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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
##### Keywords:
$$k$$-uniformly convex; $$k$$-starlike functions
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##### References:
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