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Subclasses of Starlike Functions Subordinate to Convex Functions

Published online by Cambridge University Press:  20 November 2018

H. Silverman  and
E. M. Silvia
H. Silverman
Affiliation:
College of Charleston, Charleston, South Carolina
E. M. Silvia
Affiliation:
University of California, Davis, Davis, California
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Let S denote the class of functions of the form

that are analytic and univalent in the unit disk Δ = {z:|z| < 1}, with S*(α) and K(α) designating the subclasses of S that are, respectively, starlike of order a and convex of order α, 0 ≦ α < 1. If f(z) and g(z) are analytic in Δ, we say that f(z) is subordinate to g(z), written f ≺ g, if there exists a Schwarz function w(z), w(0) = 0 and |w(z)| < 1 in Δ, such that f(z) = g(w(z)). A function f(z) = z + … is said to be in S*[A, B] if

(1)

and in K[A, B] if

(2)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 1 , 01 February 1985 , pp. 48 - 61
Copyright
Copyright © Canadian Mathematical Society 1985

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