Certain properties of S(x,n) (Q919155)

S(x,n) is defined as follows: \(S(x,n)=-BS_ 3(x,n)+(n-x)^{n-1} S_ 4(x,n)\), where \(B=(n-1)^{n-1}\) for \(n>1\), \[ S_ 3(x,n)=(8n^ 2- 5)x^ 3-2(8n^ 3+20n^ 2-15n+20)x^ 2+3(24n^ 3-68n^ 2+42n- 5)x+4n(4n-1)(4n-3), \] \[ S_ 4(x,n)=(n-1)(4n^ 2-10n+5)x^ 4+(8n^ 3-52n^ 2+87n-40)x^ 3+ \] \[ +3(12n^ 3-42n^ 2+37n-5)x^ 2+3n(16n^ 2-32n+9)x+12n^ 2(2n-1). \] The main theorem states: S(x,n) is increasing with respect to x in \(1<x<n\) with \(n\geq 2\). The author studies a nonlinear differential equation: \[ nx(1-x^ 2)d^ 2x/dt^ 2+(dx/dt)^ 2+(1-x^ 2)(nx^ 2-1)=0, \] which is the equation of geodesics of the 2-dimensional Riemannian manifold with the metric: \[ ds^ 2=(1-u^ 2-v^ 2)^{n-2}\{(1-v^ 2)du^ 2+2uv du dv+(1-u^ 2)dv^ 2\} \] on \(u^ 2+v^ 2<1\) for which any solution x(t) with \(x^ 2+x^{'2}<1\) for any \(n>1\) is periodic. Regarding its period T as a function of \(\tau =(x_ 1-1)/(n-1)\) and n, where \(x_ 1=n\times \{\max x(t)\}^ 2\), T is monotone decreasing with respect to n (\(\geq 2)\) for any fixed \(\tau\) \((0<\tau <1)\). In the course of the verification of this fact, it was necessary to prove the main theorem stated above for especially \(2\leq n\leq (11+\sqrt{77})/4=4.9437... \).