In mathematics, a function f is logarithmically convex or superconvex if , the composition of the logarithm with f, is itself a convex function. Definition Let be a convex subset of a real vector space, and let be a function taking non-negative values. Then is: Logarithmically convex if is convex, and Strictly logarithmically convex if is strictly convex. Here we interpret as . Explicitly, is logarithmically convex if and only if, for all and all , the two following equivalent conditions hold: Similarly, is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all . The above definition permits to be zero, but if is logarithmically convex and vanishes anywhere in , then it vanishes everywhere in the interior of . Equivalent conditions If is a differentiable function defined on an interval , then is logarithmically convex if and only if the following condition holds for all and in : This is equivalent to the condition that, whenever and are in and , Moreover, is strictly logarithmically convex if and only if these inequalities are always strict. If is twice differentiable, then it is logarithmically convex if and only if, for all in , If the inequality is always strict, then is strictly logarithmically convex. However, the converse is false: It is possible that is strictly logarithmically convex and that, for some , we have . For example, if , then is strictly logarithmically convex, but . Furthermore, is logarithmically convex if and only if is convex for all . Sufficient conditions If are logarithmically convex, and if are non-negative real numbers, then is logarithmically convex. If is any family of logarithmically convex functions, then is logarithmically convex. If is convex and is logarithmically convex and non-decreasing, then is logarithmically convex. Properties 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. For example, the squaring function is convex, but its logarithm is not. Therefore the squaring function is not logarithmically convex. Examples is logarithmically convex when and strictly logarithmically convex when . is strictly logarithmically convex on for all Euler's gamma function is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the Bohr–Mollerup theorem, this property can