# Sharp inequality for real numbers

Let $n\geq 3$, $\frac{1}{2} > c_1 \geq c_2 \geq \dots \geq c_n \geq 0$ real, with $c_1 + \dotsc + c_n \leq 1$. Assume moreover that for $j = 1, \dotsc, n-2$ $$c_j < c_{j+1} + \dotsb + c_n$$ and define $$c_j^{*} = 1 - \frac{c_j}{c_{j+1} + \dotsb + c_n}.$$

Then (claim) $$c_1 + c_1^{*}c_2 + c_1^{*}c_2^{*}c_3 + \dotsb + c_1^{*}\dotsm c_{n-2}^{*}c_{n-1} < \frac{1}{2}.$$

Any suggestion/proof/counterexample is welcome!

## 1 Answer

By induction. I omit the base case ($n = 3$): I checked it and it works.

Suppose the hypothesis is true for $n-1$.

Let $x := c_2 + \dotsc + c_n$. We know that $c_2 < \frac{x}{2}$. By the induction hypothesis applied to the $(n-1)$-tuple $\bigl(\frac{c_2}{x},\dotsc ,\frac{c_n}{x}\bigr)$ we obtain

$$c_2 + c_2^{\ast} c_3 + \dotsc + c_2^{\ast}\dotsc c_{n-2}^{\ast} c_{n-1} < \frac{x}{2}.$$

Then

\begin{align}&\quad c_1+c_1^*c_2+\dotsc+c_1^*\dotsc c_{n-2}^*c_{n-1} \\ &=c_1+c_1^*\bigg(c_2+c_2^*c_3+\dotsc+c_2^*\dotsc c_{n-2}^*c_{n-1}\bigg) \\ &<c_1+c_1^*\frac{x}{2}\le\frac{1}{2}.\end{align}

• When $n=3$, we choose $c_1=c_2=c_3=\frac 1 3$, and then $c_1^*=\frac 1 2$. Thus, $$c_1+c_1^*c_2=\frac 1 3 + \frac 1 6=\frac 1 2$$ – Aforest Oct 18 '16 at 14:08
• What does ensure that choosing $c_1 = c_2 = c_3$ gives the maximum value? – Hugo Oct 18 '16 at 16:22