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

$$f_n(x_1, x_2, \ldots, x_n) = {x_1\over x_2+x_3}+{x_2\over x_3+x_4}+\ldots+{x_{n-1}\over x_n+x_1}+{x_n\over x_1+x_2},\enqo$$ (1)

where the $x_j$s are Nonnegative and the Denominators are Positive. Shapiro (1954) asked if

$$f_n(x_1, x_2, \ldots, x_n) \geq {\textstyle{1\over 2}}n$$ (2)

for all $n$. It turns out (Mitrinovic et al. 1993) that this Inequality is true for all Even $n\leq 12$ and Odd $n\leq 23$. Ranikin (1958) proved that for

$$f(n)=\inf_{x\geq 0} f_n(x_1, x_2, \ldots, x_n),$$ (3)

$$\lambda=\lim_{n\to\infty} {f(n)\over n}=\inf_{n\geq 1}{f(n)\over n}<{\textstyle{1\over 2}}-7\times 10^{-8}.$$ (4)

$\lambda$ can be computed by letting $\phi(x)$ be the Convex Hull of the functions

$$y_1 = e^{-x}$$ (5)

$$y_2 = {2\over e^x+e^{x/2}}.$$ (6)

Then

$$\lambda={\textstyle{1\over 2}}\phi(0)=0.4945668\ldots$$ (7)

(Drinfeljd 1971).

A modified sum was considered by Elbert (1973):

$$g_n(x_1, x_1, \ldots, x_n)={x_1+x_3\over x_1+x_2}+{x_2+x_4\o... ...+\ldots+{x_{x-1}+x_1\over x_{n-1}+x_n}+{x_n+x_2\over x_n+x_1}.$$ (8)

Consider

$$\mu=\lim_{n\to\infty} {g(n)\over n},$$ (9)

where

$$g(n)=\inf_{x\geq 0} g_n(x_1, x_2, \ldots, x_n),$$ (10)

and let $\psi(x)$ be the Convex Hull of

$$y_1 = {\textstyle{1\over 2}}(1+e^x)$$ (11)

$$y_2 = {1+e^x\over 1+e^{x/2}}.$$ (12)

Then

$$\mu=\psi(0)=0.978012\ldots.$$ (13)

See also Convex Hull

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.