# A Refinement of Turkevich's Inequality

### Problem

### Solution 1

Turkevich's inequality states:

If $a,b,c,d\ge 0,\,$ then

$a^2+b^2+c^2+d^2+2\sqrt{abcd}\ge ab+bc+cd+da+ac+bd.$

Why is the problem a refinement? Because, by the AM-GM inequality, $\displaystyle 2\sqrt{abcd}\ge\frac{32abcd}{(a+b+c+d)^2}.$

Assume, WLOG, $a+b+c+d=4.\,$ Clearly, $\displaystyle \sum_{all}ab=6(1-t^2),\,$ with $t\in [0,1].\,$ From here, $\displaystyle \sum_{cycl}a^2=4(1+3t^2).\,$ We need to prove that $2(1+3t^2)+abcd\ge 3(1-t^2).$

I (AB: Leo Giugiuc) have shown very often that

$\min (abcd)=\begin{cases} (1+t)^3(1-3t),& if\;\displaystyle 0\le t\le \frac{1}{3}\\ 0,& if\; \displaystyle \frac{1}{3}\le t \le 1. \end{cases}$

$\mathbf{Case\,1:\,\displaystyle 0\le t\le \frac{1}{3}}$

Suffice it to show that $2(1+3t^2)+(1+t^3)(1-3t)\ge 3(1-t^2),\,$ which is equivalent to $(1=t)^3(1-3t)\ge(1+3t)(1-3t),\,$ or, $(1+t)^3\ge 1+3t.\,$ The latter is obviously true.

$\mathbf{Case\,2:\,\displaystyle \frac{1}{3}\le t \le 1}$

Suffice it to show that $2(1+t^2)\ge 3(1-t^2),\,$ which is equivalent to $0\ge (1+3t)(1-3t),\,$ which is true.

Let's remark that equality holds at $(a,a,a,a)\,$ and $(a,a,a,0)\,$ and permutations, $a\gt 0.$

### Solution 2

We start with an observation that

$\displaystyle\begin{align} &\left(\sum_{sym}a^2\right)\left(\sum_{sym}a\right)^2+32abcd-\left(\sum_{sym}a\right)\left(\sum_{sym}ab\right)\\ &\qquad\qquad+\sum_{sym}ab(a^2+b^2)-[(ab+cd)(ac+bd)\\ &\qquad\qquad\qquad+(ab+cd)(ad+bc)+(ad+bc)(ac+bd)]. \end{align}$

We thus have that the required inequality is equivalent to

(1)

$\displaystyle\begin{align} &\sum_{sym}a^4+20abcd+\sum_{sym}ab(a^2+b^2)\ge 3[(ab+cd)(ac+bd)\\ &\qquad\qquad\qquad+(ab+cd)(ad+bc)+(ad+bc)(ac+bd)]. \end{align}$

We are going to make use of another of Turkevich's inequalities:

(2)

$\displaystyle \sum_{sym}a^4+2abcd\ge\sum_{sym}a^2b^2.$

Using the AM-GM inequality,

(3)

$\displaystyle \sum_{sym}ab(a^2+b^2)\ge 2\sum_{sym}a^2b^2.$

From (2)&(3),

$\displaystyle \begin{align} &\sum_{sym}a^4+20abcd+\sum_{sym}ab(a^2+b^2)\ge 18abcd+\sum_{sym}a^2b^2+2\sum_{sym}a^2b^2\\ &\qquad\qquad=3[(ab+cd)^2+(ad+bc)^2+(ac+bd)^2]\\ &\qquad\qquad\ge 3[(ab+cd)(ac+bd)+(ab+cd)(ad+bc)+(ad+bc)(ac+bd)] \end{align}$

so that (1) does imply the required inequality.

### Acknowledgment

The problem and Solution 1 are by Leo Giugiuc who kindly posted the problem at the CutTheKnotMath facebook page and then also mailed me his solution. I could not be more appreciative. Solution 2 is by Marian (Gabi Cuc) Cucoaneş.

- An Inequality for Grade 8
- An Extension of the AM-GM Inequality
- Schur's Inequality
- Newton's and Maclaurin's Inequalities
- Rearrangement Inequality
- Chebyshev Inequality
- Jensen's Inequality
- Muirhead's Inequality
- Bergström's inequality
- Radon's Inequality and Applications
- Jordan and Kober Inequalities, PWW
- A Mathematical Rabbit out of an Algebraic Hat
- An Inequality With an Infinite Series
- An Inequality: 1/2 * 3/4 * 5/6 * ... * 99/100 less than 1/10
- A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
- An Inequality: Easier to prove a subtler inequality
- Inequality with Logarithms
- An inequality: 1 + 1/4 + 1/9 + ... less than 2
- Inequality with Harmonic Differences
- An Inequality by Uncommon Induction
- Hlawka's Inequality
- An Inequality in Determinants
- Application of Cauchy-Schwarz Inequality
- An Inequality from Tibet
- An Inequality with Constraint
- An Inequality from Morocco
- An Inequality for Mixed Means
- An Inequality in Integers
- An Inequality in Integers II
- An Inequality in Integers III
- An Inequality with Exponents
- Exponential Inequalities for Means
- A Simple Inequality in Three Variables
- An Asymmetric Inequality
- Linear Algebra Tools for Proving Inequalities
- An Inequality with a Generic Proof
- A Generalization of an Inequality from a Romanian Olympiad
- Area Inequality in Trapezoid
- Improving an Inequality
- RomanoNorwegian Inequality
- Inequality with Nested Radicals II
- Inequality with Powers And Radicals
- Inequality with Two Minima
- Simple Inequality with Many Faces And Variables
- An Inequality with Determinants
- An Inequality with Determinants II
- An Inequality with Determinants III
- An Inequality with Determinants IV
- An Inequality with Determinants V
- An Inequality with Determinants VI
- An Inequality with Determinants VII
- An Inequality in Reciprocals
- An Inequality in Reciprocals II
- An Inequality in Reciprocals III
- Monthly Problem 11199
- A Problem from the Danubius Contest 2016
- A Problem from the Danubius-XI Contest
- An Inequality with Integrals and Rearrangement
- An Inequality with Cot, Cos, and Sin
- A Trigonometric Inequality from the RMM
- An Inequality with Finite Sums
- Hung Viet's Inequality
- Hung Viet's Inequality II
- Hung Viet's Inequality III
- Inequality by Calculus
- Dorin Marghidanu's Calculus Lemma
- An Area Inequality
- A 4-variable Inequality from the RMM
- An Inequality from RMM with Powers of 2
- A Cycling Inequality with Integrals
- A Cycling Inequality with Integrals II
- An Inequality with Absolute Values
- An Inequality from RMM with a Generic 5
- An Elementary Inequality by Non-elementary Means
- Inequality in Quadrilateral
- Marian Dinca's Refinement of Nesbitt's Inequality
- An Inequality in Cyclic Quadrilateral
- An Inequality in Cyclic Quadrilateral II
- An Inequality in Cyclic Quadrilateral III
- An Inequality in Cyclic Quadrilateral IV
- Inequality with Three Linear Constraints
- Inequality with Three Numbers, Not All Zero
- An Easy Inequality with Three Integrals
- Divide And Conquer in Cyclic Sums
- Wu's Inequality
- A Cyclic Inequality in Three Variables
- Dorin Marghidanu's Inequality in Complex Plane
- Dorin Marghidanu's Inequality in Integer Variables
- Dorin Marghidanu's Inequality in Many Variables
- Dorin Marghidanu's Inequality in Many Variables Plus Two More
- Dorin Marghidanu's Inequality with Radicals
- Dorin Marghidanu's Light Elegance in Four Variables
- Dorin Marghidanu's Spanish Problem
- Two-Sided Inequality - One Provenance
- An Inequality with Factorial
- Wonderful Inequality on Unit Circle
- Quadratic Function for Solving Inequalities
- An Inequality Where One Term Is More Equal Than Others
- An Inequality and Its Modifications
- Complicated Constraint - Simple Inequality
- Distance Inequality
- Two Products: Constraint and Inequality
- The power of substitution II: proving an inequality with three variables
- Algebraic-Geometric Inequality
- One Inequality - Two Domains
- Radicals, Radicals, And More Radicals in an Inequality
- An Inequality in Triangle and In General
- Cyclic Inequality with Square Roots
- Dan Sitaru's Cyclic Inequality In Many Variables
- An Inequality on Circumscribed Quadrilateral
- An Inequality with Fractions
- An Inequality with Complex Numbers of Unit Length
- An Inequality with Complex Numbers of Unit Length II
- Le Khanh Sy's Problem
- An Inequality Not in Triangle
- An Acyclic Inequality in Three Variables
- An Inequality with Areas, Norms, and Complex Numbers
- Darij Grinberg's Inequality In Three Variables
- Small Change Makes Big Difference
- Inequality with Two Variables? Think Again
- A Problem From a Mongolian Olympiad for Grade 11
- Sitaru--Schweitzer Inequality
- An Inequality with Cyclic Sums And Products
- Problem 1 From the 2016 Pan-African Math Olympiad
- An Inequality with Integrals and Radicals
- Twin Inequalities in Four Variables: Twin 1
- Twin Inequalities in Four Variables: Twin 2
- Simple Inequality with a Variety of Solutions
- A Partly Cyclic Inequality in Four Variables
- Dan Sitaru's Inequality by Induction
- An Inequality in Three (Or Is It Two) Variables
- An Inequality in Four Weighted Variables
- An Inequality in Fractions with Absolute Values
- Inequalities with Double And Triple Integrals
- An Old Inequality
- Dan Sitaru's Amazing, Never Ending Inequality
- Leo Giugiuc's Exercise
- Another Inequality with Logarithms, But Not Really
- A Cyclic Inequality of Degree Four
- An Inequality Solved by Changing Appearances
- Distances to Three Points on a Circle
- An Inequality with Powers And Logarithm
- Four Integrals in One Inequality
- Same Integral, Three Intervals
- Dorin Marghidanu's Inequality with Generalization
- Dan Sitaru's Inequality with Three Related Integrals and Derivatives
- An Inequality in Two Or More Variables
- An Inequality in Two Or More Variables II
- A Not Quite Cyclic Inequality
- Dan Sitaru's Inequality: From Three Variables to Many in Two Ways
- An Inequality with Sines But Not in a Triangle
- An Inequality with Angles and Integers
- Sladjan Stankovik's Inequality In Four Variables
- An Inequality with Two Pairs of Triplets
- A Refinement of Turkevich's Inequality
- Dan Sitaru's Exercise with Pi and Ln
- Problem 4165 from Crux Mathematicorum
- Leo Giugiuc's Cyclic Quickie in Four Variables
- Dan Sitaru's Cyclic Inequality in Four Variables
- A Not Quite Cyclic Inequality from Tibet
- Three Variables, Three Constraints, Two Inequalities (Only One to Prove) - by Leo Giugiuc
- An inequality in 2+2 variables from SSMA magazine
- Kunihiko Chikaya's Inequality with Parameter
- Dorin Marghidanu's Permuted Inequality

|Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2017 Alexander Bogomolny62681701 |