On the order of starlikeness of the shifted Gauss hypergeometric function
For a function \(f\), analytic in \textbf{D}\(=\{z\in\mathbf{C}:| z| <1\}\), with \(f(0)=0\not= f'(0)\), the \textit{order of starlikeness} (with respect to zero) is \[ \sigma(f):=\inf_{z\in\mathbf{D}}\,\text{Re}\,{zf'(z)\over f(z)}\in [-\infty,1], \] and if at least \(f'(0)\not=0\), the \textit{order of convexity} of \(f\) is \[ \kappa(f):=\sigma(zf')=1+\inf_{z\in\mathbf{D}}\,\text{Re}\,{zf''(z)\over f'(z)}\in [-\infty,1]. \] The author studies the `shifted' hypergeometric function \[ g(z):=zF(a,b,c,z)=z\,{}_2F_1(a,b;c;z) \] and gives several new results, for instance \vskip0.2cm \textbf{1.} \(a<0<b,\,c\geq b-a+1\): \[ \sigma(g)=1+{F'(a,b,c,1)\over F(a,b,c,1)}=1+{ab\over c-a-b-1}\geq 1-{b\over 2}. \] \textbf{2.} \(0<a<c<b\leq c-a+1\): \[ \sigma(g)=-\infty. \] \textbf{3.} \(0<a\leq c\leq b\leq c+1<a+b\): \[ \sigma(g)=1+{(c-b)(c-a)\over a+b-c-1}+{c-a-b\over 2}<{1\over 2}. \] \textbf{4.} \(1\leq a\leq c\leq b\leq c+a-1\): \[ 1-{b\over 2}\leq\sigma(g)=1+{(c-b)(c-a)\over a+b-c-1}+{c-a-b\over 2}\leq 1-{a\over 2}<{1\over 2}. \] In particular, in all cases \(F(a,b,c,z)\not= 0\) for \(z\in\)\textbf{D} and in the cases 2--4 the function \(z\mapsto g(z)\) is not convex. \textbf{5.} Let \(a,b,c\in\)\textbf{R}, such that \(F(a,b,c,z)\not= 0\) for \(z\in\)\textbf{D}. If either (a) \(a,b\not\in\text{-\textbf{N}},\,c-a-b<0\) or (b) \(c-b,c-a\not\in\text{-\textbf{N}},\,c-a-b<0\), then \[ \sigma(zF(a,b,c,z))=\sigma(zF(c-b,c-a,c,z))+{1\over 2}(c-a-b). \] Using known results, the author then derives information about the order of convexity for quite a number of new cases. \vskip0.2cm A densely written paper.
scientific article; zbMATH DE number 5173026
scientific article