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A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the function , which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex.
Convex function - a function in which the line segment between any two points on the graph of the function lies above the graph. Closed convex function - a convex function all of whose sublevel sets are closed sets. Proper convex function - a convex function whose effective domain is nonempty and it never attains minus infinity. Concave ...
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. [1] Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold.
An example of a function which is convex but not strictly convex is (,) = +. This function is not strictly convex because any two points sharing an x coordinate will have a straight line between them, while any two points NOT sharing an x coordinate will have a greater value of the function than the points between them.
A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right).
The rows of Pascal's triangle are examples for logarithmically concave sequences. In mathematics, a sequence a = (a 0, a 1, ..., a n) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a i 2 ≥ a i−1 a i+1 holds for 0 < i < n.
In mathematics, the scale convolution of two functions () ... also known as their logarithmic convolution or log-volution [1] is defined as the function [2] ...
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.