What is a convex inequality?

f(x) = x2 is a convex function. To make this definition precise consider two real numbers x1 and x2. f is convex if the line between f(x1) and f(x2) stays above the function f. Jensen’s inequality states that this line is everywhere at least as large as f(x).

How do you use Jensen inequality?

To use Jensen’s inequality, we need to determine if a function g is convex. A useful method is the second derivative. A twice-differentiable function g:I→R is convex if and only if g″(x)≥0 for all x∈I.

How do you pronounce Jensen’s inequality?

jensen’s inequality Pronunciation. jensen’s in·equal·i·ty.

Is the expectation a convex function?

It is well known that expectation preserves convexity: If f(x) is convex and Y is a random variable, then E[f(x−Y)] is convex.

What does convex mean in geometry?

A convex polygon is a closed figure where all its interior angles are less than 180° and the vertices are pointing outwards. The term convex is used to refer to a shape that has a curve or a protruding surface. In geometry, there are many shapes that can be classified as convex polygons.

What does the Jensen alpha measure?

The Jensen’s measure, or Jensen’s alpha, is a risk-adjusted performance measure that represents the average return on a portfolio or investment, above or below that predicted by the capital asset pricing model (CAPM), given the portfolio’s or investment’s beta and the average market return.

Is absolute value convex?

The absolute value function f(x)=|x| is convex (as reflected in the triangle inequality), even though it does not have a derivative at the point x=0.

Does expectation preserve concavity?

In recent years Karlin [ 4] has shown that the concavity of a function is preserved under the expectation transformation with respect to a class of totally positive distributions of order S. This note extends a result of Karlin to a bigger class of distributions.

Can a convex function be discontinuous?

There exist convex functions which are not continuous, but they are very irregular: If a function f is convex on the interval (a,b) and is bounded from above on some interval lying inside (a,b), it is continuous on (a,b). Thus, a discontinuous convex function is unbounded on any interior interval and is not measurable.

Which functions are convex?

A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

Is concave down the same as convex?

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

What is the significance of the Jensen inequality?

Jensen’s inequality generalizes the statement that a secant line of a convex function lies above the graph. In mathematics, Jensen’s inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proven by Jensen in 1906.

Can Jensen’s inequality fix our intuition about nonlinear transformation?

Unfortunately, we bring this intuition with us when using nonlinear transformations of variables where this relationship no longer holds. Fixing this intuition involves the discovery of Jensen’s Inequality, which provides a standard mathematical tool used in function analysis, probability, and statistics.

What is the convexity inequality?

In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.

Are there different forms of the inequality?

Given its generality, the inequality appears in many forms depending on the context, some of which are presented below.