I’ve just received a book named Development of Elliptic Functions According to Ramanujan
by K Venkatachaliengar (deceased) , edited by: ShaunCooper , Shaun Cooper

Development of Elliptic Functions According to Ramanujan

This unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter.

The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan’s work. It can be read by anyone with some undergraduate knowledge of real and complex analysis.

## Contents:

• The Basic Identity
• The Differential Equations of $$P$$, $$Q$$ and $$R$$
• The Jordan-Kronecker Function
• The Weierstrassian Invariants
• The Weierstrassian Invariants, II
• Development of Elliptic Functions
• The Modular Function $$\lambda$$

Readership: Graduate students and researchers in Number Theory and Classical Analysis, as well as scholars and aficionados of Ramanujan’s work.

## Review

It is obvious that every arithmetician should want to own a copy of this book, and every modular former should put it on his ‘to be handled-with-loving-care-shelf.’ Reader of Venkatachaliengar’s fine, fine book should be willing to enter into that part of the mathematical world where Euler, Jacobi, and Ramanujan live: beautiful formulas everywhere, innumerable computations with infinite series, and striking manouevres with infinite products.

— MAA Reviews

The author was acquainted with many who knew Ramanujan, and so historical passages offer information not found in standard biographical sources. The author has studied Ramanujan’s papers and notebooks over a period of several decades. His keen insights, beautiful new theorems, and elegant proofs presented in this monograph will enrich readers.

— MathSciNet

The author has studied Ramanujan’s papers and notebooks over a period of several decades. His keen insights, beautiful new theorems, and elegant proofs presented in this monograph will enrich readers. italic Zentralblatt MATH

— Zentralblatt MATH

• Series: Monographs in Number Theory (Book 6)
• Hardcover: 184 pages
• Publisher: World Scientific Publishing Company (September 28, 2011)
• Language: English
• ISBN-10: 9814366455
• ISBN-13: 978-9814366458

# 定义

• $$\sum\limits_{i=1}^k \overline{p_i} \geq \sum\limits_{i=1}^k \overline{q_i}$$, 当 $$k=1,2,\dotsc,n-1$$时;
•  $$\sum\limits_{i=1}^n p_i = \sum\limits_{i=1}^n q_i$$.

•  $$\int_a^xf(t)\mathrm{d}t \geq \int_a^x g(t)\mathrm{d}t$$, 当$$a<x<b$$时;
•  $$\int_a^bf(t)\mathrm{d}t = \int_a^bg(t)\mathrm{d}t$$.

$[p]= \frac{1}{n!} \sum_{\sigma \in S_n}\prod_{i=1}^nx_{\sigma(i)}^{p_i},$

# 主要结果

Majorization inequality (控制不等式)

1. 离散形式   设 $$p=(p_1,p_2,\dotsc,p_n), q=(q_1,q_2,\dotsc,q_n)$$,  所有的 $$p_i,q_i \in (a,b)$$. 若 $$p \succ q, \varphi(x)$$ 为区间 $$(a,b)$$ 内的凸函数(Convex function)，则下式为真 $$\sum_{i=1}^n\varphi(p_i)\geq \sum_{i=1}^n\varphi(q_i).$$
2. 积分形式    $$f(x), g(x)$$ 是区间 $$[a,b]$$ 上的递增函数, $$f \succ g, \varphi(x)$$ 在区间 $$a,b]$$ 上是连续凸函数，则下式为真 $$\int_a^b\varphi(f(x))\mathrm{d}x \geq \int_a^b\varphi(g(x))\mathrm{d}x.$$

Muirhead’s Inequality   设 $$x_1,x_2,\dotsc,x_n$$ 为正实数, $$p,q \in \Bbb R^n$$. 如果 $$p \succ q$$, 则 $$[p] \geq [q]$$; 并且当 $$p \ne q$$ 时, 等号成立当且仅当 $$x_1=x_2= \dotsc =x_n$$.

# 主要结果的证明

Majorization inequality 的离散形式证明  记 $$r_i=\frac{\varphi(q_i)-\varphi(p_i)}{q_i-p_i}$$, 则 $$r_i$$ 递减，这是因为 $$\varphi(x)$$ 是凸函数.

$$P_k=\sum_{i=1}^kp_i,Q_k=\sum_{i=1}^kq_i, k=1,2,\dotsc, n; P_0=0,Q_0=0.$$

$$\begin{split}\sum_{i=1}^n\varphi(p_i)-\sum_{i=1}^n\varphi(q_i) & = \sum_{i=1}^n(\varphi(p_i)-\varphi(q_i) ) \\& = \sum_{i=1}^n r_i (p_i-q_i) \\& = \sum_{i=1}^n r_i (P_i-P_{i-1}-Q_i+Q_{i-1}) \\& = \sum_{i=1}^n r_i (P_i-Q_i) – \sum_{i=1}^n r_i (P_{i-1}-Q_{i-1}) \\& =\sum_{i=1}^{n-1} r_i (P_i-Q_i) – \sum_{i=0}^{n-1} r_{i+1} (P_i-Q_i)\\&= \sum_{i=1}^{n-1}(r_i- r_{i+1})(P_i-Q_i),\end{split}$$

Muirhead’s Inequality 的证明不少, 先看一个比较传统的:

Muirhead’s Inequality 的归纳证明   先看 $$n=2$$的情况.

$$x^ny^m+x^my^n \geq x^py^q + x^qy^p,$$

$$\begin{split}x^ny^m+x^my^n & = wx^ny^m + (1-w)x^my^n +wx^my^n +(1-w)x^ny^m \\& \geq x^{wn} y^{wm}x^{(1-w)m}y^{(1-w)m} +x^{wm}y^{wn}x^{(1-w)n}y^{(1-w)m} \\& = x^{wn +(1-w)m}y^{wm+ (1-w)m}+ x^{wm+(1-w)n}y^{wn+(1-w)m} \\& = x^py^q + x^qy^p .\end{split}$$

Let $$\Bbb P$$ be the set of all primes, the set

$\{\sin p \mid p\in\Bbb P\}$

is dense  in $$[-1,1]$$.

This problem appeared in artofproblemsolving  on 22 May 2004, and  has been solved in a french math forum. The solution used tools as Fourier analysis, Vinogradov’s theorem, Erdos-Turan inequality and many others things.