In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

, found by and . It was used in the proof of the

Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was ...

, as it shows that the Milin conjecture implies the Robertson conjecture. They state that if :

\sum_ \beta_kz^k = \exp\left(\sum_ \alpha_kz^k\right)

for complex numbers

\beta_k

and

\alpha_k

, and

n

is a positive integer, then :

\sum_^, \beta_k, ^2 \le
\exp\left(\sum_^\infty k, \alpha_k, ^2\right),

\sum_^, \beta_k, ^2 \le
(n+1)\exp\left(\frac\sum_^\sum_^m(k, \alpha_k, ^2 - 1/k)\right),

, \beta_n, ^2 \le
\exp\left(\sum_^n(k, \alpha_k, ^2 -1/k)\right).

References

* * *. * * * (Translation of the 1971 Russian edition, edited by P. L. Duren). {{DEFAULTSORT:Lebedev-Milin inequality Inequalities (mathematics)