# An Inequality for Mixed Means

### Statement

Professor Dorin Marghidanu has posted the following problem and the solution at the CutTheKnotMath facebook page:

If $a_j\gt 0,$ $j=1,\ldots,m$ and $x_i\ge 0$, $i=1,\ldots,n.\;$ Then prove that:

$\displaystyle\begin{align} \sqrt[m]{\prod_{j=1}^{m}\left[a_j+\frac{1}{n}\sum_{i=1}^{n}x_i\right]}&\ge\frac{1}{n}\sum_{i=1}^{n}\sqrt[m]{\prod_{j=1}^{m}(a_j+x_i)}\\ &\ge\prod_{i=1}^{n}\prod_{j=1}^{m}\sqrt[mn]{a_j+x_i}. \end{align}$

### Solution

Define function $f:\;[0,\infty)$ as $f(x)=\sqrt[m]{(a_1+x)(a_2+x)\ldots (a_m+x)}.$ We have successively,

$\displaystyle f'(x)=\frac{1}{m}f(x)\left(\frac{1}{a_1+x}+\frac{1}{a_2+x}+\ldots+\frac{1}{a_m+x}\right)\gt 0,$

and

$\displaystyle\begin{align} f''(x)&=\frac{1}{m^2}f(x)\left(\frac{1}{a_1+x}+\frac{1}{a_2+x}+\ldots+\frac{1}{a_m+x}\right)^2\\ &\;\;-\frac{1}{m}f(x)\left(\frac{1}{(a_1+x)^2}+\ldots+\frac{1}{(a_m+x)^2}\right)\le 0, \end{align}$

by the Cauchy-Schwarz (often Cauchy-Bunyakovsky-Schwarz) inequality.

Hence function $f$ is *concave* and by Jensen's inequality for concave functions,

$\displaystyle f\left(\frac{1}{n}\sum_{i=1}^{n}x_{i}\right)\ge\frac{1}{n}\sum_{i=1}^{n}f(x_i).$

Specifically,

$\displaystyle \sqrt[m]{\left(a_1+\frac{1}{n}\sum_{i=1}^{n}x_i\right)\cdot\ldots\cdot\left(a_m+\frac{1}{n}\sum_{i=1}^{n}x_i\right)}\ge\frac{1}{n}\sum_{i=1}^{n}\sqrt[m]{(a_1+x_i)\cdot\ldots\cdot (a_m+x_i)},$

or, in shorthand,

$\displaystyle \sqrt[m]{\prod_{j=1}^{m}\left(a_j+\frac{1}{n}\sum_{i=1}^{n}x_i\right)}\ge\frac{1}{n}\sum_{i=1}^{n}\sqrt[m]{\prod_{j=1}^{m}(a_j+x_i)}$

which proves the left inequality in the problem. The right inequality is the direct consequence of the AM-GM inequality:

$\displaystyle\begin{align} \frac{1}{n}\sum_{i=1}^{n}\sqrt[m]{\prod_{j=1}^{m}(a_j+x_i)}&\ge\sqrt[n]{\prod_{i=1}^{n}\sqrt[m]{\prod_{j=1}^{m}(a_j+x_i)}}\\ &=\prod_{i=1}^{n}\sqrt[n]{\sqrt[m]{\prod_{j=1}^{m}(a_j+x_i)}}\\ &=\prod_{i=1}^{n}\prod_{j=1}^{m}\sqrt[mn]{a_j+x_i}. \end{align}$

### References and a Remark

$\displaystyle G_m[a+A_n[x]]\ge A_n[G_m[a+x]]\ge G_n[G_m[a+x]]=G_m[G_n[a+x]].$

The inequality appeared in

- Dorin Marghidanu,
__Asupra unei inegalitati pentru medii compozite__,*Revista de matematica si informatica*, pp 3-5, Anul XIII, n. 2, April 2013.

- 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 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

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

Copyright © 1996-2017 Alexander Bogomolny62055933 |