OFFSET
0,1
LINKS
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.
Eric Weisstein's World of Mathematics, Shapiro's cyclic sum constant.
FORMULA
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)).
This sequence gives the decimal expansion of B = (1 + exp(b))/2 = (1 + exp(A335809))/2.
The constant B - 1/2 = exp(b)/2 = 0.35924447642... gives the slope of the common tangent.
EXAMPLE
0.8592444764213993132937304712...
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)))/2
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Petros Hadjicostas, Jun 25 2020
STATUS
approved