﻿ 广义GrO¨tzsch环函数的精确界及应用 Sharp Inequalities and Application of the Generalized GrO¨tzsch Ring Function

Pure Mathematics
Vol. 09  No. 03 ( 2019 ), Article ID: 30067 , 5 pages
10.12677/PM.2019.93032

Sharp Inequalities and Application of the Generalized Grötzsch Ring Function

Fei Wang1*, Peigui Zhou2, Xiaoyu Wang1

1Teaching Section of Mathematics, Zhejiang Institute of Mechanical and Electrical Engineering, Hangzhou Zhejiang

2Keyi College of Zhejiang Sci-Tech University, Shangyu Zhejiang

Received: Apr. 9th, 2019; accepted: Apr. 20th, 2019; published: May 5th, 2019

ABSTRACT

In this paper, we study some monotonicity properties of certain functions defined in term of generalized Grötzsch ring function and some elementary functions, and get new sharp inequalities. Furthermore, we also obtain the lower bound of generalized Hersch-Pfluger distortion function by applying these results in Ramanujan modular equation theory.

Keywords:Generalized Grötzsch Ring Function, Sharp Inequality, Ramanujan Modular Equations, Generalized Hersch-Pfluger Distortion Function

1浙江机电职业技术学院数学教研室，浙江 杭州

2浙江理工大学科技与艺术学院，浙江 上虞

1. 引言

$\Gamma \left(x\right)={\int }_{0}^{\infty }{t}^{x-1}{\text{e}}^{-t}\text{d}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}B\left(x,y\right)=\frac{\Gamma \left(x\right)\Gamma \left(y\right)}{\Gamma \left(x+y\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\psi \left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$ (1)

$\gamma =\underset{n\to \infty }{\mathrm{lim}}\left[\underset{k=1}{\overset{n}{\sum }}\frac{1}{k}-\mathrm{log}n\right]=\text{0}\text{.57721566}\cdots$，是Euler-Mascheroni常数，则

$\psi \left(1\right)=-\gamma$ , $\psi \left(1/2\right)=-\gamma -\mathrm{log}4$ (2)

$\left(0,\infty \right)×\left(0,\infty \right)$ 上定义Ramanujan常数 [2] [3] 为：

$R\left(a,b\right)=-\psi \left(a\right)-\psi \left(b\right)-2\gamma$ (3)

$b=1-a$ 时，式(3)记为

$R\left(a\right)=R\left(a,1-a\right)=-\psi \left(a\right)-\psi \left(1-a\right)-2\gamma$ ,

$F\left(a,b;c;z\right)={}_{2}F{}_{1}\left(a,b;c;z\right)=\underset{n=0}{\overset{\infty }{\sum }}\frac{\left(a,n\right)\left(b,n\right)}{\left(c,n\right)}\frac{{z}^{n}}{n!}$ , $|z|<1$ (4)

$a\in \left(0,1\right)$，第一类、第二类广义椭圆积分分别定义 [5] 为：

${K}_{a}={K}_{a}\left(r\right)=\frac{\text{π}}{2}F\left(a,1-a;1;{r}^{2}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{K}^{\prime }}_{a}={{K}^{\prime }}_{a}\left(r\right)={K}_{a}\left({r}^{\prime }\right)$ (5)

${E}_{a}={E}_{a}\left(r\right)=\frac{\text{π}}{2}F\left(a-1,1-a;1;{r}^{2}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{E}^{\prime }}_{a}={{E}^{\prime }}_{a}\left(r\right)={E}_{a}\left({r}^{\prime }\right)$ (6)

${\mu }_{a}\left(r\right)=\frac{\text{π}}{2\mathrm{sin}\left(a\text{π}\right)}\frac{{{K}^{\prime }}_{a}\left(r\right)}{{K}_{a}\left(r\right)}$ (7)

$\frac{F\left(a,1-a;1;1-{s}^{2}\right)}{F\left(a,1-a;1;{s}^{2}\right)}=p\frac{F\left(a,1-a;1;1-{r}^{2}\right)}{F\left(a,1-a;1;{r}^{2}\right)}$ (8)

${\mu }_{a}\left(s\right)=p{\mu }_{a}\left(r\right),p>0$ (9)

$s={\phi }_{K}\left(a,r\right)={\mu }_{a}^{-1}\left({\mu }_{a}\left(r\right)/K\right)$ , $K=1/p$ (10)

$\mu \left(r\right)+{C}_{1}\left(1-{r}^{2}\right)\le {\mu }_{a}\left(r\right)\le \mu \left(r\right)+{C}_{1}-\left(a-1/2\right){r}^{2}$

$a=1/2$ 时等号成立，其中 ${C}_{1}=\left[R\left(a\right)-\mathrm{log}16\right]/2$

${\phi }_{1/K}\left(a,r\right)>{r}^{K}\mathrm{exp}\left\{\left(1-K\right)c\left(r\right)\right\}$ (11)

2. 引理

$\frac{\text{d}{K}_{a}}{\text{d}r}=2\left(1-a\right)\frac{{E}_{a}-{{r}^{\prime }}^{2}{K}_{a}}{r{{r}^{\prime }}^{2}}$ , $\frac{\text{d}{E}_{a}}{\text{d}r}=2\left(1-a\right)\frac{{E}_{a}-{K}_{a}}{r}$ (12)

$\frac{\text{d}{\mu }_{a}\left(r\right)}{\text{d}r}=-\frac{{\text{π}}^{2}}{4r{{r}^{\prime }}^{2}{K}_{a}^{2}}$ (13)

$F\left(x\right)=\frac{f\left(x\right)-f\left(a\right)}{g\left(x\right)-g\left(a\right)}$

$G\left(x\right)=\frac{f\left(x\right)-f\left(b\right)}{g\left(x\right)-g\left(b\right)}$

(1) $\frac{{\text{π}}^{2}/4-{r}^{\prime }{K}_{a}^{2}\left(r\right)}{{r}^{2}}$$\left(0,1\right)$$\left({\text{π}}^{2}\left[{a}^{2}+{\left(1-a\right)}^{2}\right]/4,{\text{π}}^{2}/4\right)$ 上严格单调递增。

(2) ${r}^{\prime }{K}_{a}\left(r\right)$$\left(0,1\right)$$\left(0,\text{π}/2\right)$ 上严格单调递减。

3. 主要结果及证明

$F\left(r\right)=\frac{{C}_{1}-\mathrm{exp}\left[{\mu }_{a}\left(r\right)-arth\left({r}^{\prime }\right)\right]}{{r}^{2}}$

$\left(0,1\right)$$\left({\left(1/2-a\right)}^{2}{C}_{1},{C}_{1}-1\right)$ 上严格单调递增。特别地，当 $0 时，

$\mathrm{log}\left[{C}_{1}-\left({C}_{1}-1\right){r}^{2}\right]+\mathrm{log}\left(1+{r}^{\prime }\right)<{\mu }_{a}\left(r\right)+\mathrm{log}r<\mathrm{log}{C}_{1}+\mathrm{log}\left[1-{\left(1/2-a\right)}^{2}{r}^{2}\right]+\mathrm{log}\left(1+{r}^{\prime }\right)$ (14)

$\begin{array}{c}\frac{{{f}^{\prime }}_{1}\left(r\right)}{{{f}^{\prime }}_{2}\left(r\right)}=\frac{1}{2}\frac{\mathrm{exp}\left[{\mu }_{a}\left(r\right)-arth\left({r}^{\prime }\right)\right]}{{r}^{\prime }}\frac{{\text{π}}^{2}/4-{r}^{\prime }{K}_{a}^{2}}{{r}^{2}}\frac{1}{{r}^{\prime }{K}_{a}^{2}}\\ =\frac{1}{2}\mathrm{exp}\left[{\mu }_{a}\left(r\right)-arth\left({r}^{\prime }\right)+\mathrm{log}\frac{1}{{r}^{\prime }}\right]\frac{{\text{π}}^{2}/4-{r}^{\prime }{K}_{a}^{2}}{{r}^{2}}\frac{1}{{r}^{\prime }{K}_{a}^{2}}\\ =\frac{1}{2}\mathrm{exp}\left[{\mu }_{a}\left(r\right)+\mathrm{log}\frac{r}{{r}^{\prime }}-\mathrm{log}\left(1+{r}^{\prime }\right)\right]\frac{{\text{π}}^{2}/4-{r}^{\prime }{K}_{a}^{2}}{{r}^{2}}\frac{1}{{r}^{\prime }{K}_{a}^{2}}\\ ={f}_{3}\left(r\right)\end{array}$

$\underset{r\to {0}^{+}}{\mathrm{lim}}F\left(r\right)=\underset{r\to {0}^{+}}{\mathrm{lim}}\frac{{{f}^{\prime }}_{1}\left(r\right)}{{{f}^{\prime }}_{2}\left(r\right)}={\left(1/2-a\right)}^{2}{\text{e}}^{\left(R\left(a\right)-\mathrm{log}4\right)/2}={\left(1/2-a\right)}^{2}{C}_{1}$ .

$G\left(r\right)=\frac{\mathrm{exp}\left[{C}_{2}-\left({\mu }_{a}\left(r\right)-\mu \left(r\right)\right)\right]-1}{{r}^{2}}$

$\left(0,1\right)$$\left(\alpha ,\beta \right)$ 上严格单调递增。特别地，当 $0 时，

$\frac{R\left(a\right)-\mathrm{log}16}{2}-\mathrm{log}\left(1+\alpha {r}^{2}\right)<{\mu }_{a}\left(r\right)-\mu \left(r\right)<\frac{R\left(a\right)-\mathrm{log}16}{2}-\mathrm{log}\left(1+\beta {r}^{2}\right)$ (15)

$\frac{{{g}^{\prime }}_{1}\left(r\right)}{{{g}^{\prime }}_{2}\left(r\right)}=\frac{{\text{π}}^{2}}{8}\mathrm{exp}\left\{{C}_{2}-\left[{\mu }_{a}\left(r\right)-\mu \left(r\right)\right]\right\}\frac{K+{K}_{a}}{{r}^{\prime }{K}^{2}}\frac{1}{{r}^{\prime }{K}_{a}^{2}}\frac{K-{K}_{a}}{{r}^{2}}={g}_{3}\left(r\right)$

$\underset{r\to {0}^{+}}{\mathrm{lim}}G\left(r\right)=\underset{r\to {0}^{+}}{\mathrm{lim}}\frac{{{g}^{\prime }}_{1}\left(r\right)}{{{g}^{\prime }}_{2}\left(r\right)}={\left(1/2-a\right)}^{2}=\alpha$ .

${\phi }_{1/K}\left(a,r\right)>{r}^{K}{\left\{{\text{e}}^{\left[R\left(a\right)-\mathrm{log}4\right]/2}\left[1-{\left(1/2-a\right)}^{2}{r}^{2}\right]\left(1+{r}^{\prime }\right)\right\}}^{1-K}$ (16)

${\phi }_{1/K}\left(r\right)>{r}^{K}{\left\{2\left(1+{r}^{\prime }\right)\right\}}^{1-K}$ (17)

Sharp Inequalities and Application of the Generalized GrO¨tzsch Ring Function[J]. 理论数学, 2019, 09(03): 254-258. https://doi.org/10.12677/PM.2019.93032

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NOTES

*通讯作者。