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%I #69 Jul 07 2020 06:22:50
%S 9,7,8,0,1,2,4,7,8,1,8,6,6,4,6,2,2,0,2,0,1,8,2,7,9,5,9,9,7,8,6,8,2,6,
%T 8,0,9,3,2,5,3,8,6,3,5,3,4,5,9,1,4,1,8,0,9,4,9,5,3,0,4,2,0,8,3,4,5,9,
%U 9,4,4,9,2,5,8,0,7,1,0,6,9,7,5,0,0,5,5,6,6,8,9,8,5,2,0,3,9,2,6,5,9,2,4
%N Decimal expansion of Shapiro's cyclic sum constant mu.
%C From _Petros Hadjicostas_, Jun 01 2020: (Start)
%C 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.
%C 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.)
%C 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.
%C 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).
%C Eric W. Weisstein, in the link below, has a summary of the above discussion (with contributions by Steven Finch). (End)
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 209.
%H V. G. Drinfel'd, <a href="https://doi.org/10.1007/BF01316982">A cyclic inequality</a>, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.
%H Á. Elbert, <a href="https://doi.org/10.1007/BF02276104">On a cyclic inequality</a>, Periodica Mathematica Hungarica, 4 (1973), 163-168.
%H Á. Elbert, <a href="https://akjournals.com/view/journals/10998/4/2-3/article-p163.xml">On a cyclic inequality</a>, Periodica Mathematica Hungarica, 4 (1973), 163-168.
%H Petros Hadjicostas, <a href="/A086278/a086278.pdf">Plot of the curves y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)) and their common tangent</a>, 2020.
%H R. A. Rankin, <a href="https://doi.org/10.2307/3608356">2743. An inequality</a>, Mathematical Gazette, 42(339) (1958), 39-40.
%H H. S. Shapiro, <a href="https://www.jstor.org/stable/2307617">Proposed problem for solution 4603</a>, American Mathematical Monthly, 61(8) (1954), 571.
%H H. S. Shapiro, <a href="https://www.jstor.org/stable/2306671">Solution to Problem 4603: An invalid inequality</a>, American Mathematical Monthly, 63(3) (1956), 191-192; counterexample provided by M. J. Lighthill.
%H B. A. Troesch, <a href="https://doi.org/10.1090/S0025-5718-1989-0983563-0">The validity of Shapiro's cyclic inequality</a>, Mathematics of Computation, 53 (1989), 657-664.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ShapirosCyclicSumConstant.html">Shapiro's cyclic sum constant</a>.
%F From _Petros Hadjicostas_, Jun 23 2020: (Start)
%F Solve the following system of equations to find the x-coordinates of the two points where the common tangent touches the two curves:
%F exp(b) = (-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2 and
%F (exp(b)*(c - b + 1) + 1)*(1 + exp(c/2)) = 2*(1 + exp(c)).
%F Then the constant equals (1 + exp(b)*(1 - b))/2. (End)
%F 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
%e 0.97801247818664622020182795997868268... = 1 - 0.02198752181335377979817204...
%t 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
%o (PARI)
%o default("realprecision",200)
%o b(c) = log((-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2);
%o a = solve(c=-1, 1, (exp(b(c))*(c - b(c) + 1) + 1)*(1 + exp(c/2)) - 2*(1 + exp(c)));
%o (1 + exp(b(a))*(1 - b(a)))/2 \\ _Petros Hadjicostas_, Jun 23 2020
%Y Cf. A086277, A243261, A245330, A319568, A319569, A335809, A335810.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Jul 14 2003
%E More terms from _Jean-François Alcover_, Jun 02 2014
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