OFFSET
0,1
COMMENTS
From Petros Hadjicostas, Jun 01 2020: (Start)
The calculations in this sequence and in A319569 are needed for the estimation of the Shapiro cyclic sum constant lambda = A086277 = phi(0)/2 = A245330/2. This was done in Drinfel'd (1971).
Here phi(x) is the convex hull of y = exp(-x) and y = 2/(exp(x) + exp(x/2)); i.e., phi(x) = 2/(exp(x) + exp(x/2)) for x <= b; phi(x) = exp(-c) + ((2/(exp(b) + exp(b/2)) - exp(-c))/(b - c)) * (x - c) for b <= x <= c; and phi(x) = exp(-x) for x >= c. (For b <= x <= c, we have the equation of the line segment tangent to both curves.)
As stated below, b = -0.200811... (current sequence) and c = 0.15519... (A319569).
It follows that phi(0) = exp(-c) - c * ((2/(exp(b) + exp(b/2)) - exp(-c))/(b - c)) (where the y-axis crosses the line segment). Or by using the tangent line at x = c to the curve y = exp(-x), we find phi(0) = exp(-c)*(c + 1). Or by using the tangent line at x = b to the curve y = 2/(exp(x) + exp(x/2)), we may get a third formula for phi(0).
By using the above information, we get a system of two equations with two unknowns (b and c). See the PARI program below that may be used to calculate b.
Similar calculations were done by Elbert (1973) for the Shapiro cyclic sum constant mu = psi(0) = A086278. The corresponding curves are y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)), while the corresponding x-coordinates at the tangent points are b' = -0.33060... and c' = 0.38875... (not in the OEIS yet). Here psi(x) is the convex hull of these two curves (and it becomes a line segment tangent to both curves for b' <= x <= c').
Eric W. Weisstein, in the link below, has a summary of the above discussion (with contributions by Steven Finch). (End)
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld Constant, p. 209.
LINKS
V. G. Drinfel'd, A cyclic inequality, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.
Á. Elbert, On a cyclic inequality, Periodica Mathematica Hungarica, 4 (1973), 163-168.
Á. Elbert, On a cyclic inequality, Periodica Mathematica Hungarica, 4 (1973), 163-168.
Petros Hadjicostas, Plot of the curves y = exp(-x) and y = 2/(exp(x) + exp(x/2)) and their common tangent, 2020.
R. A. Rankin, 2743. An inequality, Mathematical Gazette, 42(339) (1958), 39-40.
H. S. Shapiro, Proposed problem for solution 4603, American Mathematical Monthly, 61(8) (1954), 571.
H. S. Shapiro, Solution to Problem 4603: An invalid inequality, American Mathematical Monthly, 63(3) (1956), 191-192; counterexample provided by M. J. Lighthill.
B. A. Troesch, The validity of Shapiro's cyclic inequality, Mathematics of Computation, 53 (1989), 657-664.
Eric Weisstein's MathWorld, Shapiro's Cyclic Sum Constant.
FORMULA
b = -0.2008110658618... .
c = 0.1551949747226... (A319569).
exp(c) = (1 + exp(b/2))^2/(2 + exp(-b/2)).
EXAMPLE
-0.2008110658618...
PROG
(PARI) c(b) = b + exp(b/2)/(2*exp(b)+exp(b/2))
solve(b=-2, 2, exp(-c(b))*(1-b+c(b))-2/(exp(b)+exp(b/2))) \\ Petros Hadjicostas, Jun 02 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Seiichi Manyama, Sep 23 2018
EXTENSIONS
More terms from Vaclav Kotesovec, Sep 26 2018
STATUS
approved