I was reading Zdravko Cvetkovski's excellent book Inequalities: Theorems, Techniques, and selected problems, when I arrived at the $16$th chapter: the $ABC-$Method. I had some questions related to this topic, which I could post separately; yet, since they are closely related, I decided to share them at the same time. The first sentence of the section reads
In this section we will present three theorems without proofs (the proofs can be found in Diamonds in Mathematical Inequalities (Poung T., $2007$)) which are the basis of a very useful method [...]
This was the first problem: it's almost impossible to find the referenced book — at least, I have failed doing so —. The Mathlinks-user @bitrak claims here that «this book is gift for all contestants on IMO $2007$, [...] in Hanoi-Vietnam. I don't have this book, because all printed books have only one edition». (Q0: Does anybody know if it's possible to get one copy or pdf-version?)
Q1: So, I would be very grateful if someone could share the proof of the theorem:
For this purpose we’ll consider the function $f(abc,ab+bc+ca,a+b+c)$ as a one-variable function with variable $abc$ on $\mathbb R$, i.e. on $\mathbb R^+$.
Theorem 16.1 If the function $f(abc,ab+bc+ca,a+b+c)$ is monotonic, then $f$ achieves its maximum and minimum values on $\mathbb R$ when $(a-b)(b-c)(c-a)=0$, and on $\mathbb R^+$ when $(a-b)(b-c)(c-a)=0$ or $abc=0$.
Consequence 16.1 Let $f(abc,ab+bc+ca,a+b+c)$ be a linear function with variable $abc$. then $f$ achieves its maximum and minimum values on $\mathbb R$ if and only if $(a-b)(b-c)(c-a)=0$, and on $\mathbb R^+$ if and only if $(a-b)(b-c)(c-a)=0$ or $abc=0$.
Also, there are some aspects of this formulation I do not really understand.
Q2: First of all, what does the author mean when he states that «we'll consider the function $f(abc,ab+bc+ca,a+b+c)$ as a one-variable function with variable $abc$»? Should I only consider the parts of $f$'s term where $abc$ appears, and think of $ab+bc+ca, a+b+c$ as if they were constant, or are there any further conditions on them? (This question is especially inspired by this comment on the previously linked mathlinks-post.)
Q3: And what does it mean for $f(abc,ab+bc+ca,a+b+c)$ to be monotonic wrt $abc$?
Finally, I noticed that, while Cvetkovski uses «when» in the statement of the Theorem, this becomes «if and only if» in its consequence.
Q4: Are they both meant to be «if and only if», or is there a reason for the Theorem to be stated with «when», and the consequence with «if and only if»? If so, what's the reason?
Thanks in advance :)
For more information about the $ABC-$Method, have a look here. If you know Vietnamese — I don't —, this seems to be the first article published regarding the $ABC-$Method.
Update: @Martin Hansen found «a copy of "Diamonds in Mathematical Inequalities" on the number wonder website for pupils preparing for the BMO.» Thanks! It is however unclear, if there was a more developed version...
