Two Approximation Formulas for Bateman’s G-Function with Bounded Monotonic Errors

Abstract

Two new approximation formulas for Bateman’s G-function are presented with strictly monotonic error functions and we deduced their sharp bounds. We also studied the completely monotonic (CM) degrees of two functions involving $G\left(r\right)$, deducing two of its inequalities and improving some of the recently published results.

1. Introduction and Preliminaries

Bateman’s G-function is defined as [1]

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

(1)

where $\psi \left(r\right)=\frac{d}{dr}ln\mathrm{\Gamma }\left(r\right)$ is the digamma function and $\mathrm{\Gamma }$ is the Euler gamma function [2]. The function $G\left(r\right)$ has several inequalities, such as

Qiu and Vuorinen [3]:	$$\begin{array}{cc}4(1.5-ln4)r^{-2}<G\left(r\right)-r^{-1}<\frac{1}{2}{r}^{-2}, \hspace{1em}\hspace{1em}& \hfill r>\frac{1}{2}\end{array}$$ (2)
Mortici [4]:	$$\begin{array}{cc}0<G\left(r\right)\le \gamma +\psi (1/2)+3/2,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& \hfill r\ge 2\end{array}$$ (3)
Mahmoud and et al. [5]:	$$\begin{array}{cc}ln\left(\frac{r+3}{r+2}\right)+\frac{2}{r(r+1)}<G\left(r\right)<ln\left(\frac{r+2}{r+1}\right)+\frac{2}{r(r+1)} \hspace{1em}\hspace{1em}\hspace{1em}& \hfill r>0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}$$ (4)
Nantomah [6]:	$$\begin{array}{cc}G\left(r\right)>\frac{1}{r}+\frac{1}{2{(r+1)}^2},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& \hfill r>0\end{array}$$ (5)
	$$\begin{array}{cc}2-2e^{\frac{1}{r+1}}<G\left(r\right)-\frac{2}{r}<2e-2ln2-2e^{\frac{1}{r+1}}, \hspace{0.17em}& \hfill r>0\end{array}$$ (6)

where $\gamma$ is the Euler–Mascheroni constant and the constants $c_1=3$ and $c_2=\frac{e^4-16}{12}$ are the best possible.

Mahmoud and Almuashi [7] presented a generalization of Bateman’s G-function by

$$G_{\rho }\left(r\right)=\psi \left(\frac{\rho +r}{2}\right)-\psi \left(\frac{r}{2}\right),r\ne -2m,-2m-\rho ;0<\rho <2;m=0,1,2,\dots$$

and they proved the following inequality:

$$ln\left(\frac{\rho }{r+\epsilon }+1\right)<G_{\rho }\left(r\right)-\frac{2\rho }{(r+\rho )r}<ln\left(\frac{\rho }{r+\sigma }+1\right),r>0;\rho \in (0,2)$$

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

Recently, Ahfaf, Mahmoud, and Talat [8] introduced the following rational approximations

$$G\left(r\right)=\frac{1}{r}+{\left[2r^2+\sum _{m=1}^{s}4A_m{r}^{2-2m}\right]}^{-1}+O\left(\frac{1}{r^{2s+2}}\right),$$

with

$$A_1=\frac{1}{4},\mathrm{and}{A}_m=\frac{(1-2^{2m+2})B_{2m+2}}{1+m}+\sum _{s=1}^{m-1}\frac{(1-2^{2m-2s+2})B_{2m-2s+2}{A}_s}{1-s+m},m>1$$

where $B_j^{\prime }s$ are Bernoulli numbers. As a consequence, they presented the new bounds

$$\frac{r^4}{2\left(r^6+\frac{1}{2}{r}^4-\frac{3}{4}{r}^2+\frac{27}{8}\right)}<G\left(r\right)-1/r<\frac{r^2}{2\left(r^4+\frac{1}{2}{r}^2-\frac{3}{4}\right)},$$

(7)

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

$$\frac{1}{2r^2{M}_1\left(r\right)}<G\left(r\right)-1/r<\frac{1}{2r^2{M}_2\left(r\right)},$$

(8)

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

$$M_1\left(r\right)=1+\frac{1}{2r^2}-\frac{3}{4r^4}+\frac{27}{8r^6}-\frac{423}{16r^8}+\frac{9927}{32r^{10}}-\frac{324423}{64r^{12}}+\frac{14098527}{128r^{14}}$$

and

$$M_2\left(r\right)=1+\frac{1}{2r^2}-\frac{3}{4r^4}+\frac{27}{8r^6}-\frac{423}{16r^8}+\frac{9927}{32r^{10}}-\frac{324423}{64r^{12}}+\frac{14098527}{128r^{14}}-\frac{787622823}{256r^{16}}.$$

Bateman’s $G$-function is useful in summing certain numerical and algebraic series [9]. For example:

$$\sum _{m=0}^{\infty }\frac{{(-1)}^m}{\alpha +\beta m}=\frac{1}{2\beta }G\left(\frac{\alpha }{\beta }\right),\alpha \ne 0,-\beta ,-2\beta ,\dots$$

(9)

and hence we obtain $\pi =G(1/2)$ and $ln4=G\left(1\right)$. The function $G\left(r\right)$ and its generalization $G_{\rho }\left(r\right)$ are related to the generalized hypergeometric functions by the relations [7]

$$G\left(r\right)=r^{-1}{}_2{F}_1\left(1,1;1+r;\frac{1}{2}\right),r>0$$

and

$$G_{\rho }\left(r\right)=\frac{\rho }{r+\rho }{}_3{F}_2\left(1,1,\frac{\rho +2}{2};2,\frac{r+\rho +2}{2};1\right),r>0;0<\rho <2.$$

There is a relation between the function $G\left(r\right)$ and the Wallis’s ratio $\frac{\mathrm{\Gamma }(n+1/2)}{\sqrt{\pi }\mathrm{\Gamma }(n+1)}$, $n\in \mathbb{N}$. Furthermore, the sequence

$$L_m=\frac{m-1}{2}\left[{\int }_0^{\pi /2}{\left(sinu\right)}^{m-1}du\right],m\in \mathbb{N},$$

which appears in the computation of the intersecting probability between a plane couple and a convex body [10], is related to the function $G\left(r\right)$ (see Ref. [11]).

The outline of the paper is as follows: Section 1 provides the definition of the Bateman’s G-function with some of its inequalities. In Section 2, we studied the CM degrees of two functions involving $G\left(r\right)$ and, consequently, we presented two new inequalities of $G\left(r\right)$, which improve some recently published results. Additionally, we proved that the function

$${\theta }_1\left(r\right)=r^3\left[\frac{2}{r(1+r)}+ln\left(\frac{2+r}{r+1}\right)-G\left(r\right)\right],r>0$$

is strictly increasing with the sharp bounds $0<{\theta }_1\left(r\right)<\frac{1}{3}$, and the function

$${\theta }_2\left(r\right)=r^4\left[\frac{2}{r(1+r)}+ln\left(\frac{2+r}{r+1}\right)-\frac{1}{3r^3}-G\left(r\right)\right],r>0$$

is strictly decreasing with the sharp bounds $\frac{-3}{2}<{\theta }_1\left(r\right)<0$.

2. Main Results

Recall that a function $H\left(r\right)$ on $r>0$ is called CM if its derivatives exist for all orders, such that

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

From Bernstein’s well-known theory, the convergence of the following improper integral determines the necessary and sufficient condition for $H\left(r\right)$ to be CM on $r\ge 0$ [12]

$$H\left(r\right)={\int }_0^{\infty }{e}^{-ur}d\xi \left(u\right),r\ge 0$$

(10)

where $\xi \left(u\right)$ is non-decreasing and bounded for $u\ge 0$. Let $H\left(r\right)$ be a CM function for $r>0$ and consider the notation $H(\infty )={lim}_{r\to \infty }H\left(r\right)$. If $r^{\delta }\left[H\left(r\right)-H(\infty )\right]$ is a CM function for $r>0$ if and only if $\delta \in [0,\varpi ]$; then, the number $\varpi \in R$ is called the CM degree of $H\left(r\right)$ for $r>0$ and is denoted by ${\mathrm{deg}}_{CM}^r\left[H\left(r\right)\right]=\varpi$. This concept gives more accuracy in measuring the complete monotonicity property [13,14].

Theorem 1. 

The function

$$F_1\left(r\right)=G\left(r\right)-\frac{2}{r(1+r)}-ln\left(\frac{2+r}{r+1}\right)+\frac{1}{3r^3},r>0$$

(11)

satisfies that $2\le {\mathrm{deg}}_{CM}^r\left[F_1\left(r\right)\right]\le 3$.

Proof. 

Using the relation [5]

$$G\left(r\right)=2{\int }_0^{\infty }\frac{e^{-ru}}{1+e^{-u}}du,r>0$$

(12)

we obtain

$$F_1\left(r\right)={\int }_0^{\infty }\frac{e^{-2u}{\kappa }_2\left(u\right)}{6\left(e^u+1\right)u}{e}^{-ru}du,r>0$$

where

$$\begin{array}{ccc}\hfill {\kappa }_2\left(u\right)& =& e^{2u}{u}^3+e^{3u}{u}^3+12e^{u}u-6e^{2u}+6\hfill \\ & =& 3u^4+\sum _{m=5}^{\infty }\frac{2^{m-3}\left(m^3-3m^2+2m-48\right)+3^{m-3}\left(m^3-3m^2+2m\right)+12m}{m!}{u}^m\hfill \\ & >& 0,u>0.\hfill \end{array}$$

Then, $F_1\left(r\right)$ is the CM function. Furthermore, using the asymptotic formula [5]

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

(13)

we have

$$F_1(\infty )=\underset{r\to \infty }{lim}{F}_1\left(r\right)=\underset{r\to \infty }{lim}\left(\frac{3}{2r^4}-\frac{21}{5r^5}+O\left(r^{-6}\right)\right)=0.$$

Now,

$$r^2{F}_1\left(r\right)={\int }_0^{\infty }\frac{e^{-2u}}{3{\left(e^u+1\right)}^3{u}^3}{\chi }_1\left(u\right)e^{-ru}du,r>0$$

where

$$\begin{array}{ccc}\hfill {\chi }_1\left(u\right)& =& e^{5u}{u}^3+e^{2u}\left(19u^2+27u+18\right)u+6\left(2u^2+2u+1\right)+3e^{4u}\left(u^3-u^2-2u-2\right)\hfill \\ & +& 3e^u\left(2u^3+11u^2+10u+4\right)+3e^{3u}\left(9u^3+u^2-2u-4\right)\hfill \\ & =& 36u^4+\frac{378u^5}{5}+\frac{468u^6}{5}+\frac{6073u^7}{70}+\frac{133u^8}{2}+\frac{42001u^9}{945}+\frac{504139u^{10}}{18900}\hfill \\ & +& \sum _{m=11}^{\infty }\frac{f_m}{2400m!}{u}^m\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill f_m& =& -144000\left(2\left(3^m\right)+4^m-2\right)+192(m-1)(m-2)m{5}^m+125m\left(576(2m+3)(m+1)\right.\hfill \\ & +& \left.9(-26+(m-7)m)4^m+64(m(3m-8)-1)3^m+3(m(19m-3)+56)2^{m+3}\right)\hfill \\ & =& 144000m^3+360000m^2+216000m+288000+5^m\left(192m^3-576m^2+384m\right)\hfill \\ & +& 2^{2m}\left(1125m^3-7875m^2-29250m-144000\right)\hfill \\ & +& 3^m\left(24000m^3-64000m^2-8000m-288000\right)\hfill \\ & +& 2^m\left(57000m^3-9000m^2+168000m\right)>0,m\ge 10.\hfill \end{array}$$

Then, $2\le {\mathrm{deg}}_{CM}^r\left[F_1\left(r\right)\right]$. However,

$$r^3{F}_1\left(r\right)={\int }_0^{\infty }\frac{e^{-2u}}{{\left(e^u+1\right)}^4{u}^4}{\chi }_2\left(u\right)e^{-ru}du,r>0$$

where

$$\begin{array}{ccc}\hfill {\chi }_2\left(u\right)& =& -2e^{4u}\left(8u^4+2u^3-6u-9\right)+e^{5u}\left(u^3+3u^2+6u+6\right)-2\left(4u^3+6u^2+6u+3\right)\hfill \\ & & -4e^{2u}\left(2u^4+11u^3+15u^2+12u+3\right)-e^u\left(2u^4+31u^3+45u^2+42u+18\right)\hfill \\ & & -2e^{3u}\left(5u^4+13u^3+15u^2+6u-6\right)\hfill \end{array}$$

with ${\chi }_2\left(0.5\right)=2.08162$ and ${\chi }_2\left(0.9\right)=-21.5214$. Then, $r^3{F}_1\left(r\right)$ is not a CM function; hence, ${\mathrm{deg}}_{CM}^r\left[F_1\left(r\right)\right]<3$. □

From Theorem 1, the function $F_1\left(r\right)$ is a decreasing function and $F_1(\infty )=0$; then, we obtain the following result:

Corollary 1. 

The function $G\left(r\right)$ satisfies that

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

(14)

Theorem 2. 

The function

$$F_2\left(r\right)=\frac{2}{r(1+r)}+ln\left(\frac{2+r}{r+1}\right)-\frac{1}{3r^3}+\frac{3}{2r^4}-G\left(r\right),r>0$$

(15)

satisfies that $3\le {\mathrm{deg}}_{CM}^r\left[F_2\left(r\right)\right]\le 4$.

Proof. 

Using the relation (12), we have

$$F_2\left(r\right)={\int }_0^{\infty }\frac{{\kappa }_1\left(u\right)}{12\left(e^u+1\right)u}{e}^{-ru}du,r>0$$

where

$$\begin{array}{ccc}\hfill {\kappa }_1\left(u\right)& =& 3u^4+e^u(3u-2)u^3-2u^3-24e^{-u}u-12e^{-2u}+12\hfill \\ & =& \frac{21u^5}{5}+\sum _{m=6}^{\infty }\frac{b_m}{108m!}{u}^m>0,u>0\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill b_m& =& 2^{m-2}\left(81m^4-594m^3+1215m^2-702m+5184\right)\hfill \\ & & +3^m\left(4m^4-32m^3+68m^2-40m\right)-2592m\hfill \\ & >& (m-2)\left(85m^3-456m^2+371m-2592\right)>0,m\ge 6.\hfill \end{array}$$

Then, $F_2\left(r\right)$ is a CM function. Furthermore, using the asymptotic Formula (13), we obtain

$$F_2(\infty )=\underset{r\to \infty }{lim}{F}_2\left(r\right)=\underset{r\to \infty }{lim}\left(\frac{21}{5r^5}-\frac{9}{r^6}+O\left(r^{-7}\right)\right)=0.$$

Now,

$$r^3{F}_2\left(r\right)={\int }_0^{\infty }\frac{e^{-2u}}{2{\left(e^u+1\right)}^4{u}^4}{\chi }_3\left(u\right)e^{-ru}du,r>0$$

where

$$\begin{array}{ccc}\hfill {\chi }_3\left(u\right)& =& 3e^{6u}{u}^4+e^{4u}\left(50u^4+8u^3-24u-36\right)+4\left(4u^3+6u^2+6u+3\right)\hfill \\ & +& e^u\left(4u^4+62u^3+90u^2+84u+36\right)+2e^{5u}\left(6u^4-u^3-3u^2-6u-6\right)\hfill \\ & +& 4e^{3u}\left(8u^4+13u^3+15u^2+6u-6\right)+e^{2u}\left(19u^4+88u^3+120u^2+96u+24\right)\hfill \\ & =& \frac{672u^5}{5}+\frac{1968u^6}{5}+\frac{68512u^7}{105}+\frac{82507u^8}{105}+\frac{717838u^9}{945}\hfill \\ & +& \sum _{m=10}^{\infty }\frac{h_m}{6480000m!}{u}^m\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill h_m& =& 2560000m^4{3}^m+1265625m^4{4}^m+124416m^4{5}^m+625m^4{2}^{m+3}{3}^{m+1}+961875m^4{2}^{m+3}\hfill \\ & +& 25920000m^4-850176m^3{5}^m-625m^3{2}^{m+4}{3}^{m+1}-625m^3{2}^{m+5}{3}^{m+1}-320000m^3{3}^{m+2}\hfill \\ & +& 1569375m^3{2}^{m+4}-3391875m^3{2}^{2m+1}+246240000m^3+33920000m^2{3}^m\hfill \\ & +& 11491875m^2{4}^m+124416m^2{5}^m+625m^2{2}^{m+6}{3}^{m+1}+625m^2{2}^{m+3}{3}^{m+2}\hfill \\ & +& 8150625m^2{2}^{m+3}-336960000m^2-14950656m{5}^m+6080000m{3}^{m+1}\hfill \\ & -& 625m{2}^{m+4}{3}^{m+2}+13314375m{2}^{m+4}-22426875m{2}^{2m+1}+609120000m\hfill \\ & -& \left(124416\right)5^{m+4}-\left(640000\right)3^{m+5}+\left(151875\right)2^{m+10}-\left(455625\right)2^{2m+9}+233280000\hfill \\ & =& 25920000m^4+246240000m^3-336960000m^2+609120000m+233280000\hfill \\ & +& 5^m\left(124416m^4-850176m^3+124416m^2-14950656m-77760000\right)\hfill \\ & +& 2^{2m}\left(1265625m^4-6783750m^3+11491875m^2-44853750m-233280000\right)\hfill \\ & +& 2^m\left(7695000m^4+25110000m^3+65205000m^2+213030000m+155520000\right)\hfill \\ & +& 6^m\left(15000m^4-90000m^3+165000m^2-90000m\right)\hfill \\ & +& 3^m\left(2560000m^4-2880000m^3+33920000m^2+18240000m-155520000\right)>0,m\ge 9.\hfill \end{array}$$

Then, $3\le {\mathrm{deg}}_{CM}^r\left[F_2\left(r\right)\right]$. However,

$$r^4{F}_2\left(r\right)={\int }_0^{\infty }\frac{e^{-2u}}{{\left(e^u+1\right)}^5{u}^5}{\chi }_4\left(u\right)e^{-ru}du,r>0$$

where

$$\begin{array}{ccc}\hfill {\chi }_4\left(u\right)& =& e^{6u}\left(u^4+4u^3+12u^2+24u+24\right)-8\left(2u^4+4u^3+6u^2+6u+3\right)\hfill \\ & -& 2e^{3u}u\left(11u^4+75u^3+140u^2+180u+120\right)+2e^{4u}\left(u^5-35u^4-60u^3-60u^2+60\right)\hfill \\ & -& 5e^{2u}\left(2u^5+31u^4+60u^3+84u^2+72u+24\right)\hfill \\ & -& e^u\left(2u^5+79u^4+156u^3+228u^2+216u+96\right)\hfill \\ & +& e^{5u}\left(-32u^5-11u^4-12u^3+12u^2+72u+96\right)\hfill \end{array}$$

with ${\chi }_4\left(1.2\right)=1268.84$ and ${\chi }_4\left(1.3\right)=-1981.21$. Then, $r^4{F}_2\left(r\right)$ is not a CM function and, hence, ${\mathrm{deg}}_{CM}^r\left[F_1\left(r\right)\right]<4$. □

From Theorem 2, the function $F_2\left(r\right)$ is a decreasing function and $F_2(\infty )=0$; then, we obtain the following result:

Corollary 2. 

The function $G\left(r\right)$ satisfies that

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

(16)

Lemma 1. 

The function

$${\theta }_1\left(r\right)=r^3\left[\frac{2}{r(1+r)}+ln\left(\frac{2+r}{r+1}\right)-G\left(r\right)\right],r>0$$

(17)

is a strictly increasing function with sharp bounds $0<{\theta }_1\left(r\right)<\frac{1}{3}$.

Proof. 

Using the relation (12), we have

$${\theta }_1\left(r\right)=-r^3{\int }_0^{\infty }\frac{e^{-2u}\left(2e^{u}u-e^{2u}+1\right)}{\left(e^u+1\right)u}{e}^{-ru}du,r>0$$

and

$$\frac{d}{dr}{\theta }_1\left(r\right)=r^2{\int }_0^{\infty }\frac{{\kappa }_2\left(u\right)}{{\left(e^u+e^{2u}\right)}^{2}u}{e}^{-ru}du$$

where

$$\begin{array}{ccc}\hfill {\kappa }_2\left(u\right)& =& -\left(2u^2+7u+3\right)e^u+\left(3-4u-4u^2\right)e^{2u}+(3+u)e^{3u}-2u-3\hfill \\ & =& \sum _{m=4}^{\infty }\frac{m+9}{m!}\left[3^{m-1}-\frac{2^m\left(m^2+m-3\right)+(1+m)(3+2m)}{9+m}\right]u^m,u>0.\hfill \end{array}$$

Using the induction, we obtain

$$3^{m-1}>\frac{\left(m^2+m-3\right)2^m+(m+1)(3+2m)}{9+m},m\ge 4$$

with the aid of the relation

$$\begin{array}{c}\frac{3\left(2^m\left(m^2+m-3\right)+(2m+3)(1+m)\right)}{m+9}-\frac{2^{m+1}\left(m+{(m+1)}^2-2\right)+(m+2)(5+2m)}{m+10}\\ =\frac{4m\left(m^2+12m+17\right)+2^m\left(m^3+9m^2-31m-72\right)}{(m+9)(m+10)}>0,m\ge 4.\end{array}$$

Then, ${\theta }_1\left(r\right)$ is a strictly increasing function on $r>0$. Furthermore,

$$\underset{r\to 0}{lim}{\theta }_1\left(r\right)=\underset{r\to 0}{lim}\left[\left(-\gamma -2+ln\left(2\right)-\psi \left(1/2\right)\right)r^3+\left(\frac{3}{2}-\frac{{\pi }^2}{6}\right)r^4+O\left(r^5\right)\right]=0,$$

and

$$\underset{r\to \infty }{lim}{\theta }_1\left(r\right)=\underset{r\to \infty }{lim}\left[\frac{1}{3}-\frac{3}{2}{r}^{-1}+O\left(r^{-2}\right)\right]=\frac{1}{3},$$

where $\gamma =-{\mathrm{\Gamma }}^{\prime }\left(1\right)$ is the Euler–Mascheroni constant. Hence, $0<{\theta }_1\left(r\right)<\frac{1}{3}$ with sharp bounds. □

Lemma 2. 

The function

$${\theta }_2\left(r\right)=r^4\left[\frac{2}{r(1+r)}+ln\left(\frac{2+r}{r+1}\right)-\frac{1}{3r^3}-G\left(r\right)\right],r>0$$

(18)

is a strictly decreasing function with sharp bounds $\frac{-3}{2}<{\theta }_1\left(r\right)<0$.

Proof. 

Using the relation

$${\theta }_2\left(r\right)=-r^4{F}_1\left(r\right),r>0$$

we have

$$\frac{d}{dr}{\theta }_2\left(r\right)=-r^3{\int }_0^{\infty }\frac{{\kappa }_3\left(u\right)}{6{\left(e^u+1\right)}^{2}u}{e}^{-ru}du$$

where

$$\begin{array}{ccc}\hfill {\kappa }_3\left(u\right)& =& e^{4u}{u}^3+2e^{3u}\left(u^3-3u-12\right)+12(u+2)+6e^u(u+4)(2u+1)\hfill \\ & +& e^{2u}(u(u(u+24)+36)-24)\hfill \\ & =& 870912u^5+11197440u^6+93747456u^7+645470208u^8+3970944000u^9\hfill \\ & +& \sum _{m=10}^{\infty }\frac{a_m}{1728m!}{u}^{m}u>0\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill a_m& =& 4^m\left(27m^3-81m^2+54m\right)+3^m\left(128m^3-384m^2-3200m-41472\right)\hfill \\ & & +2^m\left(216m^3+9720m^2+21168m-41472\right)+72576m+41472+20736m^2\hfill \\ & >& 0,m\ge 10.\hfill \end{array}$$

Then, ${\theta }_2\left(r\right)$ is a strictly decreasing function on $r>0$. Furthermore,

$$\underset{r\to 0}{lim}{\theta }_2\left(r\right)=\underset{r\to 0}{lim}\left[r^4\left(-\gamma -2+ln\left(2\right)-\psi \left(1/2\right)\right)-\frac{r}{3}+O\left(r^5\right)\right]=0,$$

and

$$\underset{r\to \infty }{lim}{\theta }_2\left(r\right)=\underset{r\to \infty }{lim}\left[-\frac{3}{2}+\frac{21}{5r}+O\left(r^{-2}\right)\right]=-\frac{3}{2}.$$

Hence, $\frac{-3}{2}<{\theta }_1\left(r\right)<0$ with sharp bounds. □

Remark 1. 

The lower bound of (14) is better than the lower bound of (4) for $r>1.62$. Furthermore, the upper bound of (16) is better than the lower bound of (4) for $r>\frac{9}{2}$.

Remark 2. 

The lower bound of (14) is better than the lower bound of (5) for $r>1.2$.

Remark 3. 

The lower bound of (14) is better than the lower bound of (6) for $r>0.73$. Furthermore, the upper bound of (16) is better than the upper bound of (6) for $r>0.97$.

Remark 4. 

The lower bound of (14) is better than the lower bound of (8) for $0.86745<r<2.45$.

Remark 5. 

The Upper bound of (16) is better than the upper bound of (8) for $0<r<2.77879$.

3. Conclusions

The main conclusions of this paper are stated in Lemmas 1 and 2. Concretely speaking, the authors studied two approximations for Bateman’s G-function. The approximate formulas are characterized by one strictly increasing towards $G\left(r\right)$ as a lower bound, and the other strictly decreasing as an upper bound with the increases in r values. Furthermore, our new two-sided inequality for $G\left(r\right)$ improved some of the recently published results. The results enable us to obtain the bounds of some alternating series, some generalized hypergeometric functions, Wallis’s ratio, and some other functions.