Do logarithmic functions have concavity?
Do logarithmic functions have concavity?
From Logarithm is Strictly Increasing, lnx is strictly increasing on x>0. So from Real Function is Strictly Concave iff Derivative is Strictly Decreasing, lnx is strictly concave on x>0.
Is a log-concave function concave?
Properties. A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex.
Is log function always convex?
, the composition of the logarithm with f, is itself a convex function.
How do you explain concavity?
Concavity relates to the rate of change of a function’s derivative. A function f is concave up (or upwards) where the derivative f′ is increasing. This is equivalent to the derivative of f′ , which is f′′f, start superscript, prime, prime, end superscript, being positive.
Is Gaussian concave?
The Gaussian functions are thus those functions whose logarithm is a concave quadratic function.
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.
Is log of a convex function convex?
The Bohr–Mollerup theorem characterizes the Gamma function Γ(x) as the unique function f(x) on the positive reals such that f(1)=1, f(x+1)=xf(x), and f is logarithmically convex, i.e. log(f(x)) is a convex function.
Which is an example of a log concave function?
Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function . Similarly, a function is log-convex if it satisfies the reverse inequality for all x,y ∈ dom f and 0 < θ < 1 .
When does the concavity of a function change?
Let’s look at the sign of the second derivative to work out where the function is concave up and concave down: For x > − 1 4, 24 x + 6 > 0, so the function is concave up. Note: The point where the concavity of the function changes is called a point of inflection. This happens at x = − 1 4. Finally, what about straight lines?
When is a convex function a logarithmic function?
In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality for all x,y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm…
Which is a concave up and concave down function?
A straight line f ( x) = m x + b satisfies the definitions of both concave up and concave because we always have f ( t a + ( 1 − t) b) = t f ( a) + ( 1 − t) f ( b) . y = − 2 x + 1 is a straight line. It is both concave up and concave down.