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Shapiro's Cyclic Sum Constant

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Consider the sum

f_n(x_1,x_2,...,x_n)=(x_1)/(x_2+x_3)+(x_2)/(x_3+x_4)+...+(x_(n-1))/(x_n+x_1)+(x_n)/(x_1+x_2),
(1)

where the x_js are nonnegative and the denominators are positive. Shapiro (1954) asked if

f_n(x_1,x_2,...,x_n)>=1/2n
(2)

for all n. It turns out (Mitrinovic et al. 1993) that this inequality is true for all even n<=12 and odd n<=23.

Define

f(n)=inf_(x>=0)f_n(x_1,x_2,...,x_n),
(3)

and let

lambda=lim_(n->infty)(f(n))/n=inf_(n>=1)(f(n))/n.
(4)

Then Rankin (1958) proved that

lambda<1/2-7×10^(-8).
(5)

lambda can be computed by letting phi(x) be the function convex hull of the functions

y_1=e^(-x)
(6)
y_2=2/(e^x+e^(x/2)).
(7)

Then

lambda=1/2phi(0)=0.4945668...
(8)

(OEIS A086277; Drinfeljd 1971).

A modified sum was considered by Elbert (1973):

g_n(x_1,x_1,...,x_n)=(x_1+x_3)/(x_1+x_2)+(x_2+x_4)/(x_2+x_3)+...+(x_(n-1)+x_1)/(x_(n-1)+x_n)+(x_n+x_2)/(x_n+x_1).
(9)

Consider

mu=lim_(n->infty)(g(n))/n,
(10)

where

g(n)=inf_(x>=0)g_n(x_1,x_2,...,x_n),
(11)

and let psi(x) be the convex hull of

y_1=1/2(1+e^x)
(12)
y_2=(1+e^x)/(1+e^(x/2)).
(13)

Then

mu=psi(0)=0.978012...
(14)

(OEIS A086278).

See also

Convex Hull

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References

Drinfeljd, V. G. "A Cyclic Inequality." Math. Notes. Acad. Sci. USSR 9, 68-71, 1971.Elbert, A. "On a Cyclic Inequality." Period. Math. Hungar. 4, 163-168, 1973.Finch, S. R. "Shapiro-Drinfeld Constant." §3.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 208-211, 2003.Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical and New Inequalities in Analysis. New York: Kluwer, 1993.Rankin, R. A. "An Inequality." Math. Gaz. 42, 39-40, 1958.Shapiro, H. S. "Problem 4603." Amer. Math. Monthly 61, 571, 1954.Sloane, N. J. A. Sequences A086277 and A086278 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Shapiro's Cyclic Sum Constant

Cite this as:

Weisstein, Eric W. "Shapiro's Cyclic Sum Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ShapirosCyclicSumConstant.html

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