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On generalized \(q\)-Bazilevic functions. (English) Zbl 1378.30004

Summary: Let \(S^*\) and \(C\), respectively, be the classes of starlike and convex univalent functions. The \(q\)-derivative of a function, analytic in the open unit disk is defined by \(D_q f(z)=\frac{f(qz)-f(z)}{(q-1)z}\), \(z \neq 0\), and \(D_q f(0)=f'(0)\), \(q \in (0, 1)\). As \(q \to 1^-\), \(D_q f(z) \to f'(z)\). This concept has been used to define certain classes of analytic functions such as \(S^*_q\) of \(q\)-starlike and \(C_q\) of \(q\)-convex functions with the property \(\bigcap\limits_{0<q<1}= S^*_q=S^*\) and \(\bigcap\limits_{0<q<1} C_q=C\). In this paper, we introduce a new class of \(q\)-Bazilevic functions of type \(\beta\) which contains \(S^*_q\) and \(C_q\) as special cases. Inclusion results and mappings properties of certain \(q\)-integral operators are studied.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C10 Polynomials and rational functions of one complex variable
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