Lebedev–Milin inequality

Template:Short description In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by Template:Harvs and Template:Harvs. 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

$\sum _{k\ge 0}{\beta }_k{z}^k=\exp \left(\sum _{k\ge 1}{\alpha }_k{z}^k\right)$

for complex numbers ${\beta }_k$ and ${\alpha }_k$, and $n$ is a positive integer, then

$\sum _{k=0}^{\mathrm{\infty }}|{\beta }_k{|}^2\le \exp (\sum _{k=1}^{\mathrm{\infty }}k|{\alpha }_k{|}^2),$

$\sum _{k=0}^n|{\beta }_k{|}^2\le (n+1)\exp (\frac{1}{n+1}\sum _{m=1}^n\sum _{k=1}^m(k|{\alpha }_k{|}^2-1/k)),$

$|{\beta }_n{|}^2\le \exp (\sum _{k=1}^n(k|{\alpha }_k{|}^2-1/k)).$

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

References

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• Template:Citation (Translation of the 1971 Russian edition, edited by P. L. Duren).