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International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 795-805

Applications of Ruscheweyh derivatives and Hadamard product to analytic functions

Department of Mathematics, University of Bahrain, P. O. Box 32038, Isa Town, Bahrain

Received 31 August 1995; Revised 10 February 1998

Copyright © 1999 Hindawi Publishing Corporation. 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.


For given analytic functions ϕ(z)=z+m=2λmzm,ψ(z)=z+m=2μmzm in U={z||z|<1} with λm0,μm0 and λmμm, let En(ϕ,ψ;A,B) be the class of analytic functions f(z)=z+m=2amzm in U such that (f*Ψ)(z)0 and Dn+1(f*ϕ)(z)Dn(f*Ψ)(z)1+Az1+Bz,1A<B1,zU, where Dnh(z)=z(zn1h(z))(n)/n!,nN0={0,1,2,} is the nth Ruscheweyh derivative; and * denote subordination and the Hadamard product, respectively. Let T be the class of analytic functions in U of the form f(z)=zm=2amzm,am0, and let En[ϕ,ψ;A,B]=En(ϕ,ψ;A,B)T. Coefficient estimates, extreme points, distortion theorems and radius of starlikeness and convexity are determined for functions in the class En[ϕ,ψ;A,B]. We also consider the quasi-Hadamard product of functions in En[z/(1z),z/(1z);A,B] and En[z/(1z)2,z/(1z)2;A,B].