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Starlike and convexity properties for \(p\)-valent hypergeometric functions. (English) Zbl 1212.30041

Summary: Given the hypergeometric function \(F(a,b;c;z)=\sum ^\infty _{n=0} \frac {(a)_n(b)_n}{(c)_n(1)_n}z^n\), we place conditions on \(a\), \(b\) and \(c\) to guarante that \(z^pF(a, b; c; z)\) will be in various subclasses of \(p\)-valent starlike and \(p\)-valent convex functions. Operators related to the hypergeometric function are also examined.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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