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On the order of starlikeness of the shifted Gauss hypergeometric function. (English) Zbl 1118.30010

For a function f, analytic in D={zC:|z|<1}, with f(0)=0f ' (0), the order of starlikeness (with respect to zero) is

σ(f):=inf zD Rezf ' (z) f(z)[-,1],

and if at least f ' (0)0, the order of convexity of f is

κ(f):=σ(zf ' )=1+inf zD Rezf '' (z) f ' (z)[-,1]·

The author studies the ‘shifted’ hypergeometric function

g(z):=zF(a,b,c,z)=z 2 F 1 (a,b;c;z)

and gives several new results, for instance

1. a<0<b,cb-a+1:

σ(g)=1+F ' (a,b,c,1) F(a,b,c,1)=1+ab c-a-b-11-b 2·

2. 0<a<c<bc-a+1:


3. 0<acbc+1<a+b:

σ(g)=1+(c-b)(c-a) a+b-c-1+c-a-b 2<1 2·

4. 1acbc+a-1:

1-b 2σ(g)=1+(c-b)(c-a) a+b-c-1+c-a-b 21-a 2<1 2·

In particular, in all cases F(a,b,c,z)0 for zD and in the cases 2–4 the function zg(z) is not convex. 5. Let a,b,cR, such that F(a,b,c,z)0 for zD. If either (a) a,b-N,c-a-b<0 or (b) c-b,c-a-N,c-a-b<0, then

σ(zF(a,b,c,z))=σ(zF(c-b,c-a,c,z))+1 2(c-a-b)·

Using known results, the author then derives information about the order of convexity for quite a number of new cases.

A densely written paper.

30C45Special classes of univalent and multivalent functions
33C05Classical hypergeometric functions, 2 F 1