Functions whose derivatives take values in a half-plane
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- by Fernando Gray and Stephan Ruscheweyh PDF
- Proc. Amer. Math. Soc. 104 (1988), 215-218 Request permission
Abstract:
We derive sharp upper and lower bounds for $\left | {zf’(z)/f(z)} \right |$ where $f \in R$, i.e. $f$ analytic in ${\mathbf {D}}$ with $f(0) = 0,f’(0) = 1$ and ${e^{i\alpha }}f’(z){\text { > }}0$ in ${\mathbf {D}}$ for a certain $\alpha = \alpha (f) \in {\mathbf {R}}$. The extremal function is $k(z) = - z - 2\log (1 - z)$. This result improves an earlier one of D. K. Thomas.References
- Stephan Ruscheweyh, Nichtlineare Extremalprobleme für holomorphe Stieltjesintegrale, Math. Z. 142 (1975), 19–23 (German). MR 374406, DOI 10.1007/BF01214844
- Stephan Ruscheweyh, Convolutions in geometric function theory, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 83, Presses de l’Université de Montréal, Montreal, Que., 1982. Fundamental Theories of Physics. MR 674296
- S. Ruscheweyh, Coefficient conditions for starlike functions, Glasgow Math. J. 29 (1987), no. 1, 141–142. MR 876158, DOI 10.1017/S0017089500006753
- D. K. Thomas, On functions whose derivative has positive real part, Proc. Amer. Math. Soc. 98 (1986), no. 1, 68–70. MR 848877, DOI 10.1090/S0002-9939-1986-0848877-2
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 215-218
- MSC: Primary 30C45
- DOI: https://doi.org/10.1090/S0002-9939-1988-0958069-6
- MathSciNet review: 958069