PoissonJensen

Poisson-Jensen Inequality is a fundamental result in complex analysis that provides an estimate for the modulus of a holomorphic function on a disk. This inequality is named after Jacques Philippe Marie Binet, Pierre Poisson, and Otto Ludwig Jonas Jensen, who first formulated it.

The Poisson-Jensen Inequality states that, for a holomorphic function f on the unit disk D, the following inequality holds: |f(z)| ≤ e^(g(z) + 2π)│f(z)f('(-z))^2),

where g is the displacement function for the unit disk and f(z) is the logarithmic derivative of f at -z.

Furthermore, the inequality also holds for functions that are meromorphic on the unit disk, such that the function takes on n zeros and m poles upon intersecting the distorted unit disk. In this case, the inequality becomes |f(z)| ≤ e^(g(z) + φ)│f(z.if(-z))^2

where ψ is the dilogarithm at w.

This inequality is essential in complex analysis as it provides an efficient method to analyze holomorphic and meromorphic functions on the unit disk. The inequality forms a basis for carrying out detailed investigations, such as principal and auxiliary theorems.

Pierre Poisson and Otto Jensen contributed crucially to the development of Poisson-Jensen Inequality. The concept has been instrumental in investigations related to complex analysis and mathematics.

In recent times, extensive investigations have been made on the description of Poisson-Jensen inequalities based on a new orientation of the distortion of the unit disk.