Jensen formula
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) |
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