Nesbitt’s Inequality

Nesbitt’s inequality is one of the simplest inequalities in mathematics. According to the statement of the inequality, for any 3 given real numbers they satisfy the mathematical condition,

{a / (b + c)} + {b / (a + c)} + {c / (a + b)} >= 1.5 for all a>0, b>0, c>0.

Illustrative Examples:

The 3 numbers satisfying Nesbitts inequality are real numbers.

For a = 1, b = 2, c = 3,
the condition of the inequality
{1 / (2 + 3)} + {2 / (1 + 3)} + {3 / (1 + 2)} >= 1.5 holds true.

For a = 1.5, b = 5.6, c = 4.9,
the condition of the inequality
{1.5 / (5.6 + 4.9)} + {5.6 / (1.5 + 4.9)} + {4.9 / (1.5 + 5.6)} >= 1.5 holds true.

For a = 4, b = 6, c = 7,
the condition of the inequality
{4 / (6 + 7)} + {6 / (4 + 7)} + {7 / (4 + 6)} >= 1.5 holds true.

For a = 459, b = 62, c = 783,
the condition of the inequality
{459 / (62 + 783)} + {62 / (459 + 783)} + {783 / (459 + 62)} >= 1.5 holds true.

For a = 9, b = 6, c = 83,
the condition of the inequality
{9 / (6 + 83)} + {6 / (9 + 83)} + {83 / (9 + 6)} >= 1.5 holds true.

C++

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// CPP code to verify Nesbitt's Inequality
#include <bits/stdc++.h>
using namespace std;
  
bool isValidNesbitt(double a, double b, double c)
{
    // 3 parts of the inequality sum
    double A = a / (b + c);
    double B = b / (a + c);
    double C = c / (a + b);
    double inequality = A + B + C;
  
   return (inequality >= 1.5);
}
  
int main()
{
    double a = 1.0, b = 2.0, c = 3.0;
    if (isValidNesbitt(a, b, c)) 
        cout << "Nesbitt's inequality satisfied."
             << "for real numbers " << a << ", " 
             << b << ", " << c << "\n";
    else
       cout << "Not satisfied";
    return 0;
}

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Java

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// Java code to verify Nesbitt's Inequality
class GFG {
      
    static boolean isValidNesbitt(double a,
                          double b, double c)
    {
          
        // 3 parts of the inequality sum
        double A = a / (b + c);
        double B = b / (a + c);
        double C = c / (a + b);
        double inequality = A + B + C;
  
        return (inequality >= 1.5);
    }
  
    // Driver code
    public static void main(String args[])
    {
        double a = 1.0, b = 2.0, c = 3.0;
        if(isValidNesbitt(a, b, c) == true
        {
            System.out.print("Nesbitt's inequality"
                                  + " satisfied.");
            System.out.println("for real numbers "
                         + a + ", " + b + ", " + c);
        
        else
            System.out.println("Nesbitts inequality"
                                + " not satisfied");
    }
}
  
// This code is contributed by JaideepPyne.

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Python3

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# Python3 code to verify 
# Nesbitt's Inequality
  
def isValidNesbitt(a, b, c):
      
    # 3 parts of the
    # inequality sum
    A = a / (b + c);
    B = b / (a + c);
    C = c / (a + b);
    inequality = A + B + C;
  
    return (inequality >= 1.5);
  
# Driver Code
a = 1.0
b = 2.0;
c = 3.0;
if (isValidNesbitt(a, b, c)):
    print("Nesbitt's inequality satisfied."
          " for real numbers ",a,", ",b,", ",c);
else:
    print("Not satisfied");
  
# This code is contributed by mits

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C#

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// C# code to verify 
// Nesbitt's Inequality
using System;
  
class GFG
{
    static bool isValidNesbitt(double a,
                               double b, 
                               double c)
    {
          
        // 3 parts of the
        // inequality sum
        double A = a / (b + c);
        double B = b / (a + c);
        double C = c / (a + b);
        double inequality = A + B + C;
  
        return (inequality >= 1.5);
    }
  
    // Driver code
    static public void Main ()
    {
    double a = 1.0, b = 2.0, c = 3.0;
    if(isValidNesbitt(a, b, c) == true
    {
        Console.Write("Nesbitt's inequality"
                               " satisfied ");
        Console.WriteLine("for real numbers "
                      a + ", " + b + ", " + c);
    
    else
        Console.WriteLine("Nesbitts inequality" +
                               " not satisfied");
    }
}
  
// This code is contributed by ajit

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PHP

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<?php
// PHP code to verify 
// Nesbitt's Inequality
  
function isValidNesbitt($a, $b, $c)
{
      
    // 3 parts of the
    // inequality sum
    $A = $a / ($b + $c);
    $B = $b / ($a + $c);
    $C = $c / ($a + $b);
    $inequality = $A + $B + $C;
  
    return ($inequality >= 1.5);
}
  
    // Driver Code
    $a = 1.0; 
    $b = 2.0;
    $c = 3.0;
    if (isValidNesbitt($a, $b, $c)) 
        echo"Nesbitt's inequality satisfied."
            "for real numbers ", $a, ", ", $b
                               ", ", $c, "\n";
    else
    cout <<"Not satisfied";
  
  
// This code is contributed by Ajit.
?>

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Output :

Nesbitt's inequality satisfied.for real numbers 1, 2, 3


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