“Shapiro's Cyclic Sum Constant”的概念、定义、习题解题方法及翻译-数学辞典

单词

Shapiro's Cyclic Sum Constant

释义

Shapiro's Cyclic Sum Constant

N.B. A detailed on-line essay by S. Finchwas 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)

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.

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