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Linear operators associated with \(k\)-uniformly convex functions. (English) Zbl 0959.30007
In terms of the Hadamard product (or convolution), define the operator \(I_{a,b,c}\) by \[ \bigl[I_{a,b,c} (f)\bigr](z) =f(z)*z_2 F_1(a,b;c;z), \] where the function \(f\) is analytic in the unit disk. The classes of \(k\)-uniformly convex and \(k\)-starlike functions \((0\leq k<\infty)\) denoted by \(k\)-UCV and \(k\)-ST, respectively, were introduced recently (see, for example, S. Kanas and A. Wiśniowska [Folia Sci. Univ. Tech. Resoviensis Mat. 22(170), 65-78 (1998)]. The object of the present paper is to find conditions on the parameter \(a,b,c\) and \(k\), for which the linear operator \(I_{a,b,c}\) maps the classes of starlike and univalent functions onto \(k\)-UCV and \(k\)-ST.

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33E05 Elliptic functions and integrals
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