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| {{distinguish|Jensen's inequality}}
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| In the mathematical field known as [[complex analysis]], '''Jensen's formula''', introduced by {{harvs|txt|authorlink=Johan Jensen (mathematician)|first=Johan|last= Jensen|year=1899}}, relates the average magnitude of an [[analytic function]] on a circle with the number of its [[root of a function|zeros]] inside the circle. It forms an important statement in the study of [[entire function]]s.
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| == The statement ==
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| Suppose that ''ƒ'' is an analytic function in a region in the [[complex plane]] which contains the [[closed disk]] '''D''' of radius ''r'' about the origin, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> are the zeros of ''ƒ'' in the interior of '''D''' repeated according to multiplicity, and ''ƒ''(0) ≠ 0. '''Jensen's formula''' states that
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| :<math>\log |f(0)| = \sum_{k=1}^n \log \left( \frac{|a_k|}{r}\right) + \frac{1}{2\pi} \int_0^{2\pi} \log|f(re^{i\theta})| \, d\theta.</math>
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| This formula establishes a connection between the moduli of the zeros of the function ''ƒ'' inside the disk '''D''' and the average of ''log |f(z)|'' on the boundary circle |''z''| = ''r'', and can be seen as a generalisation of the mean value property of [[harmonic function]]s. Namely, if ''f'' has no zeros in '''D''', then Jensen's formula reduces to
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| :<math>\log |f(0)| = \frac{1}{2\pi} \int_0^{2\pi} \log|f(re^{i\theta})| \, d\theta,</math>
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| which is the mean-value property of the harmonic function <math>\log |f(z)|</math>.
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| An equivalent statement of Jensen's formula that is frequently used is
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| :<math>\frac{1}{2\pi} \int_0^{2\pi} \log |f(re^{i\theta})| \; d\theta
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| - \log |f(0)| = \int_0^r \frac{n(t)}{t} \; dt | |
| </math>
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| where <math>n(t)</math> denotes the number of zeros of <math>f</math> in the disc of radius <math>t</math> centered at the origin.
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| Jensen's formula may be generalized for functions which are merely meromorphic on '''D'''. Namely, assume that
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| :<math>f(z)=z^l \frac{g(z)}{h(z)},</math>
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| where ''g'' and ''h'' are analytic functions in '''D''' having zeros at <math>a_1,\ldots,a_n \in \mathbb D\backslash\{0\}</math>
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| and
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| <math>b_1,\ldots,b_m \in \mathbb D\backslash\{0\}</math>
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| respectively, then Jensen's formula for meromorphic functions states that
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| :<math>\log \left|\frac{g(0)}{h(0)}\right| = \log \left |r^{m-n} \frac{a_1\ldots a_n}{b_1\ldots b_m}\right| + \frac{1}{2\pi} \int_0^{2\pi} \log|f(re^{i\theta})| \, d\theta.</math>
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| Jensen's formula can be used to estimate the number of zeros of analytic function in a circle. Namely, if ''f'' is a function analytic in a disk of radius ''R'' centered at ''z<sub>0</sub>'' and if ''|f|'' is bounded by ''M'' on the boundary of that disk, then the number of zeros of ''f'' in a circle of radius ''r''<''R'' centered at the same point ''z<sub>0</sub>'' does not exceed
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| :<math>
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| \frac{1}{\log (R/r)} \log \frac{M}{|f(z_0)|}.
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| </math>
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| Jensen's formula is an important statement in the study of value distribution of entire and meromorphic functions. In particular, it is the starting point of [[Nevanlinna theory]].
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| ==Poisson–Jensen formula==
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| Jensen's formula is a consequence of the more general Poisson–Jensen formula, which in turn follows from Jensen's formula by applying a [[Möbius transformation]] to ''z''. It was introduced and named by [[Rolf Nevanlinna]]. If ''f'' is a function which is analytic in the unit disk, with zeros ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> located in the interior of the unit disk, then for every <math>z_0=r_0e^{i\varphi_0}</math> in the unit disk the '''Poisson–Jensen formula''' states that
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| :<math>\log |f(z_0)| = \sum_{k=1}^n \log \left|\frac{z_0-a_k}{1-\bar {a}_k z_0} \right| + \frac{1}{2\pi} \int_0^{2\pi} P_{r_0}(\varphi_0-\theta) \log |f(e^{i\theta})| \, d\theta.</math>
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| Here,
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| :<math>
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| P_{r}(\omega)= \sum_{n\in \mathbb Z} r^{|n|} e^{i n\omega}
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| </math>
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| is the [[Poisson kernel]] on the unit disk.
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| If the function ''f'' has no zeros in the unit disk, the Poisson-Jensen formula reduces to
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| :<math>\log |f(z_0)| = \frac{1}{2\pi} \int_0^{2\pi} P_{r_0}(\varphi_0-\theta) \log |f(e^{i\theta})| \, d\theta,</math> | |
| which is the [[Poisson formula]] for the harmonic function <math>\log |f(z)|</math>.
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| == References ==
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| * {{citation | authorlink=Lars Ahlfors | first = Lars V. | last=Ahlfors | title = Complex analysis. An introduction to the theory of analytic functions of one complex variable | edition=3rd | publisher = McGraw–Hill | year = 1979 | isbn=0-07-000657-1 | series=International Series in pure and applied Mathematics | location=Düsseldorf | zbl=0395.30001 }}
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| * {{citation | last1=Jensen | first1=J. | title=Sur un nouvel et important théorème de la théorie des fonctions | language=French | publisher=Springer Netherlands | year=1899 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=22 | pages=359–364 | doi=10.1007/BF02417878 | jfm=30.0364.02 }}
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| * {{citation | last=Ransford | first=Thomas | title=Potential theory in the complex plane | series=London Mathematical Society Student Texts | volume=28 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-46654-7 | zbl=0828.31001 }}
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| {{DEFAULTSORT:Jensen's Formula}}
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| [[Category:Complex analysis]]
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| [[Category:Theorems in complex analysis]]
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