论文标题
关于广义的Grötzsch环功能和广义HERSCH-PFLUGER失真功能的注释
Notes on Generalized Grötzsch Ring Function and Generalized Hersch-Pfluger Distortion Function
论文作者
论文摘要
对于$ a \ in（0,1）$，$ r \ in（0,1）$和$ k \ in（1，\ infty）$，让$μ__{a}（r）$和$φ_{k}^{a}^{a}（a}（r）$ be be Generalizedgrötzsch环形环函数和全身hersch hersch-plffluger-plortplort-plorts-plortplort distrortpluger-distrortpluger-distrorts Dermittern函数。在过去的几年中，功能$μ__{a}（r）$和$φ_{k}^{a}（r）$及其特殊情况$μ__{1/2}（r）$和$φ_{k} {k}^{1/2}（1/2}（r）$在Quasic的概述中起着非常重要的作用（R）。方程式。在本文中，我们提出了$μ_{a}（r）$的系列扩展，因此证明了功能$ r \ mapsto-- [μ_{a}（r） - \ log {（e^{r（a）/2}）/r}/r}] $在$（0,1）$上是绝对单调的。这里$ r（a）$是Ramanujan常数。此外，我们还研究了$φ__{k}^{a}（r）$的副本和功率副业属性，并在基本功能中为$φ_{k}^{a}（r）$建立一些新的不等式。
For $a\in(0,1)$, $r\in(0,1)$ and $K\in(1,\infty)$, let $μ_{a}(r)$ and $φ_{K}^{a}(r)$ be the generalized Grötzsch ring function and generalized Hersch-Pfluger distortion function. In the past few years, the functions $μ_{a}(r)$ and $φ_{K}^{a}(r)$, and their special cases $μ_{1/2}(r)$ and $φ_{K}^{1/2}(r)$ have been playing the very important role on the theory of quasiconformal mappings and (generalized) Ramanujan's modular equations. In this paper, we present a series expansion of $μ_{a}(r)$, and thus prove that the function $r\mapsto -[μ_{a}(r)-\log{(e^{R(a)/2})/r}]$ is absolutely monotonic on $(0,1)$. Here $R(a)$ is the Ramanujan constant. In addition, we also investigate the submultiplicative and power submultiplicative properties of $φ_{K}^{a}(r)$, and establish some new inequalities for $φ_{K}^{a}(r)$ in terms of elementary functions.
