The radii starlikeness of certain rational functions with real simple poles (Q798794)

The author determines the radius of starlikeness for the following rational function \(f(z)=\sum^{n}_{k=1}A_ k(z-a_ k)^{-1}\) where \(A_ k>0\), \(\sum^{n}_{k=1}A_ k=1,\quad -1\leq a_ k\leq a_{k+1}\leq 1,\quad 1\leq k\leq n-1,\quad n\geq 2\) and the function \(S(z)=f(1/z).\) The radius of starlikeness \(r_ s\) is determined to be \(r_ s=[3(6-33^{{1\over2}})]^{{1\over2}}\) and the corresponding extremal function is found. In the proof, the minimum of the logarithmic derivative \(\sum^{n}_{k=1}(1-a_ kz)^{-1}-\sum^{n-1}_{k=1}(1- b_ kz)^{-1}\) is calculated. The extremal functions are found. This estimate represents an improvement over the known case where the \(a_ k\) are allowed to lie in the disc \(| z| <1\).