OFFSET
0,2
COMMENTS
Let a_i > 0 for i = 1, ..., n, and define the cyclic sums E_n = a_1/(a_2 + a_3) + a_2/(a_3 + a_4) + ... + a_n/(a_1 + a_2) and F_n = a_1/(a_1 + a_2) + a_2/(a_2 + a_3) + ... + a_n/(a_n + a_1). Gauchman (1998) proved that E_n - F_n >= lambda*n for all n >= 1, where lambda = -0.02198... (this constant) is the best constant. His proof was not published, however, as a solution to Problem 10528(b) in the American Mathematical Monthly (see the link below). Only a comment was made on p. 474. - Petros Hadjicostas, Jun 02 2020
Named after the mathematicians Harold Seymour Shapiro (1928-2021) and Vladimir Drinfeld (b. 1954). Alternatively, named after the American mathematician Hillel V. Gauchman (1937-2016). - Amiram Eldar, May 29 2021
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 210.
Hillel Gauchman, Solution to Problem 10528(b), unpublished note, 1998.
LINKS
Vasile Cârtoaje, Jeremy Dawson and Hans Volkmer, Solution to Problem 10528(a,b), American Mathematical Monthly, 105 (1998), 473-474. [A comment was made about Hillel Gauchman's solution to part (b) of the problem that involves this constant, but no solution was published.]
Petros Hadjicostas, Plot of the curves y = (1 - exp(x/2))/(exp(x) + exp(x/2)) and y = (exp(-x) - 1)/2 and their common tangent, 2020.
FORMULA
From Petros Hadjicostas, Jul 04 2020: (Start)
We solve the following system of equations:
exp(-c) = (exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2 and
2*(1 - exp(b/2)) = (exp(b) + exp(b/2))*(exp(-c)*(1 + c - b) - 1).
Then the constant equals (exp(-c)*(1 + c) - 1)/2.
EXAMPLE
-0.02198752181335377979817204... = -1 + 0.9780124781866462202018...
MATHEMATICA
eq = E^(u/2) + 2*E^u + E^(3*u/2) + E^(u + v) == E^v + 2*E^(u/2 + v) && 2*(y + E^(u/2 - v) + 1) == (u - 2)/E^v + 4/E^(u/2) && u + 6*E^(u/2) + 4*E^u + 4*E^(u/2 + v) + 1 == v + 9*E^v; y0 = y /. FindRoot[eq , {{y, 0}, {u, -1/3}, {v, 1/3}}, WorkingPrecision -> 105]; RealDigits[y0, 10, 99] // First
PROG
(PARI) default("realprecision", 200)
c(b) = -log((exp(b/2) + 2*exp(b) - exp(3*b/2))/(exp(b) + exp(b/2))^2);
a = solve(b=-2, 0, (exp(b) + exp(b/2))*(-1 + exp(-c(b))*(1 + c(b) - b)) - 2*(1 - exp(b/2)));
(exp(-c(a))*(1 + c(a)) - 1)/2 \\ Petros Hadjicostas, Jul 04 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 02 2014
EXTENSIONS
Name edited by Petros Hadjicostas, Jun 02 2020
STATUS
approved