View source for Jensen formula

A relation connecting the values of a [[Meromorphic function|meromorphic function]] inside a disc with its boundary values on the circumference and with its zeros and poles. Let $f(z)$ be a meromorphic function in the disc $\lvert z\rvert\leq R$; let $a_\mu$, $\lvert a_\mu\rvert\leq R$, and $b_\nu$, $\lvert b_\nu\rvert\leq R$, be all the zeros and poles of $f(z)$, respectively, where each pole or zero is counted as many times as its order or multiplicity. If $f(0)\neq 0$, then Jensen's formula holds:

$$\ln\lvert f(0)\rvert=\frac{1}{2\pi}\int_0^{2\pi}\ln\lvert f(Re^{i\phi})\rvert\,\mbox{d}\phi+\sum_{\lvert b_\nu\rvert< R}\ln\frac{R}{\lvert b_\nu\rvert}-\sum_{\lvert a_\mu\rvert<R}\ln \frac{R}{\lvert a_\mu\rvert},$$

in which the sums extend over all zeros and poles of $f(z)$ inside the disc $\lvert z\rvert<R$; formula (1) was obtained by J.L. Jensen in [[#References|[1]]]. A small modification is necessary to adapt (1) to the case $f(0)=0$.

A more general formula holds, called by R. Nevanlinna the Poisson–Jensen formula, giving the values of $\ln \lvert f(z)\rvert$ at an arbitrary point $z=re^{i\theta}$ other than a zero or a pole:

$$\ln\lvert f(z)\rvert=\frac{1}{2\pi}\int_0^{2\pi}\ln\lvert f(Re^{i\phi})\rvert P(z,Re^{i\phi})\,\mbox{d}\phi+\sum_{\lvert b_\nu\rvert< R}\ln\left\lvert\frac{R^2-\overline{b_\nu}z}{R(z-b_\nu)}\right\rvert-\sum_{\lvert a_\mu\rvert<R}\ln\left\lvert \frac{R^2-\overline{a_\mu}z}{R(z-a_\mu)}\right\rvert,$$

$$P(z,Re^{i\phi})=\frac{R^2-r^2}{R^2+r^2-2Rr\cos(\theta-\phi)},\,\,\,r<R.$$

Formula (2) can be regarded as a generalization of the [[Poisson integral|Poisson integral]] for a disc. Generalization of the [[Schwarz integral|Schwarz integral]] for the disc in exactly the same way gives the Schwarz–Jensen formula:

$$\ln f(z)=\frac{1}{2\pi}\int_0^{2\pi}\ln\lvert f(Re^{i\phi})\rvert \frac{Re^{i\phi}+z}{Re^{i\phi}-z}\,\mbox{d}\phi+\sum_{\lvert b_\nu\rvert< R}\ln\frac{R^2-\overline{b_\nu}z}{R(z-b_\nu)}-\sum_{\lvert a_\mu\rvert<R}\ln\frac{R^2-\overline{a_\mu}z}{R(z-a_\mu)},\,\,\,r<R.$$

Formulas of the type (1)–(3) can be constructed for half-planes and other domains. The formulas (1)–(3) play an important part in [[Value-distribution theory|value-distribution theory]].

A wide generalization of the formulas (1)–(3) has been obtained by M.M. Dzhrbashyan in his theory of classes of meromorphic functions (see [[#References|[3]]]). He succeeded in obtaining a whole family of such formulas, depending on a certain continuous parameter $\alpha$, $-1<\alpha<\infty$, that is connected with an integro-differential operator $D^\alpha$; for example, formula (3) turns out to be the special case $\alpha=0$.

Formulas (1) and (2) can be generalized for subharmonic functions $u(x)$ in a ball $\lvert x\rvert\leq R$ in a Euclidean space $\mathbb{R}^n$, $n\geq 2$, as follows:

$$u(x)=\frac{1}{\sigma(R)}\int_{\lvert y\rvert=R}u(y)\frac{R^{n-2}(R^2-\lvert x\rvert^2)}{\lvert x-y\rvert^n}\,\mbox{d}\sigma(y)+\int_{\lvert y\rvert<R}G(x,y)\,\mbox{d}\mu(y),$$

where $\sigma(R)$ is the area of the sphere $\lvert y\rvert=R$ in $\mathbb{R}^n$, $G(x,y)$ is the [[Green function|Green function]] for the ball $\lvert y\rvert<R$ with pole at $x$, and $\mu$ is the positive measure associated with the [[Subharmonic function|subharmonic function]] $u(x)$. The first summand in (4) is the least [[Harmonic majorant|harmonic majorant]] of $u(x)$ in the ball $\lvert x\rvert\leq R$, expressed in the form of a Poisson integral over the boundary values; the second summand is a Green potential, which reduces in special cases to the logarithm of the modulus of the [[Blaschke product|Blaschke product]] figuring in (2). Formula (2) is obtained from (4), taking into account that for a meromorphic function $f(z)$, $\ln\lvert f(z)\rvert$ is the difference of two subharmonic functions; formula (4) is applicable to functions of this type.

Now let $f(z)$ be a holomorphic function of several complex variables $z=(z_1,...,z_n)$, $n\geq 1$, in a closed polydisc

$$\overline{U^n}=\{ z : \lvert z_j\rvert\leq R_j, j=1,...n\}.$$

Of great importance also is the Jensen inequality, which can be deduced easily from the properties of plurisubharmonic functions (cf. [[Plurisubharmonic function|Plurisubharmonic function]]), and which for $n=1$ follows immediately from formula (2):

$$\ln\lvert f(z)\rvert\leq \int\ln\lvert f(R_1e^{i\phi_1},...,R_ne^{i\phi_n})\rvert P_n(z,Re^{i\phi})\,\mbox{d}m_n(\phi),$$

where

$$ P_n(z,Re^{i\phi})=P(z_1,R_1e^{i\phi_1})\cdots P(z_n,R_ne^{i\phi_n})$$

is the Poisson kernel for $U^n$, and $m_n$ is the normalized [[Haar measure|Haar measure]] on the distinguished boundary

$$ T^n = \{ z : \lvert z_j\rvert=R_j, j=1,...,n\},\,\,\,m_n(T^n)=1$$

(see [[#References|[5]]], [[#References|[6]]]). Inequality (5), and certain higher-dimensional analogues of formula (2), find application in modern higher-dimensional value-distribution theory, see [[#References|[7]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Jensen,   "Sur un nouvel et important théorème de la théorie des fonctions"  ''Acta Math.'' , '''22'''  (1899)  pp. 359–364</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanilinna,   "Analytic functions" , Springer  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Dzhrbashyan,   "Integral transforms and representation of functions in the complex domain" , Moscow  (1966)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. Privalov,   "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P. Griffiths,   J. King,   "Nevanlinna theory and holomorphic mappings between algebraic manifolds"  ''Acta Math.'' , '''130'''  (1973)  pp. 145–220</TD></TR></table>



====Comments====
For a generalization of Jensen's formula to sectors and for connections with functions of regular growth and distribution of zeros, see [[#References|[a3]]]. For higher-dimensional versions and applications, see also [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.A. Griffith,   "Entire holomorphic mappings in one and several complex variables" , ''Annals Math. Studies'' , '''85''' , Princeton Univ. Press  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Lelong,   L. Gruman,   "Entire functions of several complex variables" , Springer  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.Ya. Levin,   "Distribution of zeros of entire functions" , Amer. Math. Soc.  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.I. Ronkin,   "Introduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.K. Hayman,   P.B. Kennedy,   "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>