Linear operators associated with \(k\)-uniformly convex functions. (English) Zbl 0959.30007

Kanas, S.; Srivastava, H. M.

Integral Transforms Spec. Funct. 9, No. 2, 121-132 (2000).

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.

Reviewer: H.M.Srivastava (Victoria)

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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