General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ?, then every asymptotic value of f, except at most 2? of them, is a limit point of critical values of f.
We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f'fn with n = 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions

https://www.math.purdue.edu/~eremenko/dvi/bergo.pdf

On the singularities of the inverse to a meromorphic function of finite order

The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography.

/pdf/mathematics-of-thermoacoustic-tomography-4454wdhd1p.pdf

Mathematics of thermoacoustic tomography

This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on C which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree d ≥ 2 is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle. Over twenty years ago, on the way to a partial explication of the phenomenon known as Bloch’s Principle, I proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains [68]. Over the years, the lemma has grown and, in dextrous hands, proved amazingly versatile, with applications to a wide variety of topics in function theory and related areas. With the renewed interest in normal families (arising largely from the important role they play in complex dynamics), it seems sensible to survey some of the most striking of these applications to the one-variable theory, with the aim of making this technique available to as broad an audience as possible. That is the purpose of this report. One pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity. In such cases, we have made no effort to resist the temptation to write out complete proofs. Hardly anything beyond a basic knowledge of function theory is required to understand what follows, so the reader is urged to take courage and plough on through. And now we turn to our tale. 1. Let D be a domain in the complex plane C. We shall be concerned with analytic maps (i.e., meromorphic functions) f : (D, | |R2) → (Ĉ, χ) Received by the editors October 15, 1997, and, in revised form, May 26, 1998. 1991 Mathematics Subject Classification. Primary 30D45; Secondary 30D35, 34A20, 58F23.

/pdf/normal-families-new-perspectives-2jbc3t1pl3.pdf

Normal families: New perspectives

A Jacobi function ${\phi _\lambda }^{\left( {\alpha ,\beta } \right)}\left( {\alpha ,\beta ,\lambda \in C,\alpha \ne - 1, - 2,} \right)$ is defined as the even C∞-function on ℝ which equals 1 at 0 and which satisfies the differential equation 

$$\left( {{d^2}/d{t^2} + \left( {\left( {2\alpha + 1} \right)cotht + \left( {2\beta + 1} \right)tht} \right)d/dt + + {\lambda ^2} + {{\left( {\alpha + \beta + 1} \right)}^2}} \right){f_\lambda }^{\left( {\alpha ,\beta } \right)}\left( t \right) = 0$$

(11)

Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups

Let k ≥ 2 be an integer and let ℱ be a family of functions meromorphic on a domain D in ℂ, all of whose poles are multiple and whose zeros all have multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each f ∈ ℱ, f (k) (z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D.

Normal families and omitted functions II

LetF be families of meromorphic functions in a domainD, and letR be a rational function whose degree is at least 3. If, for anyf∈F, the composite functionR(f) has no fixed-point inD, thenF is normal inD. The number 3 is best possible. A new and much simplified proof of a result of Pang and Zalcman concerning normality and, shared values is also given.

Normality and fixed-points of meromorphic functions

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) n assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given.

On the value distribution of ƒ + a(ƒ') n

Complex Analysis and Dynamical Systems IV: Part 2. General Relativity, Geometry, and PDE

Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

/pdf/normality-and-shared-sets-3a1wil3z66.pdf

Lawrence Zalcman

Papers

Normal families and omitted functions II

Normality and fixed-points of meromorphic functions

On the value distribution of ƒ + a(ƒ') n

Complex Analysis and Dynamical Systems IV: Part 2. General Relativity, Geometry, and PDE

Normality and shared sets