Lebedev–Milin inequality

In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by Lebedev and Milin (1965) and Isaak Moiseevich Milin (1977). It was used in the proof of the Bieberbach conjecture, as it shows that the Milin conjecture implies the Robertson conjecture. They state that if

[math]\displaystyle{ \sum_{k\ge 0} \beta_kz^k = \exp\left(\sum_{k\ge 1} \alpha_kz^k\right) }[/math]

for complex numbers [math]\displaystyle{ \beta_k }[/math] and [math]\displaystyle{ \alpha_k }[/math], and [math]\displaystyle{ n }[/math] is a positive integer, then

[math]\displaystyle{ \sum_{k=0}^{\infty}|\beta_k|^2 \le \exp\left(\sum_{k=1}^\infty k|\alpha_k|^2\right), }[/math]

[math]\displaystyle{ \sum_{k=0}^{n}|\beta_k|^2 \le (n+1)\exp\left(\frac{1}{n+1}\sum_{m=1}^{n}\sum_{k=1}^m(k|\alpha_k|^2 - 1/k)\right), }[/math]

[math]\displaystyle{ |\beta_n|^2 \le \exp\left(\sum_{k=1}^n(k|\alpha_k|^2 -1/k)\right). }[/math]

See also exponential formula (on exponentiation of power series).

References

• Conway, John B. (1995), Functions of One Complex Variable II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94460-9, OCLC 32014394 
• Grinshpan, A. Z. (1999), "The Bieberbach conjecture and Milin's functionals", The American Mathematical Monthly 106 (3): 203–214, doi:10.2307/2589676 
• Grinshpan, Arcadii Z. (2002), "Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains", in Kuhnau, Reiner, Geometric Function Theory, Handbook of Complex Analysis, 1, Amsterdam: North-Holland, pp. 273–332, ISBN 0-444-82845-1 .
• Korevaar, Jacob (1986), "Ludwig Bieberbach's conjecture and its proof by Louis de Branges", The American Mathematical Monthly 93 (7): 505–514, doi:10.2307/2323021, ISSN 0002-9890 
• Lebedev, N. A.; Milin, I. M. (1965), An inequality, 20, Vestnik Leningrad University. Mathematics, pp. 157–158, ISSN 0146-924X 
• Milin, I. M. (1977), Univalent functions and orthonormal systems, Translations of Mathematical Monographs, 49, Providence, R.I.: American Mathematical Society, pp. iv+202, ISBN 0-8218-1599-7  (Translation of the 1971 Russian edition, edited by P. L. Duren).