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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542101.png" /> be a meromorphic function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542102.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542106.png" />, be all the zeros and poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542107.png" />, respectively, where each pole or zero is counted as many times as its order or multiplicity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542108.png" />, then Jensen's formula holds:
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{{TEX|done}}
 +
{{MSC|30E20}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j0542109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421010.png" /></td> </tr></table>
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$$\ln\lvert f(0)\rvert=\frac{1}{2\pi}\int_0^{2\pi}\ln\lvert f(Re^{i\phi})\rvert\,\mbox{d}\phi+\sum_{\lvert b_\nu\rvert< R}\ln\frac{R}{\lvert b_\nu\rvert}-\sum_{\lvert a_\mu\rvert<R}\ln \frac{R}{\lvert a_\mu\rvert},$$
  
in which the sums extend over all zeros and poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421011.png" /> inside the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421012.png" />; formula (1) was obtained by J.L. Jensen in [[#References|[1]]]. A small modification is necessary to adapt (1) to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421013.png" />.
+
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 {{Cite|Je}}. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421014.png" /> at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421015.png" /> other than a zero or a pole:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\ln\lvert f(z)\rvert=\frac{1}{2\pi}\int_0^{2\pi}\ln\lvert f(Re^{i\phi})\rvert P(z,Re^{i\phi})\,\mbox{d}\phi+\sum_{\lvert b_\nu\rvert< R}\ln\left\lvert\frac{R^2-\overline{b_\nu}z}{R(z-b_\nu)}\right\rvert-\sum_{\lvert a_\mu\rvert<R}\ln\left\lvert \frac{R^2-\overline{a_\mu}z}{R(z-a_\mu)}\right\rvert,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421017.png" /></td> </tr></table>
+
$$P(z,Re^{i\phi})=\frac{R^2-r^2}{R^2+r^2-2Rr\cos(\theta-\phi)},\,\,\,r<R.$$
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421018.png" /></td> </tr></table>
 
  
 
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:
 
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$\ln f(z)=\frac{1}{2\pi}\int_0^{2\pi}\ln\lvert f(Re^{i\phi})\rvert \frac{Re^{i\phi}+z}{Re^{i\phi}-z}\,\mbox{d}\phi+\sum_{\lvert b_\nu\rvert< R}\ln\frac{R^2-\overline{b_\nu}z}{R(z-b_\nu)}-\sum_{\lvert a_\mu\rvert<R}\ln\frac{R^2-\overline{a_\mu}z}{R(z-a_\mu)},\,\,\,r<R.$$
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421020.png" /></td> </tr></table>
 
  
 
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]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421022.png" />, that is connected with an integro-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421023.png" />; for example, formula (3) turns out to be the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421024.png" />.
+
A wide generalization of the formulas (1)–(3) has been obtained by M.M. Dzhrbashyan in his theory of classes of meromorphic functions (see {{Cite|Dz}}). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421025.png" /> in a ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421026.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421028.png" />, as follows:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$u(x)=\frac{1}{\sigma(R)}\int_{\lvert y\rvert=R}u(y)\frac{R^{n-2}(R^2-\lvert x\rvert^2)}{\lvert x-y\rvert^n}\,\mbox{d}\sigma(y)+\int_{\lvert y\rvert<R}G(x,y)\,\mbox{d}\mu(y),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421030.png" /></td> </tr></table>
+
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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421031.png" /> is the area of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421034.png" /> is the [[Green function|Green function]] for the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421035.png" /> with pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421037.png" /> is the positive measure associated with the [[Subharmonic function|subharmonic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421038.png" />. The first summand in (4) is the least [[Harmonic majorant|harmonic majorant]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421039.png" /> in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421040.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421042.png" /> 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 $z=(z_1,\ldots,z_n)$, $n\geq 1$, in a closed polydisc
  
Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421043.png" /> be a holomorphic function of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421045.png" />, in a closed polydisc
+
$$\overline{U^n}=\{ z : \lvert z_j\rvert\leq R_j, j=1,\ldots,n\}.$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421046.png" /></td> </tr></table>
+
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):
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421047.png" /> follows immediately from formula (2):
+
$$\ln\lvert f(z)\rvert\leq \int\ln\lvert f(R_1e^{i\phi_1},\ldots,R_ne^{i\phi_n})\rvert P_n(z,Re^{i\phi})\,\mbox{d}m_n(\phi),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421049.png" /></td> </tr></table>
+
$$ P_n(z,Re^{i\phi})=P(z_1,R_1e^{i\phi_1})\cdots P(z_n,R_ne^{i\phi_n})$$
  
where
+
is the Poisson kernel for $U^n$, and $m_n$ is the normalized [[Haar measure|Haar measure]] on the distinguished boundary
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421050.png" /></td> </tr></table>
+
$$ T^n = \{ z : \lvert z_j\rvert=R_j, j=1,\ldots,n\},\,\,\,m_n(T^n)=1$$
  
is the Poisson kernel for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421051.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421052.png" /> is the normalized [[Haar measure|Haar measure]] on the distinguished boundary
+
(see {{Cite|Vl}}, {{Cite|GuRo}}). Inequality (5), and certain higher-dimensional analogues of formula (2), find application in modern higher-dimensional value-distribution theory, see {{Cite|GrKi}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054210/j05421053.png" /></td> </tr></table>
 
  
(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====
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Dz}}||valign="top"| M.M. Dzhrbashyan,  "Integral transforms and representation of functions in the complex domain", Moscow  (1966)  (In Russian)
 +
|-
 +
|valign="top"|{{Ref|GrKi}}||valign="top"| P. Griffiths,  J. King,  "Nevanlinna theory and holomorphic mappings between algebraic manifolds"  ''Acta Math.'', '''130'''  (1973) pp. 145–220
 +
|-
 +
|valign="top"|{{Ref|GuRo}}||valign="top"|  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables", Prentice-Hall  (1965)
 +
|-
 +
|valign="top"|{{Ref|Je}}||valign="top"|  J.L. Jensen,  "Sur un nouvel et important théorème de la théorie des fonctions"  ''Acta Math.'', '''22'''  (1899)  pp. 359–364
 +
|-
 +
|valign="top"|{{Ref|Ne}}||valign="top"|  R. Nevanilinna,  "Analytic functions", Springer  (1970)  (Translated from German)
 +
|-
 +
|valign="top"|{{Ref|Pr}}||valign="top"|  I.I. Privalov,  "Subharmonic functions", Moscow-Leningrad  (1937)  (In Russian)
 +
|-
 +
|valign="top"|{{Ref|Vl}}||valign="top"|  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables", M.I.T.  (1966)  (Translated from Russian)
 +
|-
 +
|}
  
====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====
 
====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]]].
+
For a generalization of Jensen's formula to sectors and for connections with functions of regular growth and distribution of zeros, see {{Cite|Le}}. For higher-dimensional versions and applications, see also {{Cite|Gr}}, {{Cite|LeGr}}, {{Cite|Ro}}.
  
 
====References====
 
====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. LelongL. 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,  "Inroduction 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>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Gr}}||valign="top"| P.A. Griffith,  "Entire holomorphic mappings in one and several complex variables", ''Annals Math. Studies'', '''85''', Princeton Univ. Press  (1976)
 +
|-
 +
|valign="top"|{{Ref|HaKe}}||valign="top"| W.K. HaymanP.B. Kennedy,  "Subharmonic functions", '''1''', Acad. Press (1976)
 +
|-
 +
|valign="top"|{{Ref|Le}}||valign="top"| B.Ya. Levin,  "Distribution of zeros of entire functions", Amer. Math. Soc.  (1980)  (Translated from Russian)
 +
|-
 +
|valign="top"|{{Ref|LeGr}}||valign="top"|  P. Lelong,  L. Gruman,  "Entire functions of several complex variables", Springer  (1986)
 +
|-
 +
|valign="top"|{{Ref|Ro}}||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)
 +
|-
 +
|}

Latest revision as of 20:51, 19 April 2012

2020 Mathematics Subject Classification: Primary: 30E20 [MSN][ZBL]

A relation connecting the values of a 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 [Je]. 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 for a disc. Generalization of the 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.

A wide generalization of the formulas (1)–(3) has been obtained by M.M. Dzhrbashyan in his theory of classes of meromorphic functions (see [Dz]). 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 for the ball $\lvert y\rvert<R$ with pole at $x$, and $\mu$ is the positive measure associated with the subharmonic function $u(x)$. The first summand in (4) is the least 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 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 $z=(z_1,\ldots,z_n)$, $n\geq 1$, in a closed polydisc

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

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

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

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 on the distinguished boundary

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

(see [Vl], [GuRo]). Inequality (5), and certain higher-dimensional analogues of formula (2), find application in modern higher-dimensional value-distribution theory, see [GrKi].


References

[Dz] M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain", Moscow (1966) (In Russian)
[GrKi] P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic manifolds" Acta Math., 130 (1973) pp. 145–220
[GuRo] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables", Prentice-Hall (1965)
[Je] J.L. Jensen, "Sur un nouvel et important théorème de la théorie des fonctions" Acta Math., 22 (1899) pp. 359–364
[Ne] R. Nevanilinna, "Analytic functions", Springer (1970) (Translated from German)
[Pr] I.I. Privalov, "Subharmonic functions", Moscow-Leningrad (1937) (In Russian)
[Vl] V.S. Vladimirov, "Methods of the theory of functions of several complex variables", M.I.T. (1966) (Translated from Russian)



Comments

For a generalization of Jensen's formula to sectors and for connections with functions of regular growth and distribution of zeros, see [Le]. For higher-dimensional versions and applications, see also [Gr], [LeGr], [Ro].

References

[Gr] P.A. Griffith, "Entire holomorphic mappings in one and several complex variables", Annals Math. Studies, 85, Princeton Univ. Press (1976)
[HaKe] W.K. Hayman, P.B. Kennedy, "Subharmonic functions", 1, Acad. Press (1976)
[Le] B.Ya. Levin, "Distribution of zeros of entire functions", Amer. Math. Soc. (1980) (Translated from Russian)
[LeGr] P. Lelong, L. Gruman, "Entire functions of several complex variables", Springer (1986)
[Ro] L.I. Ronkin, "Introduction to the theory of entire functions of several variables", Transl. Math. Monogr., 44, Amer. Math. Soc. (1974) (Translated from Russian)
How to Cite This Entry:
Jensen formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jensen_formula&oldid=17842
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article