# Dan Sitaru's Cyclic Inequality in Three Variables IV

### Problem

Prove that if $x,y,z\gt 0$ and $x+y+z=3\,$ then

$\displaystyle \sum_{cycl}\frac{(x+y)z}{\sqrt{4x^2+xy+4y^2}}\le 2.$

### Solution

$\displaystyle \begin{align} 4x^2+xy+4y^2 &= 2[(x+y)2+(x-y)^2]+\frac{1}{4}[(x+y)^2-(x-y)2)]\\ &=\frac{9}{4}(x+y)^2+\frac{7}{4}(x-y)^2\ge\frac{9}{4}(x+y)^2\\ &\Rightarrow\;\sqrt{4x^2+xy+4y^2}\ge \frac{3}{2}(x+y)\\ &\Rightarrow\;\frac{(x+y)z}{\sqrt{4x^2+xy+4y^2}}\le \frac{2}{3}z\\ &\Rightarrow\;\sum_{cycl}\frac{(x+y)z}{\sqrt{4x^2+xy+4y^2}}\le \frac{2}{3}\sum_{cycl}z=2. \end{align}$

Equality is attained for $x=y=z=1.$

### Acknowledgment

Dan Sitaru has kindly posted problem of his at the CutTheKnotMath facebook page, with a solution by Ravi Prakash. The problem appeared earlier at the Romanian Mathematical Magazine.

### 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
- 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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