In mathematics, a function f is logarithmically convex or superconvex[1] if , the composition of the logarithm with f, is itself a convex function. Let X be a convex subset of a real vector space, and let f : X → R be a function taking non-negative values. Then f is: …

Jan 20, 2025 · A function f (x) is logarithmically convex on the interval [a,b] if f>0 and lnf (x) is convex on [a,b]. If f (x) and g (x) are logarithmically convex on the interval [a,b], then the functions f (x)+g (x) and f (x)g (x) are also logarithmically convex on [a,b].

Aug 23, 2018 · I consider the one-dimensional real axis. A function $f$ defined on the axis is log-convex if $f(x)f''(x)\ge (f'(x))^2$ for all $x\in\mathbb{R}$. I know that a convex function may not be log-convex, i.e., $f(x)=x$ and $f(x)=x^2$.

Mar 27, 2015 · The real function $f$ such that $\log \cdots \log (f)$ is strictly convex on its domain for any number of $\log$'s

The Bohr–Mollerup theorem characterizes the Gamma function $\Gamma(x)$ 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.

E.g.: Consider the convex function f (x) = log x. Its perspective is g(x, t) = t log x/t = t log t/x = t log t t log x which is convex in (x, t). The function g is called the relative entropy of t and x. Therefore, the relative entropy of two vectors u, v. ui, vi. For a function f : Rn ! R, its conjugate function f ⇤ : Rn ! R is defined as. (x)).

Why is $x\log(x)$ convex? According to the definition it must hold: $(tx+(1-t)y)\log(tx+(1-t)y)\le tx\log(x)+(1-t)y\log(y)$ for all positive $x,y$ and $t\in[0,1]$ edit: It is allowed to derive,...

nonnegative multiple: fif is convex if f is convex, fi ‚ 0 sum: f1 +f2 convex if f1;f2 convex (extends to inflnite sums, integrals) composition with a–ne function: f(Ax+b) is convex if f is convex examples † log barrier for linear inequalities f(x) = ¡ Xm i=1 log(bi ¡aT i …

Mar 1, 2012 · In this paper, some properties of log-convex function are researched, and integral inequalities of log-convex functions are proved. As an application, an estimation formula of remainder terms in Taylor series expansion is given.

Oct 29, 2016 · If a function is logarithmically convex, it is either identically equal to zero or is strictly positive and $\log f$ is a convex function (of a real variable).