# Sladjan Stankovik's Inequality With Constraint II

### Problem

### Solution

We need to prove that $3(a^4+b^4+c^4+d^4)+12abcd\ge 24.\,$ By Surányi's Inequality,

$3(a^4+b^4+c^4+d^4)+4abcd\ge (a+b+c+d)(a^3+b^3+c^3+d^3).$

By the Cauchy-Schwarz inequality,

$3(a+b+c+d)(a^3+b^3+c^3+d^3)\ge (a^2+b^2+c^2+d^2)^2.$

Thus, suffice it to show that

$(a^2+b^2+c^2+d^2)^2+8abcd\ge 24.$

Since $a+b+c+d=4,\,$ $a^2+b^2+c^2+d^2=4(1+3t^2),\,$ with $0\le t\le 1.\,$ Hence, we need to prove that

$2(1+3t^2)^2+abcd\ge 3.$

By one of Leonard Giugiuc and Daniel Sitaru's results (Problem 4121, *Crux Mathematicorum*), for $0\le t\le\displaystyle\frac{1}{3},\,$

$abcd\ge (1+t)^3(1-3t)=-3t^4-8t^3-6t^2+1.$

Hence, in this case, suffice it to show that

$2(1+3t^2)^2-3t^4-8t^3-6t^2+1\ge 3.$

But this is equivalent to $t^2(15t^2-8t+6)\ge 0,\,$ which is true.

For $t\ge\displaystyle\frac{1}{3},\,$ $2(1+3t^2)^2+abcd\ge 2(1+3t^2)^2\ge\displaystyle\frac{32}{9}\ge 3.\,$ The proof is complete.

### Acknowledgment

The problem and the above solution have been kindly communicated to me by Leo Giugiuc.

### 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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