A further refinement for coefficient estimates of univalent functions
HTML articles powered by AMS MathViewer
- by David Horowitz PDF
- Proc. Amer. Math. Soc. 71 (1978), 217-221 Request permission
Abstract:
The coefficient inequalities of FitzGerald are used to show that if $f(z) = z + {a_2}{z^2} + {a_3}{z^3} + \ldots$ is analytic and univalent in the unit disc, then $|{a_n}| < (1.0657)n$. The technique used to obtain this bound cannot yield a result better than $|{a_n}| < (1.0599)n$.References
- I. E. Bazilevič, Improvement of estimates for the coefficients of univalent functions, Mat. Sbornik N. S. 22(64) (1948), 381–390 (Russian). MR 0026689
- I. E. Bazilevič, On distortion theorems and coefficients of univalent functions, Mat. Sbornik N.S. 28(70) (1951), 147–164 (Russian). MR 0039804
- Carl H. Fitzgerald, Quadratic inequalities and coefficient estimates for schlicht functions, Arch. Rational Mech. Anal. 46 (1972), 356–368. MR 335777, DOI 10.1007/BF00281102
- G. M. Goluzin, On the coefficients of univalent functions, Mat. Sbornik N. S. 22(64) (1948), 373–380 (Russian). MR 0026688 D. A. Horowitz, Applications of quadratic inequalities in the theory of univalent functions, Doctoral dissertation, University of California, San Diego, 1974.
- David Horowitz, A refinement for coefficient estimates of univalent functions, Proc. Amer. Math. Soc. 54 (1976), 176–178. MR 396932, DOI 10.1090/S0002-9939-1976-0396932-X
- Edmund Landau, Über schilichte Funktionen, Math. Z. 30 (1929), no. 1, 635–638 (German). MR 1545083, DOI 10.1007/BF01187792
- I. M. Milin and N. A. Lebedev, On the coefficients of certain classes of analytic functions, Doklady Akad. Nauk SSSR (N.S.) 67 (1949), 221–223 (Russian). MR 0032740
- N. A. Lebedev and I. M. Milin, On the coefficients of certain classes of analytic functions, Mat. Sbornik N.S. 28(70) (1951), 359–400 (Russian). MR 0045816 J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. (2) 23 (1925), 481-519.
- I. M. Milin, A bound for the coefficients of schlicht functions, Dokl. Akad. Nauk SSSR 160 (1965), 769–771 (Russian). MR 0172991
- J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 71 (1978), 217-221
- MSC: Primary 30A34
- DOI: https://doi.org/10.1090/S0002-9939-1978-0480979-0
- MathSciNet review: 0480979