|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Shapiro inequality''' is an [[inequality (mathematics)|inequality]] proposed by H. Shapiro in 1954.
| | The author is known as Irwin Wunder but it's not the most masucline name out there. Puerto Rico is exactly where he's always been living but she needs to move because of her family. Body developing is 1 of the things I love most. My working day occupation is a meter reader.<br><br>Feel free to surf to my blog ... [http://www.hard-ass-porn.com/blog/111007 at home std test] |
| | |
| ==Statement of the inequality==
| |
| | |
| Suppose ''n'' is a [[natural number]] and <math>x_1, x_2, \dots, x_n</math> are [[positive number]]s and:
| |
| | |
| * ''n'' is even and less than or equal to 12, or
| |
| * ''n'' is odd and less than or equal to 23.
| |
| | |
| Then the '''Shapiro inequality''' states that
| |
| | |
| :<math>\sum_{i=1}^{n} \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}</math>
| |
| | |
| where <math>x_{n+1}=x_1, x_{n+2}=x_2</math>.
| |
| | |
| For greater values of ''n'' the inequality does not hold and the strict lower bound is <math>\gamma \frac{n}{2}</math> with <math>\gamma \approx 0.9891\dots</math>.
| |
| | |
| The initial proofs of the inequality in the pivotal cases ''n'' = 12 (Godunova and Levin, 1976) and ''n'' = 23 (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for ''n'' = 12.
| |
| | |
| The value of γ was determined in 1971 by [[Vladimir Drinfeld]], who won a [[Fields Medal]] in 1990. Specifically, Drinfeld showed that the strict lower bound ''γ'' is given by <math>\frac{1}{2} \psi(0)</math>, where ''ψ'' is the function convex hull of ''f''(''x'') = ''e''<sup>−''x''</sup> and <math>g(x) = \frac{2}{e^x+e^{\frac{x}{2}}}</math>. (That is, the region above the graph of ''ψ'' is the [[convex hull]] of the union of the regions above the graphs of ''f'' and ''g''.)
| |
| | |
| Interior local mimima of the left-hand side are always ≥ ''n''/2 (Nowosad, 1968).
| |
| | |
| ==Counter-examples for higher ''n''==
| |
| | |
| The first counter-example was found by Lighthill in 1956, for ''n'' = 20:
| |
| :<math>x_{20} = (1+5\epsilon,\ 6\epsilon,\ 1+4\epsilon,\ 5\epsilon,\ 1+3\epsilon,\ 4\epsilon,\ 1+2\epsilon,\ 3\epsilon,\ 1+\epsilon,\ 2\epsilon,\ 1+2\epsilon,\ \epsilon,\ 1+3\epsilon,\ 2\epsilon,\ 1+4\epsilon,\ 3\epsilon,\ 1+5\epsilon,\ 4\epsilon,\ 1+6\epsilon,\ 5\epsilon)</math> where <math>\epsilon</math> is close to 0.
| |
| Then the left-hand side is equal to <math>10 - \epsilon^2 + O(\epsilon^3)</math>, thus lower than 10 when <math>\epsilon</math> is small enough.
| |
| | |
| The following counter-example for ''n'' = 14 is by Troesch (1985):
| |
| :<math>x_{14}</math> = (0, 42, 2, 42, 4, 41, 5, 39, 4, 38, 2, 38, 0, 40) (Troesch, 1985)<!--this example has been double-checked by user:FvdP, 2010/01/12-->
| |
| <!-- Next counter-example is cited in A M Fink 1998 as from Troesch 1985. Sadly, it seems wrong: I compute LHS = 0.50010878... there probably is a typo in it.
| |
| :<math>x_{25}</math> = (25, 0, 29, 0, 34, 5, 35, 13, 30, 17, 24, 18, 18, 17, 13, 16, 9, 16, 5, 16, 2, 18, 0, 20, 0)
| |
| -->
| |
| | |
| ==References==
| |
| * {{cite book | zbl=0895.26001 | last=Fink | first=A.M. | chapter=Shapiro's inequality | editor=Gradimir V. Milovanović, G. V. | title=Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinović | location=Dordrecht | publisher=Kluwer Academic Publishers. | series=Mathematics and its Applications (Dordrecht) | volume=430 | pages=241–248 | year=1998 | isbn=0-7923-4845-1 }}
| |
| * {{cite journal | zbl=1018.26010 | last1=Bushell | first1=P.J. | last2=McLeod | first2=J.B. | title=Shapiro's cyclic inequality for even n | journal=J. Inequal. Appl. | volume=7 | number=3 | pages=331–348 | year=2002 | issn=1029-242X | url=http://downloads.hindawi.com/journals/jia/2002/509463.pdf}} They give an analytic proof of the formula for even ''n'' ≤ 12, from which the result for all ''n'' ≤ 12 follows. They state ''n'' = 23 as an open problem.
| |
| | |
| ==External links==
| |
| * [http://www.math.niu.edu/~rusin/known-math/99/shapiro Usenet discussion in 1999] (Dave Rusin's notes)
| |
| * [http://planetmath.org/encyclopedia/ShapiroInequality.html PlanetMath]
| |
| | |
| [[Category:Inequalities]]
| |
The author is known as Irwin Wunder but it's not the most masucline name out there. Puerto Rico is exactly where he's always been living but she needs to move because of her family. Body developing is 1 of the things I love most. My working day occupation is a meter reader.
Feel free to surf to my blog ... at home std test