International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 3, Pages 435-442

a*-families of analytic functions

G. P. Kapoor and A. K. Mishra

Department of Mathematics, Indian Institute of Technology, Kanpur 208016, U.P., India

Received 15 February 1983

Copyright © 1984 G. P. Kapoor and A. K. Mishra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Using convolutions, a new family of analytic functions is introduced. This family, called a*-family, serves in certain situations to unify the study of many previously well known classes of analytic functions like multivalent convex, starlike, close-to-convex or prestarlike functions, functions starlike with respect to symmetric points and other such classes related to the class of univalent or multivalent functions. A necessary and sufficient condition on the Taylor series coefficients so that an analytic function with negative coefficients is in an a*-family is obtained and sharp coefficents bound for functions in such a family is deduced. The extreme points of an a*-family of functions with negative coefficients are completely determined. Finally, it is shown that Zmorvic conjecture is true if the concerned families consist of functions with negative coefficients.