)<0$, ∀$w>\frac{13}{5}$, which means ${[2,16]}_f\left(w\right)>G\left(w\right)-\frac{1}{w}$, ∀$w>\frac{13}{5}$. □

Remark 1.

The lower bound of (23) improves the lower bound of (15) for $w>\sqrt{\frac{9}{2}}$.

Remark 2.

The upper bound of (23) improves the upper bound of (15) for $w\in {\mathbb{R}}^{+}-\left(\frac{\sqrt{\sqrt{13}-1}}{2},\sqrt{\frac{3}{2}}\right)$.

Remark 3.

Let

$$b_{16}\left(w\right)=\frac{1}{2w^2}-\frac{1}{4w^4}+\frac{1}{2w^6}-\frac{17}{8w^8}+\frac{31}{2w^{10}}-\frac{691}{4w^{12}}+\frac{5461}{2w^{14}}-\frac{929,569}{16w^{16}},$$

and

$$b_{18}\left(w\right)=\frac{1}{2w^2}-\frac{1}{4w^4}+\frac{1}{2w^6}-\frac{17}{8w^8}+\frac{31}{2w^{10}}-\frac{691}{4w^{12}}+\frac{5461}{2w^{14}}-\frac{929,569}{16w^{16}}+\frac{3,202,291}{2w^{18}}.$$

Consider the difference:

$${[2,14]}_f\left(w\right)-b_{16}\left(w\right)=\frac{r_3\left(w\right)}{16w^{16}{s}_3\left(w\right)},$$

where

$$\begin{array}{ccc}\hfill r_3\left(w\right)& =& 128,654,692w^{12}-262,161,780w^{10}+1,299,430,638w^8-10,170,269,640w^6+104,226,438,528w^4\hfill \\ & -& 1,219,083,574,950w^2+13,105,553,644,863,\hfill \end{array}$$

and

$$s_3\left(w\right)=128w^{14}+64w^{12}-96w^{10}+432w^8-3384w^6+39,708w^4-648,846w^2+14,098,527.$$

Since the equations:

$$128,654,692w^2-262,161,780w+299,430,638=0,$$

$$1,000,000,000w^4-10,170,269,640w^3+104,226,438,528w^2-1,219,083,574,950w+13,105,553,644,863=0,$$

$$200w^4-3384w^3+39,708w^2-648,846w+14,098,527=0,$$

and

$$128w^3+64w^2-96w+232=0$$

have no positive real roots, then $r_3\left(w\right)$ and $s_3\left(w\right)$ are positive ∀$w>0$, which shows that the lower bound of (24) improves the lower bound of (15) for $w>0$. Furthermore, consider the difference:

$$b_{18}\left(w\right)-{[2,16]}_f\left(w\right)=\frac{r_4\left(w\right)}{16w^{18}{s}_4\left(w\right)},$$

where

$$\begin{array}{ccc}& & r_4\left(w+\frac{26}{9}\right)=\hfill \\ & & 6,953,960,836w^{14}+\frac{2,531,241,744,304w^{13}}{9}+\frac{426,660,551,197,660w^{12}}{81}\hfill \\ & & +\frac{44,139,882,177,895,712w^{11}}{729}+\frac{3,131,486,526,410,399,774w^{10}}{6561}\hfill \\ & & +\frac{161,200,595,978,180,233,592w^9}{59049}+\frac{2,070,444,862,008,627,300,848w^8}{177,147}\hfill \\ & & +\frac{60,682,986,467,807,767,571,456w^7}{1,594,323}+\frac{1,359,824,925,032,346,851,475,104w^6}{14348907}\hfill \\ & & +\frac{69,571,718,596,248,162,913,883,264w^5}{387,420,489}+\frac{889190,199,144,227,314,857,593,450w^4}{3,486,784,401}\hfill \\ & & +\frac{8,253,963,639,145,423,710,931,321,616w^3}{31,381,059,609}+\frac{52,622,035,382,970,648,465,996,271,591w^2}{282,429,536,481}\hfill \\ & & +\frac{215,974,211,992,242,994,684,174,061,164w}{2,541,865,828,329}+\frac{96,838,271,852,334,314,086,590,005,044}{22,876,792,454,961},w>0\hfill \end{array}$$

and

$$\begin{array}{ccc}& & s_4\left(w+\frac{26}{9}\right)=\hfill \\ & & 256w^{16}+\frac{106,496w^{15}}{9}+\frac{6,925,696w^{14}}{27}+\frac{2,523,469,312w^{13}}{729}+\frac{213,550,796,096w^{12}}{6561}\hfill \\ & & +\frac{4,450,595,881,472w^{11}}{19,683}+\frac{637,979,089,706,848w^{10}}{531,441}+\frac{23,764,147,568,110,976w^9}{4782,969}\hfill \\ & & +\frac{77,485,950,743,106,704w^8}{4,782,969}+\frac{16,175,911,245,854,067,968w^7}{387,420,489}\hfill \\ & & +\frac{295,580,430,758,925,054,136w^6}{3,486,784,401}+\frac{1,403,341,767,918,187,371,872w^5}{10,460,353,203}\hfill \\ & & +\frac{45,820,036,824,602,356,573,316w^4}{282,429,536,481}+\frac{368,352,992,789,069,056,571,296w^3}{2,541,865,828,329}\hfill \\ & & +\frac{687,632,404,365,282,872,950,378w^2}{7,625,597,484,987}+\frac{7,208,442,438,570,664,907,699,096w}{205,891,132,094,649}\hfill \\ & & +\frac{10,611,078,269,119,878,030,645,073}{1,853,020,188,851,841},w>0.\hfill \end{array}$$

Then, $b_{18}\left(w\right)-{[2,16]}_f\left(w\right)>0$ for $w>\frac{26}{9}$, which shows that the upper bound of (24) improves the upper bound of (15) for $w>\frac{26}{9}$.

4. A Completely Monotonic Function Involving $\mathbf{G}\left(\mathbf{w}\right)$

In this section, we present another advantage of Padé approximants in formulating new completely monotonic functions involving the function $G\left(w\right)$.

Theorem 4.

The function:

$$M\left(w\right)=\frac{(w+3)\left(4w^5+34w^4+122w^3+228w^2+225w+92\right)}{w(w+1)(w+2)\left(4w^4+32w^3+98w^2+136w+69\right)}-G\left(w\right),$$

is completely monotonic for $w>0$, that is

$${(-1)}^r{M}^{\left(r\right)}\left(w\right)\ge 0w>0;r\in \mathbb{N}.$$

Proof. 

The function M satisfies

$$M\left(w\right)=-{\int }_0^{\infty }m\left(t\right)\frac{\sqrt{13}cosh\left(\frac{t}{2}\right)e^{-\frac{3t}{2}}}{\sqrt{13}\left(e^t+1\right)}{e}^{-wt}dt,$$

where

$$m\left(t\right)=2tanh\left(\frac{t}{2}\right)-\left(\frac{\sqrt{1+\sqrt{13}}sin\left(\frac{1}{2}\sqrt{1+\sqrt{13}}t\right)}{\sqrt{13}}\right.$$

$$\left.+\frac{\sqrt{\sqrt{13}-1}sinh\left(\frac{1}{2}\sqrt{\sqrt{13}-1}t\right)}{\sqrt{13}}\right).$$

Using the relation:

$$\begin{array}{ccc}& & 2tanh\left(\frac{t}{2}\right)-\left(\frac{31t^9}{362l880}-\frac{17t^7}{20l160}+\frac{t^5}{120}-\frac{t^3}{12}+t\right)\hfill \\ & & =-{\displaystyle \sum _{n=11}^{\infty }}{a}_n\frac{t^n}{362l880n!}\hfill \end{array}$$

where

$$a_n=(n-2)(31n^4-186n^3+500n^2-150n+189)(n-10)(n-8)(n-6)(n-4)$$

we obtain

$$2tanh\left(\frac{t}{2}\right)<\left(\frac{31t^9}{362l880}-\frac{17t^7}{20l160}+\frac{t^5}{120}-\frac{t^3}{12}+t\right),t>0.$$

(25)

Taylor’s formula with the remainder of the function $sint$ gives the following double inequality ([20], p. 284) for $n\in \mathbb{N}$:

$$sint={\displaystyle \sum _{\nu =1}^r}\frac{{(-1)}^{\nu -1}{t}^{2\nu -1}}{(2\nu -1)!}+E_{2r}\left(t\right),E_{2r}\left(t\right)\le \frac{{\left|t\right|}^{2r+1}}{(2r+1)!}-\infty <t<\infty$$

(26)

where $|{sin}^{(n+1)}t|\le 1$. Furthermore, using the expansion:

$$sinht={\displaystyle \sum _{\nu =0}^{\infty }}\frac{t^{2\nu +1}}{(2\nu +1)!},-\infty <t<\infty$$

we obtain $s\in \mathbb{N}$

$$sinh\left(\frac{1}{2}\sqrt{\sqrt{13}-1}t\right)>{\displaystyle \sum _{v=0}^s}\frac{1}{(2v+1)!}{\left(\frac{1}{2}\sqrt{\sqrt{13}-1}t\right)}^{2v+1},t>0$$

(27)

From the inequalities (26) with $r=3$ and (27) with $s=3$, we obtain

$$\begin{array}{ccc}\hfill & & \frac{\sqrt{1+\sqrt{13}}sin\left(\frac{1}{2}\sqrt{1+\sqrt{13}}t\right)}{\sqrt{13}}+\frac{\sqrt{\sqrt{13}-1}sinh\left(\frac{1}{2}\sqrt{\sqrt{13}-1}t\right)}{\sqrt{13}}\hfill \\ & & >\left(-\frac{t^7}{5760}+\frac{t^5}{120}-\frac{t^3}{12}+t\right),t>0.\hfill \end{array}$$

(28)

Hence, from the inequalities (25) and (28), we obtain

$$m\left(t\right)<0,0<t<9\sqrt{\frac{3}{31}}.$$

(29)

Now, the function $2tanh\left(\frac{t}{2}\right)$ is increasing and concave on $t\in (0,\infty )$ with a limit equal to 2 as $t\to \infty$. However, the line $\left(\frac{1}{2}-\frac{1}{2\sqrt{13}}\right)t>1$ for $t\ge 9\sqrt{\frac{3}{31}}$, then we obtain

$$2tanh\left(\frac{t}{2}\right)<\left(\frac{1}{2}-\frac{1}{2\sqrt{13}}\right)t+1,t\ge 9\sqrt{\frac{3}{31}}.$$

(30)

Using Inequality (28), we obtain

$$\begin{array}{ccc}& & \frac{\sqrt{1+\sqrt{13}}sin\left(\frac{1}{2}\sqrt{1+\sqrt{13}}t\right)}{\sqrt{13}}+\frac{\sqrt{\sqrt{13}-1}sinh\left(\frac{1}{2}\sqrt{\sqrt{13}-1}t\right)}{\sqrt{13}}\hfill \\ & & -\left(\frac{1}{2}-\frac{1}{2\sqrt{13}}\right)t-1>f_1\left(t\right),t>0\hfill \end{array}$$

where

$$f_1\left(t\right)=-\frac{t^7}{5760}+\frac{t^5}{120}-\frac{t^3}{12}+\left(\frac{1}{2}+\frac{1}{2\sqrt{13}}\right)t-1,t>0.$$

The function $f_1\left(t\right)$ is convex on $(t_1,t_2)$, where $t_i=2\sqrt{\frac{1}{7}\left(20+{(-1)}^i\sqrt{190}\right)}$; $i=1,2$ with the two conditions $f_1\left(9\sqrt{\frac{3}{31}}\right)>0$ and $f_1^{\prime }\left(9\sqrt{\frac{3}{31}}\right)>0$. Then, $f_1\left(t\right)$ is a positive increasing function on $\left[9\sqrt{\frac{3}{31}},t_2\right]$. Furthermore, $f_1\left(t\right)$ is a concave function on $\left(t_2,\infty \right)$ with $f_1\left(6\right)>0$, and hence, $f_1\left(t\right)$ is positive on $\left[t_2,6\right]$. Then, $f_1\left(t\right)>$ on $\left[9\sqrt{\frac{3}{31}},6\right]$. From the inequalities (26) with $n=1$ and (27) with $s=3$, we obtain

$$\begin{array}{ccc}& & \frac{\sqrt{1+\sqrt{13}}sin\left(\frac{1}{2}\sqrt{1+\sqrt{13}}t\right)}{\sqrt{13}}+\frac{\sqrt{\sqrt{13}-1}sinh\left(\frac{1}{2}\sqrt{\sqrt{13}-1}t\right)}{\sqrt{13}}\hfill \\ & & -\left(\frac{1}{2}-\frac{1}{2\sqrt{13}}\right)t-1>f_2\left(t\right),t>0\hfill \end{array}$$

where

$$f_2\left(t\right)=\frac{\left(31\sqrt{13}-91\right)t^7}{1,048,320}+\frac{\left(26-5\sqrt{13}\right)t^5}{6240}-\frac{t^3}{12}+\frac{1}{26}\left(13+\sqrt{13}\right)t-1,t>0.$$

$f_2\left(t\right)$ is a convex function on $\left(\frac{1}{3}\sqrt{2\left(\sqrt{210\left(41+11\sqrt{13}\right)}-15\left(1+\sqrt{13}\right)\right)},\infty \right)$ with $f_2\left(6\right)>0$ and $f_2^{\prime }\left(6\right)>0$. Hence, $f_2\left(t\right)>0$ for $t\ge 6$. Then, we have

$$\begin{array}{ccc}\hfill & & \frac{\sqrt{1+\sqrt{13}}sin\left(\frac{1}{2}\sqrt{1+\sqrt{13}}t\right)}{\sqrt{13}}+\frac{\sqrt{\sqrt{13}-1}sinh\left(\frac{1}{2}\sqrt{\sqrt{13}-1}t\right)}{\sqrt{13}}\hfill \\ & & >\left(\frac{1}{2}-\frac{1}{2\sqrt{13}}\right)t+1,t\ge 9\sqrt{\frac{3}{31}}.\hfill \end{array}$$

(31)

Hence, from Inequalities (30) and (31), we have

$$m\left(t\right)<0,t\ge 9\sqrt{\frac{3}{31}}.$$

(32)

The two inequalities (29) and (32) complete the proof. □

5. Conclusions

The Padé approximant method presents some rational approximations for Bateman’s G-function $G\left(w\right)$. These approximations provided us with new inequalities of the function $G\left(w\right)$ with completely monotonic functions involving it. We presented proofs to clarify the novelty of our results, which could be of interest to a large part of the readers. This method is considered a powerful tool in deducing estimates and inequalities for several other special functions.