On inequalities for zeros of entire functions

We derive inequalities for zeros of an entire function of finite order in terms of the coefficients of its Taylor series. Our results are new even for polynomials.

     

Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros

It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis, are proved. Similar results are established for polynomials with zeros in the closed unit disk. The classical theorems by Turan and Ankeny-Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of a polynomial is proved. Bibliography: 16 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 101–112.     

Transcendence measures and algebraic growth of entire functions

In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w) in ℂ² along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z,f(z)) in the disk of radius r, in terms of the degree of P and of r. Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {n _j } of degrees of polynomials. But for special classes of functions, including the Riemann ζ-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values f(E), in terms of the size of the set E.     

Zero cancellation for general rational matrix functions

The problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints. Firstly, the class of all factors that dislocate the zeros and feature minimal McMillan degree are derived. Further, necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being J-unitary, or J-inner, either with respect to the imaginary axis or to the unit circle. The main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial, proper or improper, rank deficient, with arbitrary poles and zeros including at infinity. A particular attention is paid to the numerically-sound construction of solutions by employing at each stage unitary transformations, reliable numerical algorithms for eigenvalue assignment and efficient Lyapunov equation solvers.     

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.     

The rank-generating functions of upho posets

Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that the principal order filter at every vertex is isomorphic to the whole poset. Well-known examples include k-ary trees, the grid graphs, and the Stern poset. Very little is known about upho posets in general. In this paper, we construct upho posets with Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions have rational poles and zeros. We also categorize the rank-generating functions of all planar upho posets. Finally, we prove the existence of an upho poset with an uncomputable rank-generating function.     

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL ₂(R) and such that each derivativef ⁽ⁿ⁾,n=0, 1, 2,…, has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay onR allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (asn→∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.     

Zeros of entire functions and a problem of Ramanujan

We derive representations for certain entire q-functions and apply our technique to the Ramanujan entire function (or q-Airy function) and q-Bessel functions. This is used to show that the asymptotic series of the large zeros of the Ramanujan entire function and similar functions are also convergent series. The idea is to show that the zeros of the functions under consideration satisfy a nonlinear integral equation.     

Inequalities for the Derivatives of Rational Functions with Real Zeros

We study Turán-type inequalities for the derivatives of rational functions, whose zeros are all real and lie inside [-1,1] but whose poles are outside (-1,1), in the supremum- and L2-norms respectively. We generalize several well-known results for classical polynomials. We also obtain a sharp L2 Bernstein-type inequality for the rational system with prescribed poles.     

Sums of entire functions having only real zeros

We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang Circle Theorem from statistical mechanics. Also, we state several open problems.

     

On the exact location of the zeros of certain families of rational period functions and other related rational functions

The classification of Rational Period Functions on the modular group has been of some interest recently, and was accomplished by studying the pole sets of these rational functions. We take a complex analytic point of view and begin an investigation into the location of zeros of certain families of rational period functions.

     

Close‐to‐convexity of normalized Dini functions

In this paper necessary and sufficient conditions are deduced for the close‐to‐convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.     

On the growth of entire functions with discretely measurable zeros

We solve the problem of the least possible type of entire functions of order ρ ∈ (0, 1) with positive zeros in a special class specified by certain conditions on the upper and lower averaged ρ-density of zeros.     

正规族与复合亚纯函数

本文研究了亚纯函数族涉及复合有理函数与分担亚纯函数的正规性. 证明了一个正规定则:设 α(z) 和 F 分别是区域 D 上的亚纯函数与亚纯函数族, R(z) 是一个次数不低于 3 的有理函数.如果对族 F 中函数 f(z) 和 g(z), R○f(z) 和 R○g(z) 分担 α(z) IM,并且下述 条件之一成立:
(1) 对任何 z₀ ∈ D, R(z)-α(z₀) 有至少三个不同的零点或极点;
(2) 存在 z₀ ∈ D 使得 R(z)-α(z₀):=(z-β₀)^pH(z) 至多有两个零点(或极点) β₀,同时 k ≠ l|p|,其中 l 和 k 分别是 f(z)-β₀ 和 α(z)-α(z₀) 在 z₀ 处的零点重数, H(z) 是满足 H(β₀) ≠ 0, ∞ 的有理函数, α(z) 非常数并满足 α(z₀) ∈ C ∪{∞}.
那么 F 在 D 内正规.特别地,这个结果是著名的 Montel 正规定则的一种推广.     

On differential polynomials, fixpoints and critical values of meromorphic functions

We prove some results concerning functions ? meromorphic of finite lower order in the plane, such that ??″?α(?′)² has few zeros, where α is a constant, using in part new estimates for the multipliers at fixpoints of certain functions. We go on to consider zeros of derivatives and the minimal lower growth of meromorphic functions with finitely many critical values.     

Entire functions defined by Dirichlet series

The present paper has the object of showing that entire functions defined by Dirichlet series behave in many respects like lacunary entire functions. Both have the property of being large except in very small neighbourhoods of their zeros.     

Schlicht regions for entire and meromorphic functions

Let f:C →C be a meromorpMc function. We study the size of the maximal disc inC, with respect to the spherical metric, in which a single-valued branch of f^-1 exists. This problem is related to normality and type criteria. Best possible lower estimates of the size of such discs are obtained for entire functions and a class of meromorphic functions containing all elliptic functions. An estimate for the class of rational functions is also given which is best possible for rational functions of degree 7. For algebraic functions of given genus we obtain an estimate which is precise for genera 2 and 5 and asymptotically best possible when the genus tends to infinity. Supported by a Heisenberg fellowship of the DFG. Partially supported by NSF grant DMS-950036 and by the Lady Davis Foundation.     

Approximation by rational functions in complex regions

Approximation by rational functions R_n (z) (in the C and L_p metrics) on plane compacta is investigated. The possibility is studied of the coincidence of rational and polynomial approximations for all n, and some functions are described for which this coincidence holds. Approximations on finite sets of points are investigated, and an explanation is given of why there are functions which cannot be approximated by rational functions of degree not higher than n (in the C metric).Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 121–130, February, 1971.Several questions concerning this work were discussed with the author by A. A. Gonchar. The author wishes to thank A. A. Gonchar for his help.     

Real entire functions of infinite order and a conjecture of Wiman

We prove that if f is a real entire function of infinite order, then ff has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman of 1911.