# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Certain subclasses of analytic functions associated with the generalized hypergeometric function. (English) Zbl 1040.30003

For two holomorphic functions in the unit disk $|z|<1,$ $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n}$ and $g\left(z\right)=z+{\sum }_{n=2}^{\infty }{b}_{n}{z}^{n}$ the Hadamard product (or convolution ) is defined as $\left(f☆g\right)\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_{n}{b}_{n}{z}^{n}·$ Let ${}_{q}{F}_{s}\left({\alpha }_{1},...\phantom{\rule{0.166667em}{0ex}}{\alpha }_{q};{\beta }_{1},...,{\beta }_{s};z\right)$ denote the generalized hypergeometric function with $q\le s+1$, $q,s=0,1,2,...$, ${\alpha }_{j}\in ℂ$, ${\beta }_{j}\in ℂ\setminus \left\{0,-1,-2...\right\}$. Very special subclasses of holomorphic functions $f\left(z\right)={a}_{1}z-{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n}$, ${a}_{n}\ge 0$, $n\ge 2$, $|z|<1$ defined by convolution of $f$ with ${z}_{q}{F}_{s}$ and subordinate to $\frac{1+Az}{1+Bz}$, $0\le B\le 1$, $-B\le A\le B$ are studied.

(They satisfy moreover the condition $f\left(\rho \right)=\rho$ or ${f}^{\text{'}}\left(\rho \right)=1$, $0<\rho <1$.) The necessary and sufficient conditions in the terms of coefficients for $f$ to be in the class under consideration are determined. Some distortion theorems and the radii of convexity and starlikeness are found.

##### MSC:
 30C45 Special classes of univalent and multivalent functions 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$