Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Starlikeness and convexity from integral means of the derivative


Author: Shinji Yamashita
Journal: Proc. Amer. Math. Soc. 103 (1988), 1094-1098
MSC: Primary 30C45
MathSciNet review: 954989
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $ f$ is analytic in $ \vert z\vert < 1$ and normalized: $ f(0) = f'(0) - 1 = 0$, then $ f$ is univalent and starlike in $ \vert z\vert < 1(f)$, where

$\displaystyle I(f) = \sup \,r{\left\{ {{{(2\pi )}^{ - 1}}\int_0^{2\pi } {\vert f'(r{e^{it}}){\vert^2}dt} } \right\}^{ - 1/2}},\quad 0 \leq r \leq 1.$

Furthermore, there exists a normalized $ f$ such that $ I(f) < 1$ and that $ f'$ vanishes at a point on $ \vert z\vert = I(f)$.

If $ f$ is analytic and normalized in $ \vert z\vert < 1$, then $ f$ is univalent and convex in $ \vert z\vert < I(f)/2$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C45

Retrieve articles in all journals with MSC: 30C45


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0954989-7
PII: S 0002-9939(1988)0954989-7
Article copyright: © Copyright 1988 American Mathematical Society