Consider the sum

(1)

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

(2)

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

Define

(3)

and let

(4)

Then Rankin (1958) proved that for

(5)

can be computed by letting be the function convex hull of the functions

			(6)
			(7)

Then

(8)

(Sloane's A086277; Drinfeljd 1971).

A modified sum was considered by Elbert (1973):

(9)

Consider

(10)

where

(11)

and let be the convex hull of

			(12)
			(13)

Then

(14)

(Sloane's A086278).

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."

CITE THIS AS:

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