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Zbl 1118.30010
Küstner, Reinhold
On the order of starlikeness of the shifted Gauss hypergeometric function.
(English)
[J] J. Math. Anal. Appl. 334, No. 2, 1363-1385 (2007). ISSN 0022-247X

For a function $f$, analytic in {\bf D}$=\{z\in\text {\bf C}:\vert z\vert <1\}$, with $f(0)=0\not= f'(0)$, the {\it order of starlikeness\/} (with respect to zero) is $$\sigma(f):=\inf_{z\in\text {\bf D}}\,\text {Re}\,{zf'(z)\over f(z)}\in [-\infty,1],$$ and if at least $f'(0)\not=0$, the {\it order of convexity\/} of $f$ is $$\kappa(f):=\sigma(zf')=1+\inf_{z\in\text {\bf D}}\,\text {Re}\,{zf''(z)\over f'(z)}\in [-\infty,1].$$ 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 {\bf 1.} $a<0<b,\,c\geq b-a+1$: $$\sigma(g)=1+{F'(a,b,c,1)\over F(a,b,c,1)}=1+{ab\over c-a-b-1}\geq 1-{b\over 2}.$$ {\bf 2.} $0<a<c<b\leq c-a+1$: $$\sigma(g)=-\infty.$$ {\bf 3.} $0<a\leq c\leq b\leq c+1<a+b$: $$\sigma(g)=1+{(c-b)(c-a)\over a+b-c-1}+{c-a-b\over 2}<{1\over 2}.$$ {\bf 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 $z\in${\bf D} and in the cases 2--4 the function $z\mapsto g(z)$ is not convex. {\bf 5.} Let $a,b,c\in${\bf R}, such that $F(a,b,c,z)\not= 0$ for $z\in${\bf D}. If either (a) $a,b\not\in\text {-{\bf N}},\,c-a-b<0$ or (b) $c-b,c-a\not\in\text {-{\bf N}},\,c-a-b<0$, 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.
[Marcel G. de Bruin (Haarlem)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
33C05 Classical hypergeometric functions

Keywords: Gauss hypergeometric function; univalence; order of starlikeness; order of convexity; zerofreeness in the unit disk; Carlson-Shaffer convolution

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