Logarithmically convex function

Builds on Interval Convex function Natural logarithm

Definition

Let $I$ be an interval 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, where $\ln$ is the natural logarithm.

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, $f(x)^t f(y)^{1-t} \le t f(x) + (1 - t) f(y)$ , 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 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 $\ln 0 = -\infty$ .

Reading path

To understand this page from the ground up, read these 61 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Convex function definition
Integers object
Limit point of a set definition
Local maximum definition
Ordered field definition
Rational numbers object
Sequence definition
Strictly monotone function definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Real sequence definition
Subsequence definition
Continuity at a point definition
Derivative definition
Bounded sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
The rationals form a field statement
Continuous function definition
Fermat's theorem on stationary points statement
Sum of a finite geometric series statement
Monotone convergence theorem for sequences statement
Convergent sequence definition
Squeeze theorem statement
Every sequence has a monotone subsequence statement
Cauchy product definition
Partial sum definition
Power series definition
The rationals form an ordered field statement
Intermediate value theorem statement
Bolzano–Weierstrass theorem statement
The sequence r^n tends to zero for |r| < 1 example
Order on the reals definition
Radius of convergence definition
Convergent series definition
Boundedness theorem statement
Interval definition
A series of nonnegative terms converges iff its partial sums are bounded statement
Absolute convergence definition
Sum of a series definition
Extreme value theorem statement
Comparison test statement
Geometric series example
Cauchy product of absolutely convergent series statement
Rolle's theorem statement
Ratio test statement
Mean value theorem statement
Ratio formula for the radius of convergence statement
The exponential series has infinite radius of convergence statement
A function with positive derivative is strictly increasing statement
Exponential function definition
Exponential of a sum statement
The exponential function is positive statement
The exponential function is its own derivative statement
The exponential function is strictly increasing statement
Natural logarithm definition
Logarithmically convex function definition

Tag	`E97P413DKM`
URI	`m2://E97P413DKM`
Kind	definition
Writing	W0 stub
MSC	26A51
Defines	Logarithmically convex function
Uses	Interval, Convex function, Natural logarithm
Changed	2 November 2026

Logarithmically convex function

Definition

Remarks

Reading path

Further reading

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