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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.
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 (+ ()) ()
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 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.
A common example of a sigmoid function is the ... A sigmoid function is convex for values less than ... bases have a sigmoid shape due to the logarithmic nature of ...
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).
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. 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.
An example of a function that is pseudoconvex, but not convex, is: () = +, > The figure shows this function for the case where k = 0.2 {\displaystyle k=0.2} . This example may be generalized to two variables as: