To prove $1^1\cdot2^2\cdot 3^3...\cdot n^n<(\frac{2n+1}{3})^{\frac{n(n+1)}{2}} $

Question

So we have to prove the following for $n\in N $ $$1^1\cdot 2^2\cdot 3^3...\cdot n^n<\left(\frac{2n+1}{3}\right)^{\frac{n(n+1)}{2}} $$

So I used concept of weighted means (arithmetic and geometric) used AM GM inequality.

$$AM=\frac{a_1w_1+a_2w_2+...+a_nw_n}{w_1+w_2+...+w_n}$$

$$GM=(a_1^{w_1}\cdot a_2^{w_2}\cdot...\cdot a_n^{w_n})^{\frac{1}{w_1+w_2+...+w_n}}$$ So here I let $w_1=1, w_2=2^1,w_3=3^1..$ and of course $a_1=1,a_2=2^1,a_3=3^2...$

So we get: $$\frac{1^1+ 2^2+ 3^3...+ n^n}{\frac{n(n+1)}{2}}>(1^1\cdot 2^2\cdot 3^3...\cdot n^n)^{\frac{1}{\frac{n(n+1)}{2}}}$$

However on lhs, I cant deal with numerator, and I feel that if it can be simplified, I would get the answer. So please help or if possible suggest new method.

Answer 1

AMGM is the right idea, you just applied it wrong. As you say, the inequality is

$$ \frac{w_1x_1+\cdots+w_nx_n}{w_1+\cdots+w_n}\ge(x_1^{w_1}\cdots x_n^{w_n})^{1/(w_1+\cdots+w_n)}. $$

(If we define $p_k=w_k/(w_1+\cdots+w_n)$, this reads $p_1x_1+\cdots+p_nx_n\ge x_1^{p_1}\cdots x_n^{p_n}$.)

In your case, if you define $w_k=x_k=k$ for $k=1,\cdots,n$ the inquality becomes

$$ \frac{1^2+\cdots+n^2}{1+\cdots+n}\ge(1^1\cdots n^n)^{1/(1+\cdots+n)}. $$

Using $1+\cdots+n=\frac{n(n+1)}{2}$ and $1^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ you should be able to finish.

(After writing this I read up and apparently Giant Ray pointed this out in the comments.)

Answer 2

Use the AM-GM inequality [where there are $\frac{n(n+1)}{2}$ terms; indeed for each positive integer $i \le n$, there are $i$ terms of $i$]:

$$\frac{\sum_{1=1}^n i^2}{n(n+1)/2} \ \ge \ \sqrt[\frac{n(n+1)}{2}]{1^12^2 \cdots n^n},$$ where in the above inequality, the LHS represents the arithmetic mean of the above terms and the RHS the geometric mean of the above terms.

However, the equation $$\frac{\sum_{1=1}^n i^2}{n(n+1)/2} = \frac{(n)(n+1)(2n+1)}{6} \times \frac{2}{n(n+1)}$$ $$ = \frac{2n+1}{3} $$ also holds. So then combining this string of equations with the top AM-GM inequality, yields the inequality $$\frac{2n+1}{3} \ \ge \ \sqrt[\frac{n(n+1)}{2}]{1^12^2 \cdots n^n} \ .$$

Raising each side of this to the $\frac{n(n+1)}{2}$-power yields the desired result.

Answer 3

This is not a proof since working for large values of $n$ $$\prod_{k=1}^n k^k=H(n)$$ where $H(n)$ is the hyperfactorial function.

Expanding its logarithm $$\log (H(n))=-\frac 14 n^2+\frac 1{12} \left(6 n^2+6 n+1\right)\log(n)+\log (A)+\frac{1}{720 n^2}\left(1-\frac{1}{7 n^2}+\frac{1}{14 n^4}+O\left(\frac{1}{n^6}\right) \right)$$ Doing the same for the logarithm of the rhs $$\log\left(\frac{\text{rhs}}{\text{lhs}}\right)=\log \left(\frac{4 e}{9}\right)\,\frac{n(n+1)}4-\frac 1{12}\log(n)+\left(\frac{3}{16}-\log (A)\right)+O\left(\frac{1}{n}\right)$$