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In 1925 Littlewood proved a different, looser bound. While Bieberbach conjectured that each coefficient is bound above by its index (\(|a_{n}|\leq n\)), Littlewood proved that for all \(n\), \(|a_{n}|\leq e\,n\).
**Theorem 151**.: _Let \(f(z)\in S\). Then \(|a_{n}| : The proof depends on Littlewood’s integral inequality [20] which is: for \(f\in S\) \[\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta}|d\theta\leq\frac{r}{1-r}\quad 0 \leq r<1.\] Using Cauchy’s integral formula, \[|a_{n}|=\left|\frac{f^{(n)}(0)}{n!}\right|=\left|\frac{1}{2\pi}\int_{|z|=r} \frac{f(z)}{z^{n+1}}\ dz\right|,\qquad r<1.\] Changing \(z=re^{i\theta}\) and using Littlewood’s integral inequality we obtain \[|a_{n}|=\left|\frac{1}{2\pi}\int_{0}^{2\pi}\frac{f(re^{i\theta})}{r^{n}\cdot e ^{in\theta}}\ dz\right|\leq\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}\left|f(re^{i \theta})\right|d\theta\leq\frac{1}{r^{n}}\frac{r}{1-r}.\] Thus, \[|a_{n}|\leq\frac{1}{(1-r)r^{n-1}}.\] To find the minimum value for \(|a_{n}|\), we maximize \(h(r)=(1-r)r^{n-1}\) in \([0,1]\). From \[h^{\prime}(r)=r^{n-2}((n-1)(1-r)-r),\] we see the maximum attained when \(r=1-\frac{1}{n}\). Therefore \[|a_{n}|\leq\frac{1}{(1-r)r^{n-1}}=n\!\left(\frac{n}{n-1}\right)^{n-1} [MISSING_PAGE_EMPTY:716] ## Bibliography * [1] S. Abbott. _Understanding analysis_. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2001. * [11] L. Bieberbach. Uber die koeffizienten derjenigen potenzreihen, welche eine schlichte abbildung des einheitskreises vermitteln. _Sitzungsberichte Akademie der Wissenschaften_, pages 940-955, 1916. * [25] A. Hatcher. _Algebraic topology_. Cambridge University Press, Cambridge, 2002. * [40] A. Pinkus. Weierstrass and approximation theory. _J. Approx. Theory_, 107(1):1-66, 2000. * [55] P. Zorn. The Bieberbach conjecture. _Math. Mag._, 59(3):131-148, 1986. absolute convergence, 58 accumulation point of a set, 67 algebraic number, 22, 29 almost everywhere, 250 angle between curves, 315 annulus, 280 Archimedean property, 33 arithmetic-geometric mean inequality, 154 Arzela-Ascoli theorem, 144 axiom of choice, 18 axiom of completeness, 30 Cantor-Lebesgue function, 252, 254 Cantor’s diagonalization, 15 Cantor set, 77, 120 cardinality, 10 Cauchy condensation test, 57 Cauchy criterion, 50 Cauchy initial value problem, 183 Cauchy-Riemann equations, 290 Cauchy-Schwarz inequality, 152, Cauchy sequence, 49, 134 Cauchy’s formula for derivatives, 353 Cauchy’s functional equation, 215 Cauchy’s inequality, 354 Cauchy’s integral formula, 350 Cauchy’s theorem, 338 Cesaro sums, 233 characteristic function, 243 closed ball, 131 closure, 68 compact set, 70, 93, 143 comparison test, 59 complete metric space, 135 completeness, 30 complex exponential function, 298 complex fundamental theorem of calculus, 334 complex inner product, 149 complex logarithmic function, 307 complex numbers, 277complex sequence, 53 conformal mapping, 315 connected sets, 75 continuous function, 87 continuum hypothesis, 15 contour, 338 contraction mapping, 175 contraction mapping theorem, 175 convergent sequence, 39, 134 convex functions, 221 convolution, 236 countable, 12 countably infinite, 12 d’Alembert’s wave equation, 223 deformation, 339 de Moivre’s formula, 285 De Morgan’s Laws, 21, 145 dense, 33, 35, 121 differentiable function, 98 Dini’s theorem, 197 Dirichlet formula, 232 Dirichlet function, 89, 112, 113, 241, 242 Dirichlet kernel, 231 Dirichlet problem, 320 Dirichlet test, 199 disk of convergence, 297 divergence test, 56 dominated convergence theorem, 268 Egorov’s theorem, 252 entire functions, 298, 352 equicontinuous, 98, 144 equivalence class, 17 equivalence relation, 17 essential singularity, 364 Euler’s formula, 279 exponential function, 304 extreme value theorem, 93 Fatou lemma, 264 Fejer kernel, 233 Fibonacci sequence, 38 field axioms, 25 finite intersection property, 74 fixed point, 174 Fourier coefficients, 226 Fourier series, 226 Fourier transform, 235 Frechet metric, 162 fractional dimension, 81 Fredholm integral equation, 185 Frobenius norm, 166 Fubini’s theorem, 236 function bijective, 7 holomorphic, 289 measurable, 247 one-to-one, 21, 104, 105 step, 124 surjective, 6 functional equations, 213 fundamental theorem of algebra, 284, 352 fundamental theorem of calculus, 117, 119 gamma function, 213 Gauss’ mean value theorem, 358 geometric series, 56 Gram-Schmidt procedure, 157 greatest lower bound, 31 Green’s theorem, 337 Hadamard’s formula, 297 Hamel basis, 217, 222 harmonic, 321 harmonic series, 57 Hausdorff maximality principle, 20 Heine-Borel theorem, 73 Hilbert space, 161 Holder’s Inequality, 129 holomorphic function, 290 homotopic, 339 identity map, 7 induction proof, 8 infinite series, 55 inner product, 150 inner product space, 147integral Lebesgue, 255 Riemann, 124 integral equations, 185 integration along paths, 331 interior of a set, 68 interior point, 65 intermediate value theorem, 96 intermediate value theorem for integrals, 116 inverse function theorem, 104, 108 isolated point of a set, 67 isolated singularity, 364 isometric, 137 jacobian, 292 Jensen’s equation, 219 Jordan curve theorem, 338 kernel, 185 Koebe function, 376 Koebe\(\frac{1}{4}\) – theorem, 380 Lagrange’s identity, 289 Laplace’s equation, 320 Laplacian, 237 Laurent series, 359 least upper bound, 30 Lebesgue criterion, 120Proof
* [2] J. Aczel. _Lectures on functional equations and their applications_. Mathematics in Science and Engineering, Vol. 19. Academic Press, New York-London, 1966. Translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser.
* [3] W. A. Adkins and S. H. Weintraub. _Algebra_, volume 136 of _Graduate Texts in Mathematics_. Springer-Verlag, New York, 1992. An approach via module theory.
* [4] L. V. Ahlfors. An extension of Schwarz’s lemma. _Trans. Amer. Math. Soc._, 43(3):359-364, 1938.
* [5] L. V. Ahlfors. _Complex analysis: An introduction of the theory of analytic functions of one complex variable_. Second edition. McGraw-Hill Book Co., New York-Toronto-London, 1966.
* [6] A. G. Aksoy and M. A. Khamsi. _Nonstandard methods in fixed point theory_. Universitext. Springer-Verlag, New York, 1990. With an introduction by W. A. Kirk.
* [7] E. Artin. _The gamma function_. Translated by Michael Butler. Athena Series: Selected Topics in Mathematics. Holt, Rinehart and Winston, New York-Toronto-London, 1964.
* [8] S. Axler. _Measure, integration & real analysis_, volume 282 of _Graduate Texts in Mathematics_. Springer, Cham, [2020] (c)2020.
* [9] J. A. Beachy and W. D. Blair. _Abstract Algebra_. Waveland Press, Inc., Long Grove, 2006.
* [10] R. Bernatz. _Fourier series and numerical methods for partial differential equations_. John Wiley & Sons, Inc., Hoboken, NJ, 2010.
* [12] P. Bloomfield. _Fourier analysis of time series_. Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience [John Wiley & Sons], New York, 2000. An introduction, [2013] reprint of the second (2000) edition [MR1884963].
* [13] K. C. Border. _Fixed point theorems with applications to economics and game theory_. Cambridge University Press, Cambridge, 1989.
* [14] N. L. Carothers. _Real analysis_. Cambridge University Press, Cambridge, 2000.
* [15] E. W. Cheney. _Introduction to approximation theory_. AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1982) edition.
* [16] P. J. Cohen. _Set theory and the continuum hypothesis_. W. A. Benjamin, Inc., New York-Amsterdam, 1966.
* [17] C. W. Curtis. _Linear algebra_. Undergraduate Texts in Mathematics. Springer-Verlag, New York, fourth edition, 1993. An introductory approach.
* [18] K. R. Davidson and A. P. Donsing. _Real Analysis with Real Applications_. Prentice Hall, 2002.
* [19] L. de Branges. A proof of the Bieberbach conjecture. _Acta Math._, 154(1-2):137-152, 1985.
* [20] P. Duren. _Invitation to classical analysis_, volume 17 of _Pure and Applied Undergraduate Texts_. American Mathematical Society, Providence, RI, 2012.
* [21] G. B. Folland. _Fourier analysis and its applications_. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
* [22] G. B. Folland. _Real analysis_. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999. Modern techniques and their applications, A Wiley-Interscience Publication.
* [23] K. Goebel. _Concise course on fixed point theorems_. Yokohama Publishers, Yokohama, 2002.
* [24] S. Gong. _The Bieberbach conjecture_, volume 12 of _AMS/IP Studies in Advanced Mathematics_. American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 1999. Translated from the 1989 Chinese original and revised by the author, With a preface by Carl H. FitzGerald.
* [26] M. Hoffman and M. J. E. _Elementary Classical Analysis_. W. H. Freeman and Company, 1993.
* [27] B. B. Hubbard. _The world according to wavelets_. A K Peters, Ltd., Wellesley, MA, second edition, 1998. The story of a mathematical technique in the making.
* [28] T. W. Hungerford. _Algebra_, volume 73 of _Graduate Texts in Mathematics_. Springer-Verlag, New York-Berlin, 1980. Reprint of the 1974 original.
* [29] T. Jech. _Set theory_. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded.
* [30] S. Katok. \(p\)_-adic analysis compared with real_, volume 37 of _Student Mathematical Library_. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2007.
* [31] T. Kawata. _Fourier analysis in probability theory_. Academic Press, New York-London, 1972. Probability and Mathematical Statistics, No. 15.
* [32] M. A. Khamsi and W. A. Kirk. _An introduction to metric spaces and fixed point theory_. Pure and Applied Mathematics (New York). Wiley-Interscience, New York, 2001.
* [33] T. W. Korner. _Fourier analysis_. Cambridge University Press, Cambridge, 1988.
* [34] S. G. Krantz. _Geometric function theory_. Cornerstones. Birkhauser Boston, Inc., Boston, MA, 2006. Explorations in complex analysis.
* [35] E. Kreyszig. _Introductory functional analysis with applications_. John Wiley & Sons, New York-London-Sydney, 1978.
* [36] M. Kuczma. _An introduction to the theory of functional equations and inequalities_, volume 489. Uniwersytet Slaski, Katowice; Panstwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality, With a Polish summary.
* [37] S. Lang. _Linear algebra_. Undergraduate Texts in Mathematics. Springer-Verlag, New York, third edition, 1987.
* [38] G. G. Lorentz. _Bernstein polynomials_. Mathematical Expositions, no. 8. University of Toronto Press, Toronto, 1953.
* [39] J. Pawlikowski. The Hahn-Banach theorem implies the Banach-Tarski paradox. _Fund. Math._, 138(1):21-22, 1991.
* [41] H. A. Priestley. _Introduction to complex analysis_. Oxford University Press, Oxford, second edition, 2003.
* [42] W. Rudin. _Principles of mathematical analysis_. McGraw-Hill Book Co., New York-Auckland-Dusseldorf, third edition, 1976. International Series in Pure and Applied Mathematics.
* [43] P. J. Sally, Jr. _Tools of the trade_. American Mathematical Society, Providence, RI, 2008. Introduction to advanced mathematics.
* [44] Y. A. Shashkin. _Fixed points_, volume 2 of _Mathematical World_. American Mathematical Society, Providence, RI; Mathematical Association of America, Washington, DC, 1991. Translated from the Russian by Viktor Minachin [V. V. Minakhin].
* [45] C. G. Small. _Functional equations and how to solve them_. Problem Books in Mathematics. Springer, New York, 2007.
* [46] M. Spivak. _Calculus on manifolds. A modern approach to classical theorems of advanced calculus_. W. A. Benjamin, Inc., New York-Amsterdam, 1965.
* [47] G. Springer. _Introduction to Riemann surfaces_. Addison-Wesley Publishing Co., Inc., Reading, MA, 1957.
* [48] J. M. Steele. _The Cauchy-Schwarz master class_. MAA Problem Books Series. Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 2004. An introduction to the art of mathematical inequalities.
* [49] E. M. Stein and R. Shakarchi. _Fourier analysis_, volume 1 of _Princeton Lectures in Analysis_. Princeton University Press, Princeton, NJ, 2003. An introduction.
* [50] E. M. Stein and R. Shakarchi. _Real analysis_. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ, 2005. Measure theory, integration, and Hilbert spaces.
* [51] M. H. Stone. The generalized Weierstrass approximation theorem. _Math. Mag._, 21:167-184, 237-254, 1948.
* [52] W. Wade. _An Introduction to Analysis_. Prentice Hall, 1999.
* [53] S. Wagon. _The Banach-Tarski paradox_, volume 24 of _Encyclopedia of Mathematics and its Applications_. Cambridge University Press, Cambridge, 1985. With a foreword by Jan Mycielski.
* [54] D. G. Zill and P. D. Shanahan. _A first course in Complex Analysis with Applications (second edition)_. Jones and Bartlett Publishers, 2009.
* [56] A. Zygmund. _Trigonometric series. Vol. I, II_. Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 2002. With a foreword by Robert A. Fefferman.
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