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Last modified on May 17th, 2024

Jensen’s inequality relates the expected value of a convex or concave function to the function of their expected values.

It states that if ‘X’ is an integrable random variable and f: ℝ → ℝ is a convex function such that Y = f(X) is also integrable, then E[f(X)] ≥ f(E[X])

Thus, if a convex function is applied to the mean of a variable, it is less than or equal to the mean of that function applied to the variable. However, if ‘f’ is a concave function, the inequality reverses.

Let us consider a convex function f(x).

Then, for any point x_{0,} the graph of f(x) lies entirely above the tangent at the point ‘x_{0}’

f(x) ≥ f(x_{0}) + b(x – x_{0}), where ‘b’ is the slope of the tangent

Now, considering x = X and x_{0} = E[X], we get

f(X) ≥ f(E[X]) + b(X – E[X])

Taking the expected values of both sides, we get

E[f(X)] ≥ E[f(E[X]) + b(X – E[X])]

By using the linear property of expected values, we get

E[f(X)] ≥ f(E[X]) + b(E[X] – E[X])

⇒ E[f(X)] ≥ f(E[X]) …..(i)

If the function f(x) is concave, then -f(x) is convex, and from the inequality (i), we get

E[-f(X)] ≥ -f(E[X])

Multiplying both sides by (-1), we get

E[f(X)] ≤ f(E[X]) …..(ii)

Thus, Jensen’s inequalities for convex and concave functions are proved.

However, there is an alternative form of this inequality, where the expected value or mean E[X] is expressed as E[X] = a_{1}x_{1} + a_{2}x_{2} + … + a_{n}x_{n}.

It implies:

f(E[X]) = f(a_{1}x_{1} + a_{2}x_{2} + … + a_{n}x_{n}) and E[f(X)] = a_{1}f(x_{1}) + a_{2}f(x_{2}) + … + a_{n}f(x_{n}), for all a_{1}, a_{2}, …, a_{n} ≥ 0

In particular, if a_{1}, a_{2}, …, a_{n} ≥ 0 and ${\sum ^{n}_{i=1}a _{i}=1}$ for all λ_{i}, then

${\sum ^{n}_{i=1}a _{i}f\left( x_{i}\right) \geq f\left( \sum ^{n}_{i=1}a _{i}x_{i}\right)}$

In general, if a real-valued function ‘f’ is convex on the interval ‘I,’ and x_{1}, x_{2}, …, x_{n} Є I, then for a_{1}, a_{2}, …, a_{n} ≥ 0,

To apply Jensen’s inequality, we use the following conditions of calculus.

If f: I → ℝ is a twice-differentiable function, then

- ‘f’ is convex on ‘I,’ if and only if f”(x) ≥ 0, for all x Є I
- ‘f’ is concave on ‘I,’ if and only if f”(x) ≤ 0, for all x Є I

**Use Jensen’s inequality to find the minimum of f(X) = 3X ^{2} for a random variable X with the mean μ = 5**

Solution:

Here, the function f(x) = 3X^{2}

Differentiating the function with respect to x, we get

f’(x) = 6X

f”(x) = 6 > 0

As we know, f”(x) ≥ 0 ⇔ f(x) is a convex function

Now, to find the minimum value of f(X) = 3X^{2}, we find the lower bound on the expected value E[f(X)].

Using Jensen’s inequality, we get

E[f(X)] ≥ f(E[X])

⇒ E[3X^{2}] ≥ f(E[3X^{2}])

⇒ E[3X^{2}] ≥ 3(E[X^{2}])

⇒ E[3X^{2}] ≥ 3(5)^{2} (since the mean μ = E[X] = 5)

⇒ E[3X^{2}] ≥ 75

Thus, the minimum value of f(X) = 3X^{2} for a random variable X is 75

**If p, q, r, s > 0 and p + q + r + s = 1, then find the minimum of ****${\left( p+\dfrac{1}{p}\right) ^{5}+\left( q+\dfrac{1}{q}\right) ^{5}+\left( r+\dfrac{1}{r}\right) ^{5}+\left( s+\dfrac{1}{s}\right) ^{5}}$**

Solution:

As we know, p, q, r, s > 0 and p + q + r + s = 1

Thus, 0 < p, q, r, s < 1.

Let f (x) = ${\left( x+\dfrac{1}{x}\right) ^{5}}$, x on the interval I = (0, 1),

Now, f’(x) = ${5\left( x+\dfrac{1}{x}\right) ^{4}\left( 1-\dfrac{1}{x^{2}}\right)}$

f”(x) = ${5\left( 4\left( x+\dfrac{1}{x}\right) ^{3}\left( 1-\dfrac{1}{x^{2}}\right) ^{2}+\dfrac{2\left( x^{2}+1\right) ^{4}}{x^{7}}\right)}$

Here, f”(x) > 0, then f is strictly convex on I

By Jensen’s inequality, we get

${f\left( \dfrac{p+q+r+s}{4}\right) \leq \dfrac{f\left( p\right) +f\left( q\right) +f\left( r\right) +f\left( s\right) }{4}}$

⇒ ${4f\left( \dfrac{p+q+r+s}{4}\right) \leq f\left( p\right) +f\left( q\right) +f\left( r\right) +f\left( s\right)}$

⇒ ${4\left( \dfrac{1}{4}+4\right) ^{5}\leq f\left( p\right) +f\left( q\right) +f\left( r\right) +f\left( s\right)}$

⇒ ${\dfrac{17^{5}}{4^{4}}\leq f\left( p\right) +f\left( q\right) +f\left( r\right) +f\left( s\right)}$

Thus, the minimum is ${\dfrac{17^{5}}{4^{4}}}$, attained when p = q = r = s = ${\dfrac{1}{4}}$

Last modified on May 17th, 2024