Popoviciu's inequality

The inequality of Popoviciu ( English Popoviciu’s inequality ) is a theorem of analysis , one of the branches of mathematics . The inequality , which comes from a work by the Romanian mathematician Tiberiu Popoviciu (1906–1975) from 1965, represents a characteristic property of continuous convex functions on real intervals . It can be obtained as a consequence of the majorization principle of Hardy-Littlewood-Pólya .

formulation

The theorem can be stated as follows:

An arbitrary real interval and a continuous real function are given . ${\displaystyle I\subseteq \mathbb {R} }$ ${\displaystyle f\colon I\to \mathbb {R} }$

Then the following conditions are equivalent:

(B_1) is a convex function . ${\displaystyle f}$

(B_2) Every three real numbers satisfy the inequality ${\displaystyle x,y,z\in I}$

${\displaystyle {\frac {f(x)+f(y)+f(z)}{3}}+f\left({\frac {x+y+z}{3}}\right)\geq {\frac {2}{3}}\cdot {\bigl [}f\left({\frac {x+y}{2}}\right)+f\left({\frac {y+z}{2}}\right)+f\left({\frac {z+x}{2}}\right){\bigr ]}}$  .

It is strictly convex if and only if for every three , apart from the case , the above inequality with the comparison symbol instead of the comparison symbol applies. ${\displaystyle f}$ ${\displaystyle x,y,z\in I}$ ${\displaystyle x=y=z}$ ${\displaystyle >}$ ${\displaystyle \geq }$

Two inequalities as an application

With the help of Popovicius' inequality, the following two can be derived:

For every three real numbers , which are not all the same, always applies: ${\displaystyle a,b,c>0}$

(1)  . ${\displaystyle 27(a+b)^{2}(b+c)^{2}(c+a)^{2}>64abc(a+b+c)^{3}}$

(2)  . ${\displaystyle a^{6}+b^{6}+c^{6}+3a^{2}b^{2}c^{2}>2\left(a^{3}b^{3}+b^{3}c^{3}+c^{3}a^{3}\right)}$

More general inequalities, integral version

In his work from 1965, Tiberiu Popoviciu stated his inequality in an even more general version, which was subsequently expanded - in particular by Petar M. Vasić and Ljubomir R. Stanković . Other authors found further generalizations and modifications. Last but not least, Popoviciu's inequality was also translated into an integral version.

Another inequality from Popoviciu

Several other inequalities are associated with the name of Tiberiu Popoviciu, in particular the following, which is a generalization of a well-known inequality by János Aczél :

Given are real numbers and (for a given natural number ) two tuples and positive real numbers . ${\displaystyle p,q>1{\text{ mit }}{\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle n\geq 2}$ ${\displaystyle n}$ ${\displaystyle (a_{1},\ldots ,a_{n})}$ ${\displaystyle (b_{1},\ldots ,b_{n})}$

Be further and  . ${\displaystyle {a_{1}}^{p}-{a_{2}}^{p}-\ldots -{a_{n}}^{p}>0}$ ${\displaystyle {b_{1}}^{q}-{b_{2}}^{q}-\ldots -{b_{n}}^{q}>0}$

Then:

${\displaystyle \left({a_{1}}^{p}-{a_{2}}^{p}-\ldots -{a_{n}}^{p}\right)^{\tfrac {1}{p}}\left({b_{1}}^{q}-{b_{2}}^{q}-\ldots -{b_{n}}^{q}\right)^{\tfrac {1}{q}}\leq a_{1}b_{1}-a_{2}b_{2}-\ldots -a_{n}b_{n}}$  .

literature

• Marcela V. Mihai, Flavia-Corina Mitroi-Symeonidis: New extensions of Popoviciu's inequality . In: Mediterranean Journal of Mathematics . tape 13 , 2016, p. 3121-3133 ( MR3554298 ).
• 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 ).
• Constantin Niculescu, Lars-Erik Persson: Convex Functions and Their Applications . A Contemporary Approach (=  CMS Books in Mathematics . Volume 23 ). Springer Verlag , New York 2006, ISBN 978-0-387-24300-9 ( MR2178902 ).
• Constantin P. Niculescu: The integral version of Popoviciu's inequality . In: Journal of Mathematical Inequalities . tape 3 , 2009, p. 323-328 ( MR2597657 ).
• T. Popoviciu: Sur quelques inégalités . In: Gaz. Mat. Fiz. Ser. A . tape 11 (64) , 1959, pp. 451-461 ( MR0125925 ).
• Tiberiu Popoviciu: Sur certaines inégalités qui caractérisent les fonctions convexes . In: Analele Ştiințifice Univ. “Al. I. Cuza ”, Iasi, Secția Mat. [New series] . tape 11 , 1965, pp. 155-164 ( MR0206178 ).
• Shanhe Wu: Some improvements of Aczél's inequality and Popoviciu's inequality . In: Computers & Mathematics with Applications . tape 56 , 2008, p. 1196-1205 , doi : 10.1016 / j.camwa.2008.02.021 ( MR2437287 ).

References and footnotes

1. ↑ See article Tiberiu Popoviciu in the Romanian Wikipedia!
2. ^ Constantin P. Niculescu, Lars-Erik Persson: Convex Functions and Their Applications. 2006, pp. 12, 33
3. ↑ Niculescu / Persson, op.cit., P. 12
4. ↑ Niculescu / Persson, op.cit., P. 14
5. ↑ Niculescu / Persson, op.cit., P. 60
6. ↑ See list ( =>  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. ) In MathSciNet!  
7. ↑ Constantin P. Niculescu: The integral version of Popoviciu's inequality. J. Math. Inequal. 3 (2009), no. 3, 323-328
8. ↑ Shanhe Wu: Some improvements of Aczél's inequality and Popoviciu's inequality In: Comput. Math. Appl. 56, p. 1196 ff
9. ↑ DS Mitrinović: Analytic Inequalities. 1970, pp. 58, 39
10. ↑ The inequality of Aczél obtained by reduction of  . ${\displaystyle p=q=2}$