Convexity, logarithmic

2020 Mathematics Subject Classification: Primary: 26A51 [MSN][ZBL]

The property of a non-negative function $f$, defined on some interval, that can be described as follows: If for any $x_1$ and $x_2$ in this interval and for any $p_1\ge 0$, $p_2\ge 0$ with $p_1+p_2=1$ the inequality $$f(p_1{x}_1+p_2{x}_2)\le f(x_1{)}^{p_1}\cdot f(x_2{)}^{p_2}$$ is satisfied, $f$ is called logarithmically convex. 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).