Shapiro inequality

In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954.[1]

Statement of the inequality

Suppose is a natural number and are positive numbers and:

  • is even and less than or equal to , or
  • is odd and less than or equal to .

Then the Shapiro inequality states that

where .

For greater values of the inequality does not hold and the strict lower bound is with .

The initial proofs of the inequality in the pivotal cases (Godunova and Levin, 1976) and (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for .

The value of was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound is given by , where the function is the convex hull of and . (That is, the region above the graph of is the convex hull of the union of the regions above the graphs of ' and .)[2]

Interior local minima of the left-hand side are always (Nowosad, 1968).

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for :

where is close to 0.

Then the left-hand side is equal to , thus lower than 10 when is small enough.

The following counter-example for is by Troesch (1985):

(Troesch, 1985)

References

  1. Shapiro, H. S.; Bellman, R.; Newman, D. J.; Weissblum, W. E.; Smith, H. R.; Coxeter, H. S. M. (1954). "Problems for Solution: 4603-4607". The American Mathematical Monthly. 61 (8): 571. doi:10.2307/2307617. Retrieved 2021-09-23.
  2. Drinfel'd, V. G. (1971-02-01). "A cyclic inequality". Mathematical Notes of the Academy of Sciences of the USSR. 9 (2): 68–71. doi:10.1007/BF01316982. ISSN 1573-8876. S2CID 121786805.
  • Fink, A.M. (1998). "Shapiro's inequality". In Gradimir V. Milovanović, G. V. (ed.). Recent progress in inequalities. Dedicated to Prof. Dragoslav S. Mitrinović. Mathematics and its Applications (Dordrecht). Vol. 430. Dordrecht: Kluwer Academic Publishers. pp. 241–248. ISBN 0-7923-4845-1. Zbl 0895.26001.
  • Bushell, P.J.; McLeod, J.B. (2002). "Shapiro's cyclic inequality for even n" (PDF). J. Inequal. Appl. 7 (3): 331–348. ISSN 1029-242X. Zbl 1018.26010. They give an analytic proof of the formula for even , from which the result for all follows. They state as an open problem.
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