International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 2, Pages 259-266
doi:10.1155/S0161171288000316
On a class of p-valent close-to-convex functions of order β and type α
1Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
2Department of Mathematics, Faculty of Science, University of Qatar, P. O. Box: 2713, Doha, Qatar
Received 10 October 1986; Revised 29 October 1986
Copyright © 1988 M. K. Aouf. 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.
Abstract
Let S(A,B,p,α) denote the class of functions g(z)=zp+∑n=p+1∞bnzn analytic in the unit disc U={z:|z|<1} and satisfying the conditionzg′(z)g(z)<p+[pB+(A−B)(p−α)]z1+Bz, z∈U, −1≦B<A≦1, 0≦α<p.Let C(A,B,p,β,α) denote the class of functions f(z)=zp+∑n=p+1∞anzn analytic in U, and satisfying the conditionRe{zf′(z)g(z)}>β, z∈U, g∈S(A,B,p,α).In this paper we determine the coefficient estimates and distortion theorems for the class C(A,B,p,β,α).