## 3 Special Cases of Weight Functions  
Theorem 1 clearly covers the existing case study in \(\mathbb{R}\) shown in [21]  
\[\left\{ \begin{array}{l}\gamma=\frac{2}{3},\\ T_{f}(s)=\big{(}1+\frac{1-s}{2(\beta-\lambda+\lambda s)}\big{)}H(1-s),\lambda \in\mathbb{R},\beta\in\mathbb{R}-\{0\}\\ L_{f}(s)=\frac{1}{2}(5-3s),\end{array} \right. \tag{15}\]  
where \(H(t)=1+\big{(}\frac{3\beta-2}{4\beta}\big{)}t+\big{(}\frac{9\beta^{2}-3\beta -4\lambda+2}{8\beta^{2}}\big{)}t^{2}\). Note that \(T_{f}\) in (15) is a cubic-order rational weight function. It also covers the existing case study in \(\mathbb{R}\) shown in [2]  
\[\left\{ \begin{array}{l}\gamma=\frac{2}{3},\\ T_{f}(s)=1+\frac{3(1-s)}{4}\big{(}1+\frac{3\beta(1-s)}{4-3\alpha(1-s)}\big{)}, \alpha,\beta\in\mathbb{R},\\ L_{f}(s)=\frac{1}{2}(5-3s),\end{array} \right. \tag{16}\]  
where \(T_{f}\) in (15) is a second-order rational weight function. Although the result of Theorem 1 allows an infinite number of other forms of weight functions \(T_{f}(s)\) and \(L_{f}(s)\), we are specifically interested only in second-order rational forms for the both weight functions \(T_{f}(s)\) and \(L_{f}(s)\). \[\left\{ \begin{array}{l}T_{f}(s)=\frac{a_{i}+b_{i}s+g_{i}s^{2}}{c_{i}+d_{i}s +h_{i}s^{2}},\\ L_{f}(s)=\frac{p_{i}+q_{i}s+k_{i}s^{2}}{r_{i}+\sigma_{i}s+\tau_{i}s^{2}},\text { for }1\leq i\leq 2,\end{array} \right. \tag{17}\]  
where the desired coefficients are to be determined based on the results of Theorem 1. Two cases can be considered. The first case can be given as follows:  
**Case A:**\(\{\gamma=\frac{2}{3},T_{0}=1,T_{1}=-\frac{3}{4},L_{0}=1,L_{1}=-\frac{3}{2},T_ {2}=9,|T_{3}|<\infty,|L_{2}|<\infty\}\)  
\(b_{1}=\frac{1}{3}(9a_{1}-31g_{1}+16h_{1}),c_{1}=-2a_{1}+6g_{1}-3h_{1},d_{1}= \frac{2}{3}(9a_{1}-23g_{1}+11h_{1})\), \(T_{3}=-\frac{9(9a_{1}-23g_{1}+13h_{1})}{16(3a_{1}-g_{1}+4h_{1})}\), \(p_{1}=\frac{1}{3}(-7k_{1}-5q_{1}+2\sigma_{1}+4\tau_{1})\), \(r_{1}=\frac{1}{3}(-4k_{1}-2q_{1}-\sigma_{1}+\tau_{1})\), \(L_{2}=-\frac{3(2k_{1}+3\sigma_{1}+4\tau_{1})}{4(2k_{1}+q_{1}-\sigma_{1}-2\tau _{1})}\). Hence, the desired form of (17) becomes:  
\[\left\{ \begin{array}{l}T_{f}(s)=\frac{a_{1}+\frac{1}{3}(9a_{1}-31g_{1}+16h _{1})s+g_{1}s^{2}}{-2a_{1}+6g_{1}-3h_{1}+\frac{2}{3}(9a_{1}-23g_{1}+11h_{1})s +h_{1}s^{2}},\\ L_{f}(s)=\frac{\frac{1}{3}(-7k_{1}-5q_{1}+2\sigma_{1}+4\tau_{1})+q_{1}s+k_{1}s ^{2}}{\frac{1}{3}(-4k_{1}-2q_{1}-\sigma_{1}+\tau_{1})+\sigma_{1}s+\tau_{1}s^{ 2}},\end{array} \right. \tag{18}\]  
where \(a_{1},g_{1},h_{1},k_{1},q_{1},\sigma_{1}\), and \(\tau_{1}\) are free parameters. Similarly, we can find the second case:  
**Case B:**\(\{\gamma=1,T_{0}=1,T_{1}=-\frac{1}{2},L_{0}=1,L_{1}=-1,T_{2}=\frac{3}{2},|T_{2}|< \infty,|L_{3}|<\infty\}\)  
\(b_{2}=\frac{1}{3}(-a_{2}+2d_{2}-5g_{2}+4h_{2}),c_{2}=\frac{1}{3}(2a_{2}-d_{2}-2 g_{2}+h_{2})\), \(T_{2}=\frac{-9d_{2}^{2}+6h_{2}(a_{2}-7g_{2}+2h_{2})-6d_{2}(3g_{2}+2h_{2})}{8(a_{2} +d_{2}-g_{2}+2h_{2})^{2}}\), \(p_{2}=\frac{1}{3}(7k_{2}+\sigma_{2}+2\tau_{2})\), \(r_{2}=\frac{1}{3}(2k_{2}-\sigma_{2}-\tau_{2})\), \(q_{2}=\frac{1}{3}(-8k_{2}+\sigma_{2}+4\tau_{2})\), \(L_{3}=-\frac{3(3g_{2}+4\tau_{2})}{4(2k_{2}+\sigma_{2}+\tau_{2})}\). Hence, the desired form of (17) becomes:\[\left\{ \begin{array}{ll}T_{f}(s)&=\frac{a_{2}+\frac{1}{4}(-a_{2}+2d_{2}-5a_{2}+4 h_{2})s+g_{2}s^{2}}{\frac{1}{2}2a_{2}-d_{2}-2a_{2}+h_{2}+4g_{2}+h_{2}s^{2}},\\ L_{f}(s)&=\frac{\frac{1}{4}(7k_{2}+\sigma_{2}-2r_{2})+\frac{1}{4}(-8k_{2}+ \sigma_{2}+4r_{2})s+k_{2}s^{2}}{\frac{1}{3}(2k_{2}-\sigma_{2}-r_{2})+\sigma_{2} s+\tau_{2}s^{2}},\end{array} \right. \tag{19}\]  
where \(a_{2},d_{2},g_{2},h_{2},k_{2},\sigma_{2}\), and \(\tau_{2}\) are free parameters. Although numerous forms of weight functions \(T_{f}(s)\) and \(L_{f}(s)\) satisfying (18) or (19) are applicable, we are specifically interested to the following forms:  
**Case A: \(\gamma=\frac{2}{3}\)**  
\[\left\{