The L. de Branges proof of the I. M. Milin and L. Bieberbach conjectures. (English. Russian original) Zbl 0624.30024
Sib. Math. J. 28, No. 1-2, 178-191 (1987); translation from Sib. Mat. Zh. 28, No. 2(162), 7-20 (1987).
The author begins with a general introduction to the Bieberbach conjecture and then derives the Lebedev-Milin inequalities with arguments different from the ones used by L. de Branges [Acta Math. 154, 137- 152 (1985; Zbl 0573.30014)]. The author states that he wishes to offer a solution that is widely accessible for verification.
In addition, the author finds sharp bounds for \(| f^{(n)}(z)|\) \((| z| <1\), \(f\in S)\) and coefficient estimates for the subclass of bounded functions in S.
In addition, the author finds sharp bounds for \(| f^{(n)}(z)|\) \((| z| <1\), \(f\in S)\) and coefficient estimates for the subclass of bounded functions in S.
Reviewer: Renate McLaughlin
MSC:
| 30C50 | Coefficient problems for univalent and multivalent functions of one complex variable |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |
Biographic References:
de Branges, LouisCitations:
Zbl 0573.30014References:
| [1] | I. E. Bazilevich, ?Domains of the initial coefficients of bounded univalent functions with p-fold symmetry,? Mat. Sb.,43, 409-428 (1957). |
| [2] | K. I. Babenko, ?On a theory of extremal problems for univalent functions of the class S,? Trudy Mat. Inst. Steklova Akad. Nauk SSSR,101, 1-318 (1972). |
| [3] | A. C. Schaeffer and D. C. Spencer, Coefficient Regions for Schlicht Functions, Colloquium Publ. No. 35, Am. Math. Soc., Providence (1950). · Zbl 0066.05701 |
| [4] | L. Bieberbach, ?Über die Koeffizienten derjenigen Potenzreihen welche eine schlichte Abbildung des Einheitskreises vermitteln,? Sitzber. Preuss. Akad. Wiss., 940-955 (1916). · JFM 46.0552.01 |
| [5] | K. Löwner ?Untersuchungen über schlichte konforme Abbildung des Einheitkreises. I,? Math. Ann.,89, 103-121 (1923). · JFM 49.0714.01 · doi:10.1007/BF01448091 |
| [6] | P. R. Garabedian and M. A. Schiffer, ?A proof of the Bieberbach conjecture for the fourth coefficient,? J. Rat. Mech. Anal.,4, 427-465 (1955). · Zbl 0065.06902 |
| [7] | R. N. Pederson, ?A proof of the Bieberbach conjecture for the sixth coefficient,? Arch. Rat. Mech. Anal.,31, 331-351 (1968). · Zbl 0184.10501 · doi:10.1007/BF00251415 |
| [8] | M. Ozawa, ?An elementary proof of the Bieberbach conjecture for the sixth coefficient,? Kodai Math. Semin. Repts.,21, 129-132 (1969). · Zbl 0202.07201 · doi:10.2996/kmj/1138845875 |
| [9] | R. N. Pederson and M. A. Schiffer, ?A proof of the Bieberbach conjecture for the fifth coefficient,? Arch. Rat. Mech. Anal.,45, 161-193 (1972). · Zbl 0241.30025 · doi:10.1007/BF00281531 |
| [10] | J. E. Littlewood, ?On inequalities of the theory of functions,? Proc. London Math. Soc.,23, 481-519 (1925). · JFM 51.0247.03 · doi:10.1112/plms/s2-23.1.481 |
| [11] | I. E. Bazilevich, ?On theorems of distortion and the coefficients of univalent functions,? Mat. Sb.,28, 147-164 (1951). |
| [12] | I. M. Milin, ?An estimate of the coefficients of univalent functions,? Dokl. Akad. Nauk SSSR,160, 769-771 (1965). · Zbl 0158.32201 |
| [13] | C. H. Fitzgerald, ?Exponentiation of certain quadratic inequalities for schlicht functions,? Bull. Am. Math. Soc.,78, 209-210 (1972). · Zbl 0229.30020 · doi:10.1090/S0002-9904-1972-12919-7 |
| [14] | D. Horowitz, ?A refinement for coefficient estimates of univalent functions,? Proc. Am. Math. Soc.,54, 176-178 (1976). · Zbl 0294.30006 · doi:10.1090/S0002-9939-1976-0396932-X |
| [15] | W. K. Hayman, Multivalent Functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 48, Cambridge Univ. Press (1958). |
| [16] | I. E. Bazilevich, ?On a criterion of univalency of regular functions and the dispersion of their coefficients,? Mat. Sb.,74, 133-146 (1967). · Zbl 0181.35604 |
| [17] | I. M. Milin, Univalent Functions and Orthonormal Systems, Am. Math. Soc., Providence, R.I. (1977). |
| [18] | A. Z. Grinshpan, ?Logarithmic coefficients of functions of the class S,? Sib. Mat. Zh.,13, 1145-1157 (1972). · Zbl 0249.30013 |
| [19] | L. de Branges, ?A proof of the Bieberbach conjecture,? Preprint E-5-84, Steklov Math. Inst., Leningrad Branch (1984), pp. 1-21. |
| [20] | L. de Branges, ?A proof of the Bieberbach conjecture,? Acta Math.,154, 137-152 (1985). · Zbl 0573.30014 · doi:10.1007/BF02392821 |
| [21] | I. A. Aleksandrov, Parametric Continuations in the Theory of Univalent Functions [in Russian], Nauka, Moscow (1976). · Zbl 0462.30008 |
| [22] | O. M. Fomenko and G. V. Kuz’mina, ?The last 100 days of the Bieberbach conjecture,? Math. Intell.,8, No. 1, 40-47 (1986). · Zbl 0616.30010 · doi:10.1007/BF03023920 |
| [23] | N. A. Lebedev and I. M. Milin, ?On an inequality,? Vestn. Leningr. Gos. Univ.,19, No. 4, 157-158 (1965). |
| [24] | H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York (1953). |
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