---
tag: E97P413DKM
kind: definition
name: Logarithmically convex function
msc: [26A51]
defines: 8FRWATNNMX
uses: [0TK8Y3J5RW, 0VV0HSW1ZS, 0044CPHJBV]
sources:
  - url: https://en.wikipedia.org/wiki/Logarithmically_convex_function
---

## Definition

Let $I$ be an [[interval|0TK8Y3J5RW]] of $\R$ and let $f \colon I \to (0, \infty)$ be a function.
The function $f$ is *logarithmically convex*, or *log-convex*, if the function $\ln \circ f \colon I \to \R$ is [[convex|0VV0HSW1ZS]], where $\ln$ is the [[natural logarithm|0044CPHJBV]].

## Remarks

Equivalently, $f(t x + (1 - t) y) \le f(x)^{t} f(y)^{1 - t}$ for all $x, y \in I$ and $t \in [0, 1]$.
By the [[AM–GM inequality|0DRF0F3XC9]], $f(x)^t f(y)^{1-t} \le t f(x) + (1 - t) f(y)$, so a log-convex function is convex.

Example: $f(x) = e^{x^2}$ on $\R$ is log-convex, because $\ln f(x) = x^2$ and $x^2$ is convex.
The [[gamma function|0V3FZMVB4K]] on $(0, \infty)$ is also log-convex.

Non-example: $f(x) = x^2$ on $(0, \infty)$ is convex, but $\ln f(x) = 2 \ln x$ is strictly concave, so $f$ is not log-convex.

Some authors allow $f$ to take the value $0$, with the convention $\ln 0 = -\infty$.