Co-convex interpolation by rational spline functions over a uniform grid of nodes

To solve an interpolation problem with the conditions of preserving the convexity and co-convexity of discrete functions $f(x)$ defined on uniform grids of nodes $\Delta: a=x_0<x_1<\dots<x_N=b$ $(N\geqslant 3)$ rational spline-functions $R_{N,1}(x)$ are applied. Here $R_{N,1}(x)=R_{N,1} (x, f, \Delta, g(t))= (R_i(x)(x-x_{i-1})+R_{i-1}(x)(x_i-x))/(x_i-x_{i-1})$ with $x\in [x_{i-1},x_i]$ $(i=1,2,\dots,N)$, $R_i(x)=\alpha_i+\beta_i(x-x_i)+\gamma_i/(x-g_i(t))$ $(i=1,2,\dots,N-1)$ and $R_i(x_j)=f(x_j)$ $(j=i-1,i,i+1)$, the parameter $t$ locate a position of the pole $g_i(t)$ with respect to the points $x_{i-1}$ and $x_i$. We take $R_0(x)\equiv R_1(x)$, $R_N(x)\equiv R_{N-1}(x)$.
For such splines we obtain co-convex preserving conditions $0,5<q_i<2$ or $-3,20...<q_i<-0,31...$ with $q_i=f(x_{i-2}, x_{i-1}, x_i)/f(x_{i-1},x_i, x_{i+1})$ for all corresponding intervals $(x_{i-1},x_i)$, hence for the segment $[a,b]$.