# An Inequality with Constraint VI

### Statement

Let $0\lt x_k,$ $k=1,\dots,n$ such that $\displaystyle\sum_{k=1}^{n}x_k=1,$ $n\ge 1.$ Then

$\displaystyle\prod_{k=1}^{n}\frac{1+x_k}{x_k}\ge \prod_{k=1}^{n}\frac{n-x_k}{1-x_k}.$

### Solution

Consider function $\displaystyle f(x)=\ln\left(1+\frac{1}{x}\right).$ The function is convex on $(0,1).$ By Jensen's inequality, for any fixed $k=1,2,\ldots,n,$

$\displaystyle\frac{1}{n-1}\sum_{i=1,i\ne k}^{n}\ln\left(1+\frac{1}{x_i}\right)\ge \ln\left(1+\frac{n-1}{\displaystyle\sum_{i=1,i\ne k}^{n}x_i}\right)=\ln\left(\frac{n-x_{k}}{1-x_k}\right).$

In other words, $\displaystyle\prod_{i=1,i\ne k}^{n}\left(1+\frac{1}{x_i}\right)\ge\left(\frac{n-x_k}{1-x_k}\right)^{n-1}.$ The desired inequality is obtained on multiplying all $n\;$ of these inequalities up.

### References

- Xu Jiagu,
*Lecture Notes on Mathematical Olympiad Courses*, v 8, (For senior section, v 2), World Scientific, 2012, 88-89

### Inequalities with the Sum of Variables as a Constraint

- An Inequality with Constraint
- An Inequality with Constraints II
- An Inequality with Constraint V
- An Inequality with Constraint VI
- An Inequality with Constraint XI
- Monthly Problem 11199
- Problem 11804 from the AMM
- Sladjan Stankovik's Inequality With Constraint
- Sladjan Stankovik's Inequality With Constraint II
- An Inequality with Constraint V
- An Inequality with Constraint VI
- An Inequality with Constraint XI
- An Inequality with Constraint XII
- An Inequality with Constraint XIII
- Inequalities with Constraint XV and XVI
- An Inequality with Constraint XVII
- An Inequality with Constraint in Four Variables
- An Inequality with Constraint in Four Variables II
- An Inequality with Constraint in Four Variables III
- An Inequality with Constraint in Four Variables IV
- Inequality with Constraint from Dan Sitaru's Math Phenomenon
- An Inequality with a Parameter and a Constraint
- Cyclic Inequality with Square Roots And Absolute Values
- From Six Variables to Four - It's All the Same
- Michael Rozenberg's Inequality in Three Variables with Constraints
- Michael Rozenberg's Inequality in Two Variables
- Dan Sitaru's Cyclic Inequality in Three Variables II
- Dan Sitaru's Cyclic Inequality in Three Variables IV
- Dan Sitaru's Cyclic Inequality in Three Variables VI
- An Inequality with Arbitrary Roots
- Inequality 101 from the Cyclic Inequalities Marathon
- Sladjan Stankovik's Inequality With Constraint II
- An Inequality with Constraint in Four Variables
- An Inequality with Constraint in Four Variables IV
- Cyclic Inequality with Square Roots And Absolute Values
- From Six Variables to Four - It's All the Same
- Michael Rozenberg's Inequality in Two Variables
- Dan Sitaru's Cyclic Inequality in Three Variables II
- Inequality 101 from the Cyclic Inequalities Marathon

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