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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
\end{displaymath} (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
\end{displaymath} (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),
\end{displaymath} (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}.
\end{displaymath} (4)

$\lambda$ can be computed by letting $\phi(x)$ be the Convex Hull of the functions
$\displaystyle y_1$ $\textstyle =$ $\displaystyle e^{-x}$ (5)
$\displaystyle y_2$ $\textstyle =$ $\displaystyle {2\over e^x+e^{x/2}}.$ (6)

\lambda={\textstyle{1\over 2}}\phi(0)=0.4945668\ldots
\end{displaymath} (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}.
\end{displaymath} (8)

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

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

and let $\psi(x)$ be the Convex Hull of
$\displaystyle y_1$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(1+e^x)$ (11)
$\displaystyle y_2$ $\textstyle =$ $\displaystyle {1+e^x\over 1+e^{x/2}}.$ (12)

\end{displaymath} (13)

See also Convex Hull


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.''

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

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© 1996-9 Eric W. Weisstein