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

References:

[1]	DOI: 10.1007/BF02392821 · Zbl 0573.30014
[2]	Hohlov Yu.E., Izv. Vysš. Uč. Zaved. Matematika 10 pp 83– (1978)
[3]	Kanas S., J. Comput. Appl. Math. 10 (1978)
[4]	Kanas S., Folia Sci. Univ. Tech. Resoviensis Mat. 22 pp 65– (1998)
[5]	Kanas S., Rev.Roumanie Math. Pures Appl. 22 (1998)
[6]	DOI: 10.1090/S0002-9939-1975-0367176-1
[7]	Ruscheweyh S., Convolutions in Geometric Function Theory (1982) · Zbl 0499.30001
[8]	Srivastava H.M., Current Topics in Analytic Function Theory (1992) · Zbl 0976.00007
[9]	DOI: 10.1080/00207168108803266 · Zbl 0475.33002