OFFSET
0,1
COMMENTS
From Petros Hadjicostas, Jun 01 2020: (Start)
Based on the references, it seems that this constant was first defined by Elbert (1973). We have psi(0) = mu. Two auxiliary constants, b = -0.33060494... = -A335809 and c = 0.38755227... = A335810, are needed for the estimation of mu.
Here psi(x) is the convex hull of y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)); i.e., psi(x) = (1 + exp(x))/2 for x <= b; psi(x) = (1 + exp(b))/2 + (((1 + exp(c))/(1 + exp(c/2)) - (1 + exp(b))/2)/(c - b)) * (x - b) for b <= x <= c; and psi(x) = (1 + exp(x))/(1 + exp(x/2)) for x >= c. (For b <= x <= c, we have the equation of the line segment tangent to both curves.)
It follows that mu = psi(0) = (1 + exp(b))/2 - b * (((1 + exp(c))/(1 + exp(c/2)) - (1 + exp(b))/2)/(c - b)) (where the y-axis crosses the line segment). Or by using the tangent line at x = b to the curve y = (1 + exp(x))/2, we find mu = psi(0) = (1 + exp(b))/2 - b * exp(b)/2. Or by using the tangent line at x = c to the curve y = (1 + exp(x))/(1 + exp(x/2)), we may get a third formula for mu = psi(0) in terms of c only.
Similar calculations were done by Drinfel'd (1971) for the Shapiro cyclic sum constant lambda = phi(0)/2 = A086277 = A245330/2. In this case, the corresponding curves are y = exp(-x) and y = 2/(exp(x) + exp(x/2)), while the corresponding x-coordinates at the tangent points are -A319568 = -0.20081... and A319569 = 0.15519... Here phi(x) is the convex hull of these two curves (and it becomes a line segment tangent to both curves for -A319568 <= x <= A319569).
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 = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + 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 World of Mathematics, Shapiro's cyclic sum constant.
FORMULA
From Petros Hadjicostas, Jun 23 2020: (Start)
Solve the following system of equations to find the x-coordinates of the two points where the common tangent touches the two curves:
exp(b) = (-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2 and
(exp(b)*(c - b + 1) + 1)*(1 + exp(c/2)) = 2*(1 + exp(c)).
Then the constant equals (1 + exp(b)*(1 - b))/2. (End)
It can be proved that this constant equals 1 plus the Gauchman constant (which is the negation of A243261); i.e., without negations, A086278 = 1 - A243261. - Petros Hadjicostas, Jul 04 2020
EXAMPLE
0.97801247818664622020182795997868268... = 1 - 0.02198752181335377979817204...
MATHEMATICA
eq = E^u + 2*E^(u + v/2) + E^(v/2) + E^(u + v) == 2*E^v + E^(3*v/2) && 2 + 2*E^(u + v/2) == 2*y + 2*E^v + E^u*(v - 2) && E^u*(u - v + 1) + 2*E^(u + v/2) + 1 == 2*E^v; mu = y /. FindRoot[eq , {{y, 1}, {u, -1/3}, {v, 1/3}}, WorkingPrecision -> 105]; RealDigits[mu, 10, 103] // First
PROG
(PARI)
default("realprecision", 200)
b(c) = log((-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2);
a = solve(c=-1, 1, (exp(b(c))*(c - b(c) + 1) + 1)*(1 + exp(c/2)) - 2*(1 + exp(c)));
(1 + exp(b(a))*(1 - b(a)))/2 \\ Petros Hadjicostas, Jun 23 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 14 2003
EXTENSIONS
More terms from Jean-François Alcover, Jun 02 2014
STATUS
approved