Week -8 | Inequalities – 2

1.1 Convex Functions and Jenson’s Inequality

A function \(f\) is called a convex function if the line segment between any two points on the graph of the function lies above or on the graph. The following curve illustrates this oncept of a convex function and concave function:

Fig 1.1

 

For a convex function \(f\), we have a famous inequality called Jenson’s Inequality:

For \(x_1,x_2,…,x_n \in \mathbb{R}\), and \(a_1,a_2,…a_n\ge 0\) with \(a_1+a_2…a_n=1\), we have

\(f(a_1x_1+a_2x_2+…a_nx_) \le a_1f(x_1)+a_2f(x_2)+…a_nf(x_n)\).

The proof of Jensen’s inequality is very simple. Since the graph of every convex function lies above its tangent line at every point, we can compare the function \(f\) by \(l\) whose graph is tangent to the graph of \(a_1x_1+a_2x_2+…+a_nx_n\). Then the left hand side of the inequality is the same for \(f\) and \(l\), while the right hand side is smaller for \(l\). But the equality case holds for all linear functions!

Jenson’s inequality has a lot of applications including the AM-GM inequality being one of them! We can derive AM-GM inequality from Jenson’s Inequality by considering the function \(f(x)=-\ln x\).

1 important result of convexity should be observed over here. For convex \(f\) and reals \(a_i\) the following inequality holds true:

\(\frac{f(a_1)+f(a_2)+…f(a_n)}{n} \ge f\left(\frac{a_1+a_2+..a_n}{n}\right)\)

 

1.2 Triangle Inequalitites

We often deal with inequalities regarding sides of a triangle. We’ll keep handy the following facts about triangle inequalitites, with \(a,b,c\) as the sides of the triangle:

• \(a+b>c\), \(b+c>a\), \(a+c>b\). Also, if no other conditions are specified it makes sense to assume \(a\le b\le c\).
• \(a>|b-c|\), \(b>|a-c|\), \(c>|a-b|\)
• \((a+b-c)(b+c-a)(c+a-b)>0\)
• \(a=y+z\), \(b=z+x\), \(c=x+y\) with \(x,y,z \in \mathbb{R}^+\)

 

2.1 Power Mean Inequality

Suppose that we have positive reals \(a_1,a_2,..a_n\) and reals \(k_1,k_2\) with \(k \ge k_2\), we have the following inequality:

 

The proof of this follows from the fact that \(f(x)=x^{\frac{k_2}{k_1}}\) is concave for \(x>0\),and so we can apply Jenson’s Inequality to get

 

2.2 Cauchy-Schwarz Inequality

This is one of the most ueful, but most underrated inequality of all times:

For reals \(a_i,b_i\)

\((a_1b_1+a_2b_2+…+a_nb_n)^2 \le (a_1^2+a_2^2+….a_n^2)(b_1^2+…+b_n^2)\).

Putting \(b_i=1\) for all \(i\) we have

\(\left(\sum\limits_{i=1}^na_i\right)^2 \le n\sum\limits_{i=1}^na_i^2\)