In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let
and let
be a given function having a third derivative on the range
, and such that
![{\displaystyle f'''(x)\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/818d6d93e3ba0d92592b7ff015e9f0e551606ae7)
for all
. Suppose
and
for
. Then
![{\displaystyle {\frac {\sum _{i=1}^{n}p_{i}f(x_{i})}{\sum _{i=1}^{n}p_{i}}}-f\left({\frac {\sum _{i=1}^{n}p_{i}x_{i}}{\sum _{i=1}^{n}p_{i}}}\right)\leq {\frac {\sum _{i=1}^{n}p_{i}f(2a-x_{i})}{\sum _{i=1}^{n}p_{i}}}-f\left({\frac {\sum _{i=1}^{n}p_{i}(2a-x_{i})}{\sum _{i=1}^{n}p_{i}}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/120d25175987132c1109730b5532f239401efcd1)
The Ky Fan inequality is the special case of Levinson's inequality, where
![{\displaystyle p_{i}=1,\ a={\frac {1}{2}},{\text{ and }}f(x)=\log x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5edaa41b32ea608d1d7f752620f4a79013f0f9c8)
- Scott Lawrence and Daniel Segalman: A generalization of two inequalities involving means, Proceedings of the American Mathematical Society. Vol 35 No. 1, September 1972.
- Norman Levinson: Generalization of an inequality of Ky Fan, Journal of Mathematical Analysis and Applications. Vol 8 (1964), 133–134.