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: ${\displaystyle I=[0,\infty )}$ ${\displaystyle f\colon I\to I}$

(1)  . ${\displaystyle f(0)=0}$

(2) is a continuous bijection and thereby a strictly monotonically increasing function . ${\displaystyle f}$

(3) The restriction to the interior of the interval is a Jensen convex function . ${\displaystyle f|_{(0,\infty )}}$

(4) The real function given by the assignment is also Jensen convex. ${\displaystyle t\mapsto F(t)=\ln \left(f\left(e^{t}\right)\right)}$ ${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$

Then for any natural number and two - tuple always the inequality ${\displaystyle N>0}$ ${\displaystyle N}$ ${\displaystyle (a_{1},\ldots ,a_{N}),(b_{1},\ldots ,b_{N})\in I^{N}}$

${\displaystyle f^{-1}\left(\sum _{i=1}^{N}{f(a_{i}+b_{i})}\right)\leq f^{-1}\left(\sum _{i=1}^{N}{f(a_{i})}\right)+f^{-1}\left(\sum _{i=1}^{N}{f(b_{i})}\right)}$  .

Corollary

If you take the power function above (for a given real number ) as a function , you get a version of the Minkowski inequality: ${\displaystyle p\geq 1}$ ${\displaystyle f}$ ${\displaystyle t\mapsto f(t)=t^{p}\;(t\geq 0)}$

For every natural number and every two tuples and nonnegative real numbers always applies ${\displaystyle N>0}$ ${\displaystyle N}$ ${\displaystyle (a_{1},\ldots ,a_{N})}$ ${\displaystyle (b_{1},\ldots ,b_{N})}$

${\displaystyle {\sqrt[{p}]{\sum _{i=1}^{N}{(a_{i}+b_{i})^{p}}}}\leq {\sqrt[{p}]{\sum _{i=1}^{N}{{a_{i}}^{p}}}}+{\sqrt[{p}]{\sum _{i=1}^{N}{{b_{i}}^{p}}}}}$  .

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 ).

Individual evidence

1. ^ Marek Kuczma: An Introduction to the Theory of Functional Equations and Inequalities. 2009, pp. 218-222
2. ↑ DS Mitrinović: Analytic Inequalities. 1970, p. 55 ff
3. ↑ ^{a b} Kuczma, op.cit., P. 221
4. ↑ Mitrinović, op. Cit., Pp. 56–57
5. ↑ The convention is followed . ${\displaystyle 0^{p}={\sqrt[{p}]{0}}=0}$