For $n \geq 2$, show that $\sum_{r = 1}^{n} r \sqrt{\binom{n}{r}} < \sqrt{2 ^ {n - 1} n ^ 3}$

Question

Good day,

Can someone help me with giving hints for this problem:

Show that for $n \geq 2, n \in \mathbb{Z}$, $$\sum_{r = 1}^{n} r \sqrt{\binom{n}{r}} < \sqrt{2 ^ {n - 1} n ^ 3}$$

I tried $$\sum_{r = 1}^{n} r \sqrt{\binom{n}{r}} = \sum_{r = 1}^{n} \sqrt{r ^ 2\binom{n}{r}}$$

$$= \sqrt{n} \sum_{r = 1}^{n} \sqrt{r\binom{n - 1}{r - 1}}$$

So, we need to prove $$\sum_{r = 1}^{n} \sqrt{r\binom{n - 1}{r - 1}} < n \sqrt{2 ^ {n - 1}}$$

This reminds me of $$\sum_{r = 0}^{n} (r + 1) \binom{n}{r} = n 2 ^ {n - 1} + 2 ^ n = 2 ^ {n - 1}(n + 2)$$

but I can't see how to use it.

Also, the only complex math inequality I know is AM-GM, so it's quite possible that it is my theoretical knowledge that is lacking. So, if a well-known inequality is used, please also mention it.

Thanks

Answer 1

By Cauchy-Schwarz, \begin{align*} \sum_{r = 1}^{n} r \sqrt{\binom{n}{r}} &\le \left(\sum_{r = 1}^{n} r^2\right)^{1/2}\left(\sum_{r = 1}^{n} \binom{n}{r}\right)^{1/2} \\ &= \left((2^n-1)\left(\frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}\right)\right)^{1/2} \\ &< \left(2^{n-1}\cdot2\left(\frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}\right)\right)^{1/2} \end{align*} Then, verify $2\left(\frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}\right) < n^3$ for $n \ge 4$ by induction while manually checking the $n=2, n=3$ cases.