Coconvex interpolation by splines with three-point rational interpolants

For discrete functions $f(x)$ defined on arbitrary grid nodes $\Delta: a=x_0 < x_1 < \dots < x_N=b$ $(N\geqslant 3)$, we study the issues of preserving the (upward or downward) convexity and coconvexity with a change of convexity direction by rational spline-functions $R_{N,1}(x)=R_{N,1}(x,f,\Delta,g(t))=(R_i(x)(x-x_{i-1})+R_{i-1}(x)(x_i-x))/(x_i-x_{i-1})$, where $x\in [x_{i-1},x_i]$ $(i=1,2,\dots,N)$, $R_i(x)=\alpha_i+\beta_i(x-x_i)+\gamma_i/(x-g_i(t))$ $(i=1,2,\dots,N-1)$, and $R_i(x_j)=f(x_j)$ $(j=i-1,i,i+1)$. The location of the pole $g_i(t)$ with respect to the nodes $x_{i-1}$ and $x_i$ is defined by the parameter $t$. We assume that $R_0(x)\equiv R_1(x)$ and $R_N(x)\equiv R_{N-1}(x)$. For these spines we derive the conditions $1/2 < |q_i| < 2$ of convexity preservation, where $q_i=f(x_{i-2},x_{i-1},x_i)/f(x_{i-1},x_i,x_{i+1})$ for $i=2,3,\dots,N-1$.