Mulholland inequality
The inequality of Mulholland ( English Mulholland’s inequality ) is a result of analysis , one of the branches of mathematics . The inequality is related to the Minkowski inequality , which essentially results from the Mulholland inequality as a corollary . It was founded by Hugh P. Mulholland in 1950 published and gave rise to a number of further investigations.
formulation
The result can be given as follows:
-
Given the real interval and a real function with the following properties:
- (1) .
- (2) is a continuous bijection and thereby a strictly monotonically increasing function .
- (3) The restriction to the interior of the interval is a Jensen convex function .
- (4) The real function given by the assignment is also Jensen convex.
-
Then for any natural number and two - tuple always the inequality
- .
Corollary
If you take the power function above (for a given real number ) as a function , you get a version of the Minkowski inequality:
-
For every natural number and every two tuples and nonnegative real numbers always applies
- .
literature
- Marek Kuczma : An Introduction to the Theory of Functional Equations and Inequalities . Cauchy's Equation and Jensen's Inequality. 2nd Edition. Birkhäuser Verlag , Basel 2009, ISBN 978-3-7643-8748-8 ( MR2467621 ).
- DS Mitrinović : Analytic Inequalities . In cooperation with PM Vasić (= The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 165 ). Springer Verlag , Berlin ( inter alia ) 1970, ISBN 3-540-62903-3 ( MR0274686 ).
- HP Mulholland: On generalizations of Minkowski's inequality in the form of a triangle inequality . In: Proceedings of the London Mathematical Society (2) . tape 51 , 1950, pp. 294-307 ( MR0033865 ).
- Zhen Xiao Huang, Bicheng Yang: On a half-discrete Hilbert-type inequality similar to Mulholland's inequality . In: Journal of Inequalities and Applications . 2013, p. 2013: 290 ( MR3073994 ).