I would like to ask about WLOG assumptions in inequalities.
I know that in symmetric inequality we can WLOG assume $a\ge b\ge c$, in cyclic inequality we can WLOG assume $a=\max(a,b,c)$.
What about assumptions like $a+b+c=1$ or $abc=5$ ?
When is it allowed? could you show me the rule which is used when we do something like this?
thanks in advance
Best Answer
This is not going to be a truly in-depth answer about homogeneity, but I think it should be enough for you.
The rest of the answer assumes the variables of your inequality are positive.
You can assume the things you mention when an inequality is homogenous, and that is when both LHS and RHS are of the same degree.
The degree of, e.g., $x^ay^b$, where $x,y$ are any variables and $a,b$ are any real constants, is $a+b$, while when you have addition of functions, we can name the degree of $f(x)+g(x)$ iff the degree of $f(x)$ is equal to the degree of $g(x)$, and the degree of the sum is then the degree of either $f(x)$ or $g(x)$.
When the inequality is homogenuous (and we rearrange it so that the RHS is $0$ and call the LHS $f(x)$), we know that $f(tx_1,tx_2,\ldots,tx_n)=t^df(x_1,x_2,\ldots,x_n)$, where $d$ is the degree of the function $f(x_1,\ldots,x_n)$. But we can divide both sides of the inequality by $t^d$ and the inequality does not change, as we assume that $t$ is a positive real number, which leaves the variables positive.
Thus $$f(tx_1,tx_2,\ldots,tx_n)\stackrel{\le}\ge 0$$ is equivalent to $$f(x_1,x_2,\ldots,x_n)\stackrel{\le}\ge 0$$
Call this result $(1)$.
Now, we know that there exists a particular number the sum of the variables is equal to, and it is $X_1+X_2+\cdots+X_n=s$. Since the variables are all positive, $s$ must also be positive. If any positive sum of the variables makes the inequality true, then any combination of positive variables makes it true too.
Now consider that you want to investigate the inequality when the sum is equal to a positive real constant $w$ (in your question description you wanted it to be $1$). Then re-write your variables as $X_1=tx_1, X_2=tx_2,\ldots, X_n=tx_n$, where $\frac{s}{t}=w$ and $t$ is positive.
If you find that the inequality is true when the variables are $X_1=x_1,X_2=x_2,\ldots,X_n=x_n$ so that $$X_1+X_2+\cdots+X_n=x_1+x_2+\cdots+x_n=w,$$ then because of $(1)$ the inequality must also be true when the variables are $X_1=tx_1,X_2=tx_2,\ldots,X_n=tx_n$ so that $$X_1+X_2+\cdots+X_n=t'x_1+t'x_2+\cdots+t'x_n=t'w=s$$
This means that if your inequality is true when the variables sum up to a positive real number $w$ (i.e. the number you want the sum to be for your assumption), then it must also be true when the variables sum up to any other positive real number $s$, and if any possible sum that you can create makes the inequality true, then the inequality must always be true.
We can prove it similarly with other kinds of expressions
(e.g., $abc=5, ab+bc+ca=5, a^2+b^2+c^2=5$, etc.).