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 and. 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


 * $$\sum_{k\ge 0} \beta_kz^k = \exp\left(\sum_{k\ge 1} \alpha_kz^k\right)$$

for complex numbers $$\beta_k$$ and $$\alpha_k$$, and $$n$$ is a positive integer, then


 * $$\sum_{k=0}^{\infty}|\beta_k|^2 \le

\exp\left(\sum_{k=1}^\infty k|\alpha_k|^2\right),$$


 * $$\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),$$


 * $$|\beta_n|^2 \le

\exp\left(\sum_{k=1}^n(k|\alpha_k|^2 -1/k)\right).$$

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