Recently I was trying to prove AM-GM Inequality using several different methods. When it came to Karamata’s Inequality, the proof that flashed through my mind was so simple and amazing, but it came out to be the opposite sign. Thus I wondered is it capable?
The following is my basic thought:
- Karamata’s Inequality
First, we introduce Karamata’s Inequality: Let $a=${$a_i$}$^n_{i=1}$ and $b=${$b_i$}$^n_{i=1}$ be two finite sequences of real numbers. $a$ majorizes $b$: $a\succ b$ if the two sequences satisfy $$1^\circ\space\ a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n$$ $$2^\circ \space\ a_1≥a_2≥\cdots≥a_n\space and\space b_1≥b_2≥\cdots≥b_n $$ $$3^\circ \space\ \forall k\in[1,n-1]\cap Z,\space a_1+a_2+\cdots+a_k≥b_1+b_2+\cdots+b_k$$ If $a=${$a_i$}$^n_{i=1}$ and $b=${$b_i$}$^n_{i=1}$ are two finite sequences from an interval $I$ satisfying $a\succ b$, and if $f:I\mapsto R$ is a convex function, then Karamata’s Inequality $$\sum^n_{i=1}f(a_i)≥\sum^n_{i=1}f(b_i)$$ holds. The equality holds if and only if $a_i=b_i$ for $i=1,2,\cdots,n$.
This is a definition from Algebraic Inequality by Titu Andreescu
- Proof of AM-GM Inequality
$$\frac{x_1+x_2+\cdots+x_n}n ≥(x_1x_2\cdots x_n)^{\frac1n}$$
Let $S=x_1+x_2+\cdots+x_n$, then its obvious that {$x_1,x_2,\cdots,x_n$}$\succ${$\frac Sn,\frac Sn,\cdots,\frac Sn$}
We rewrite the $RHS$ of AM-GM Inequality $$RHS=(x_1x_2\cdots x_n)^{\frac 1n}=e^{ln(x_1x_2\cdots x_n)^{\frac 1n}}=e^{\frac 1n(lnx_1+lnx_2+\cdots lnx_n)}$$ then let $f(x)=lnx$, we have $f’(x)=\frac1x$ and $f’’(x)=-\frac1{x^2}<0$, thus $f(x)$ is convex and $$\sum^n_{i=1}ln{x_i}≥\sum^n_{i=1}ln{\frac Sn}$$ which is equivalent to $$lnx_1+lnx_2+\cdots+lnx_n≥ln{\frac Sn}+ln{\frac Sn}+\cdots+ln{\frac Sn}=nln{\frac Sn}$$ therefore $$RHS=e^{\frac 1n(lnx_1+lnx_2+\cdots+lnx_n)}≥e^{\frac 1n\cdot nln{\frac Sn}}=e^{ln{\frac Sn}}=\frac Sn$$ which is $$(x_1x_2\cdots x_n)^{\frac 1n}≥\frac{x_1+x_2+\cdots+x_n}{n}$$
This has an opposite sign against AM-GM Inequality and here is where I become confused. Where did I go wrong? Or is it capable to prove AM-GM Inequality suing Karamata’s only?