Jensen's Formula Research Articles

Relations Between Mahler's Measure and Values of L-Series

Mahler's measure is a natural generalization of Jensen's formula to polynomials in several variables. Its definition is as follows:The importance of Mahler's measure for polynomials in several variables lies in its connection to Lehmer's classical question which can be phrased in terms of Mahler's measure for polynomials in one variable:Given , are there any polynomials p with integer coefficients in one variable for which ?Surprisingly, Boyd [1] has shown that to answer this question, it is necessary to investigate the larger question involving polynomials in several variables.

Algebraic values of meromorphic functions on Riemann surfaces

We extend the Schneider-Lang theorem concerning algebraic values of meromorphic functions on the complex plane to meromorphic functions on punctured Riemann surfaces by using Jensen's formula for Riemann surfaces.

On Jensen's Formula and � 2π 0 log | 1 - e iθ | dθ

On Jensen's Formula and � 2π 0 log | 1 - e iθ | dθ

On Jensen's Formula and

On Jensen's Formula and

A modular version of Jensen's formula

A well-known result of classical function theory, Jensen's formula, expresses the integral around a circle of the log modulus of a meromorphic function in terms of the log modulus of the zeros and poles of that function lying inside the circle. Explicitly, if F is a meromorphic function on the unit disc {ω ε ℂ: |ω| < 1} and F(0) = 1, then, for 0 < r < 1,where ordωF is the order of F at ω. The purpose of this note is to observe that a formula analogous to (1) holds when F is replaced by a modular function for SL2(ℤ) and the integral by a suitable double integral over a fundamental domain. We shall derive this modular variant of Jensen's formula from the usual version by applying the Rankin-Selberg method and the first Kronecker limit formula. The argument admits some extension to Fuchsian groups other than SL2(ℤ), and to modular forms of weight other than zero; this point will be discussed later.

Induced gravity and Planck zeros

Starting with an asymptotically free gauge theory with dynamical symmetry breaking and a mass hierarchy, we investigate the Adler-Zee formula for the induced gravitational constant. We study the two-point function $\ensuremath{\psi}({q}^{2})$, constructed with the trace of the energy-momentum tensor. First, we show that if the zeros of $\ensuremath{\psi}$ are at a mass scale significantly below the leading scale, then ${{G}_{\mathrm{ind}}}^{\ensuremath{-}1}=O({{m}_{\mathrm{zero}}}^{2})$ making it impossible to get a realistic ${G}_{\mathrm{ind}}$ from the Adler-Zee formula with low-mass zeros. Next we use the Jensen formula to derive a sum rule for $|{m}_{\mathrm{zero}}|$. The analysis of this sum rule coupled with the result above leads to a dilemma with only one reasonable resolution. To get a realistic ${G}_{\mathrm{ind}}$ from the Adler-Zee formula, $\ensuremath{\psi}({q}^{2})$ must have a pair of complex-conjugate zeros at ${q}^{2}={{M}_{0}}^{2}\ifmmode\pm\else\textpm\fi{}2i\ensuremath{\gamma}{M}_{0}$, where ${M}_{0}$ is large and of the maximal scale and $\frac{\ensuremath{\gamma}}{{M}_{0}}\ensuremath{\ll}1$. The presence of this zero essentially determines ${{G}_{\mathrm{ind}}}^{\ensuremath{-}1}$. It gives a lower bound, which with our previously derived general upper bound gives $[\frac{{\ensuremath{\pi}}^{2}}{4(\mathrm{ln}10)288}]{C}_{\ensuremath{\psi}}{{M}_{0}}^{2}\ensuremath{\le}{(16\ensuremath{\pi}G)}^{\ensuremath{-}1}\ensuremath{\le}[\frac{5{\ensuremath{\pi}}^{2}}{288}]{C}_{\ensuremath{\psi}}{{M}_{0}}^{2}$, where ${C}_{\ensuremath{\psi}}$ is the anomaly coefficient, a number easily determined by low-order perturbation theory for any group.

Sur les nombres de Lelong associés à l'image directe d'un courant positif fermé

From a Jensen formula in several variables, one defines generalized Lelong numbers of a closed positive current relatively to a logarithmically plurisubharmonic weight. The invariance properties of these numbers with respect to analytic morphism give precise bounds for Lelong numbers of a direct image, involving some multiplicities of the morphism. An analogous theory can be applied to study the growth of a current at infinity.

Combinatorial identities of Engelberg and Jensen's formula

Combinatorial identities of Engelberg and Jensen's formula

An inequality for analytic functions

If F F denotes the boundary value of a function f ∈ H p , 1 ≤ p ≤ ∞ f \in {H^p}, 1 \leq p \leq \infty , the infimum of the measure of { θ | | F ( θ ) | > A } \{ \theta |\;|F(\theta )| > A\} for given A , 0 > A > | f ( 0 ) | , | | f | | H p = 1 A, 0 > A > |f(0)|, ||f|{|_{{H^p}}} = 1 , is determined.

The Poisson—Jensen formula with the double integral

The abstracts (in two languages) can be found in the pdf file of the article.
 Original author name(s) and title in Russian and Lithuanian:
 В. Кабайла. Аналог формулы Пуассона — Йенсена с двойным интегралом
 V. Kabaila. Puasono—Jenseno formulės analogas su dvilypiu integralu

Estimates of Zeros of a Polynomial

Throughout this note we shall consider a fixed polynomial with complex coefficients and of degree n ≥ 2. Its zeros will be denoted by ξ1, ξ2, …, ξn where the numbering is such that Making use of Jensen's integral formula, Mahler (4) showed that, for l ≥ k < n, A slightly weaker result had been established by Feldman in an earlier publication (2). Mahler's inequality (1) is of importance in the study of transcendental numbers, and our first object is to sharpen his bound by proving the following result.

An application of Jensen's formula to polynomials

In this note new proofs will be given for two inequalities on polynomials due to N. I. Feldman [1] and A. 0. Gelfond [2], respectively; these inequalities are of importance in the theory of transcendental numbers. While the original proofs by the two authors are quite unconnected, we shall deduce both results from the same source, viz. from Jensen's integral formula in the theory of analytic functions.