Petrović's inequality

The inequality of Petrović ( English Petrović inequality ) is a result of the analysis , one of the branches of mathematics .

The inequality was of the Serbian mathematician Mihailo Petrovic in 1932 published and is related to the inequality of Jensen , from which they as a corollary can be obtained. It gives a simple estimate of certain convex functions in the field of real numbers . Petrović's publication gave rise to a number of further investigations.

formulation

The result can be given as follows:

Let be a real interval with and be a continuous function whose restriction to the interior of the interval is Jensen-convex . ${\displaystyle I=[0,a)}$ ${\displaystyle 0\leq a\leq \infty }$ ${\displaystyle f\colon I\to \mathbb {R} }$ ${\displaystyle f|_{I^{\circ }}}$

Then the inequality always holds for every natural number and every real number with ${\displaystyle n\geq 1}$ ${\displaystyle n}$ ${\displaystyle x_{1},\ldots ,x_{n}\in I}$ ${\displaystyle x_{1}+\dots +x_{n}\in I}$

${\displaystyle f\left(x_{1}+\dots +x_{n}\right)\geq f(x_{1})+\dots +f(x_{n})-(n-1)f(0)}$  .

Evidence sketch

In Marek Kuczma's monograph, two evidences are given. The first of the two uses complete induction . The essential step of this proof is to prove that the above inequality holds for the case and is done using the Jensen inequality. ${\displaystyle n=2}$

Under the conditions mentioned, one can assume and one obtains without loss of generality ${\displaystyle x_{1}+x_{2}>0}$

${\displaystyle {\begin{aligned}f(x_{1})&=f\left({\frac {x_{1}}{x_{1}+x_{2}}}\cdot \left(x_{1}+x_{2}\right)+{\frac {x_{2}}{x_{1}+x_{2}}}\cdot 0\right)\\\\&{\leq }{\frac {x_{1}}{x_{1}+x_{2}}}\cdot f\left(x_{1}+x_{2}\right)+{\frac {x_{2}}{x_{1}+x_{2}}}\cdot f(0)\\\end{aligned}}}$

and in the same way too

${\displaystyle {\begin{aligned}f(x_{2})&{\leq }{\frac {x_{2}}{x_{1}+x_{2}}}\cdot f\left(x_{1}+x_{2}\right)+{\frac {x_{1}}{x_{1}+x_{2}}}\cdot f(0)\\\end{aligned}}}$

and finally by adding the left and right sides of these two inequalities

${\displaystyle f(x_{1})+f(x_{2})\leq f\left(x_{1}+x_{2}\right)+f(0)}$  .

The latter inequality, however, is equivalent to Petrović's inequality for  . ${\displaystyle n=2}$

Sources and background literature

• M. Petrović: Sur une equation fonctionnelle . In: Publ. Math. Univ. Belgrade . tape 1 , 1932, p. 149-156 .
• 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 ).
• JE Pečarić: On the Petrović inequality for convex functions . In: Glasnik Matematički. Seria III . tape 18 (38) , 1983, pp. 77-85 ( MR0710389 ).
• Josip Pečarić, Jurica Perić: Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type means . In: Analele Universităţii din Craiova. Seria Matematică-Informatică . tape 39 , 2012, p. 65-75 ( MR2979954 ).
• Petar M. Vasić: Sur une inégalité de M. Petrović . In: Matematichki Vesnik . tape 5 (20) , 1968, pp. 473-478 ( MR0243019 ).

Individual evidence

1. ^ ^{A b} Marek Kuczma: An Introduction to the Theory of Functional Equations and Inequalities. 2009, p. 217