## Shapiro's Cyclic Sum Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

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 . Ranikin (1958) proved that for
 (3)

 (4)

can be computed by letting be the Convex Hull of the functions
 (5) (6)

Then
 (7)

(Drinfeljd 1971).

A modified sum was considered by Elbert (1973):

 (8)

Consider
 (9)

where
 (10)

and let be the Convex Hull of
 (11) (12)

Then
 (13)

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. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/shapiro/shapiro.html

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical and New Inequalities in Analysis. New York: Kluwer, 1993.