Logarithmically convex function

A logarithmic convex function is a positive function for which is convex. Since the composition of convex functions, is convex if g is monotonically increasing, and the exponential function is both convex and monotonically increasing, logarithmic convex functions are also convex. The converse is not true in general. For example, a convex function, but is not convex. Therefore, it is convex but not logarithmically convex.

From the definition of convexity, monotonicity of the exponential and logarithmic function and the Logarithmengesetzen follows that a positive function if and only logarithmically convex on the interval when

A particularly important example of a logarithmic convex function is the gamma function. After the set of Bohr - Mollerup each logarithmically convex function on which satisfies the functional equation, a multiple of the gamma function.

Swell

• Mathematical function