Shapiro's inequality

The Shapiro inequality is one of consequences of positive numbers existing inequality of mathematics . It is named after Harold Shapiro .

Inequality

Be it

{\ displaystyle x_ {1}, x_ {2}, x_ {3}, \ ldots}

a sequence of positive real numbers .

Then the inequality holds for all even numbers and all odd numbers ${\ displaystyle n \ leq 12}$ ${\ displaystyle n \ leq 23}$

{\ displaystyle {\ frac {x_ {1}} {x_ {2} + x_ {3}}} + {\ frac {x_ {2}} {x_ {3} + x_ {4}}} + \ ldots + {\ frac {x_ {n-2}} {x_ {n-1} + x_ {n}}} + {\ frac {x_ {n-1}} {x_ {n} + x_ {1}}} + {\ frac {x_ {n}} {x_ {1} + x_ {2}}} \ geq {\ frac {n} {2}}}

.

The inequality generally does not hold for even numbers and for odd numbers . ${\ displaystyle n \ geq 14}$ ${\ displaystyle n \ geq 25}$

The simplest known counterexample for is the consequence ${\ displaystyle n = 14}$

{\ displaystyle \ epsilon, 42,2,42,4,41,5,39,4,38,2,38, \ epsilon, 40}

for sufficiently small . ${\ displaystyle \ epsilon> 0}$

HS Shapiro: Advanced Problems and Solutions , Amer. Math. Monthly 61 (1954), 571-572.
BA Troesch: The validity of Shapiro's cyclic inequality. Math. Comp. 53 (1989), no. 188, 657-664.
R. Hemmecke, W. Moldenhauer: About Shapiro's inequality. Knowledge Z. pedagogue. Hochsch. Erfurt / Mühlhausen Math.-Nature. Series 26 (1990), no. 1, 33-41.
A. Clausing: A review of Shapiro's cyclic inequality. General inequalities, 6 (Oberwolfach, 1990), 17-31, Internat. Ser. Number Math., 103, Birkhäuser, Basel, 1992.
AM Fink: Shapiro's inequality. Recent progress in inequalities (Niš, 1996), 241-248, Math. Appl., 430, Kluwer Acad. Publ., Dordrecht, 1998
T. Ando: A new proof of Shapiro inequality. Math. Inequal. Appl. 16 (2013), no. 3, 611-632.