Starlikeness and convexity from integral means of the derivative
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- by Shinji Yamashita PDF
- Proc. Amer. Math. Soc. 103 (1988), 1094-1098 Request permission
Abstract:
If $f$ is analytic in $|z| < 1$ and normalized: $f(0) = fâ(0) - 1 = 0$, then $f$ is univalent and starlike in $|z| < 1(f)$, where \[ I(f) = \sup r{\left \{ {{{(2\pi )}^{ - 1}}\int _0^{2\pi } {|fâ(r{e^{it}}){|^2}dt} } \right \}^{ - 1/2}},\quad 0 \leq r \leq 1.\] Furthermore, there exists a normalized $f$ such that $I(f) < 1$ and that $fâ$ vanishes at a point on $|z| = I(f)$. If $f$ is analytic and normalized in $|z| < 1$, then $f$ is univalent and convex in $|z| < I(f)/2$.References
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Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 1094-1098
- MSC: Primary 30C45
- DOI: https://doi.org/10.1090/S0002-9939-1988-0954989-7
- MathSciNet review: 954989