Uniqueness of entire functions

Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann.     

On exceptional values of entire and meromorphic functions

Letf(z) be meromorphic function of finite nonzero orderρ. Assuming certain growth estimates onf by comparing it withr ^ρ L(r) whereL(r) is a slowly changing function we have obtained the bounds for the zeros off(z) ?g (z) whereg (z) is a meromorphic function satisfyingT (r, g)=o {T(r, f)} asr → ∞. These bounds are satisfied but for some exceptional functions. Examples are given to show that such exceptional functions exist.     

On constructing new expansions of functions using linear operators

Let T,U be two linear operators mapped onto the function f such that U(T(f))=f, but T(U(f))≠f. In this paper, we first obtain the expansion of functions T(U(f)) and U(T(f)) in a general case. Then, we introduce four special examples of the derived expansions. First example is a combination of the Fourier trigonometric expansion with the Taylor expansion and the second example is a mixed combination of orthogonal polynomial expansions with respect to the defined linear operators T and U. In the third example, we apply the basic expansion U(T(f))=f(x) to explicitly compute some inverse integral transforms, particularly the inverse Laplace transform. And in the last example, a mixed combination of Taylor expansions is presented. A separate section is also allocated to discuss the convergence of the basic expansions T(U(f)) and U(T(f)).     

Normal families and shared functions of meromorphic functions

Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f ^(k) share 0, and |f(z)| ≥ M whenever f ^(k)(z) = h(z), then F is normal in D. The condition that f and f ^(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f ^(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f ^(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.     

Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions

In this paper we prove that for an arbitrary pair {T ₁, T ₀} of contractions on Hilbert space with trace class difference, there exists a function ξ in L ¹(T) (called a spectral shift function for the pair {T ₁, T ₀}) such that the trace formula trace(f(T ₁) ? f(T ₀)) = ∫_T f′(ζ)ξ(ζ)dζ holds for an arbitrary operator Lipschitz function f analytic in the unit disk.     

Exceptional functions and normal families of meromorphic functions with multiple zeros

Let k be a positive integer with k?2; let h(?0) be a holomorphic function which has no simple zeros in D; and let F be a family of meromorphic functions defined in D, all of whose poles are multiple, and all of whose zeros have multiplicity at least k+1. If, for each function f∈F, f(k)(z)≠h(z), then F is normal in D.     

Derivatives of meromorphic functions and exponential functions

We take up a new method to prove a Picard type theorem. Let f be a meromorphic function in the complex plane, whose zeros are multiple, and let R be a Möbius transformation. If \({\overline {\lim } _{r \to \infty }}\frac{{T\left( {r,f} \right)}}{{{r^2}}} = \infty \) then f′z) = R(e ^z ) has infinitely many solutions in the complex plane.     

Majorization and some operator monotone functions

Let h(t) be a non-decreasing function on I and k(t) an increasing function on J. Then h is said to be majorized by k if k(A)≦k(B) implies h(A)≦h(B). f(t) is operator monotone, by definition, if f(t) is majorized by t. By making use of this majorization we will show that is operator monotone on [0,∞) for 0≦a,b<∞ and for 0≦r≦1; the special case of a=b=1 is the theorem due to Petz-Hasegawa.     

Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n?2, then fn(z)f(z+c) assumes every non-zero value a∈C infinitely often. This research also shows that there exist two sets S₁ with 9 (resp. 5) elements and S₂ with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj)(j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.     

Normal families of meromorphic functions and shared functions

The paper is devoted to the normal families of meromorphic functions and shared functions. Generalizing a result of Chang (2013), we prove the following theorem. Let h (≠≡ 0,∞) be a meromorphic function on a domain D and let k be a positive integer. Let F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g from F, f and g share the value 0, and f^(k) and g^(k) share the function h. If for every f ∈ F, at each common zero of f and h the multiplicities m_f for f and m_h for h satisfy m_f ≥ m_h + k + 1 for k > 1 and m_f ≥ 2m_h + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy n_f ≥ n_h + 1, then the family F is normal on D.     

Functional calculus and duality for closed operators

In this paper, we continue our spectral-theoretic study [8] of unbounded closed operators in the framework of the spectral decomposition property and decomposable operators. Given a closed operator T with nonempty resolvent set, let f → f(T) be the homomorphism of the functional calculus. We show that if T has the spectral decomposition property, then f(T) is decomposable. Conversely, if f is nonconstant on every component of its domain which intersects the spectrum of T, then f(T) decomposable implies that T has the spectral decomposition property. A spectral duality theorems follows as a corollary. Furthermore, we obtain an analytic-type property for the canonical embedding J of the underlying Banach space X into its second dual $\text{X}{}^7$.     

Entire functions that have the same fixed points with their first derivatives

Let f be a nonconstant entire function, and let k (?2) be an integer. We denote by the set consisting of all the fixed points of f. This paper proves that if f and f′ have the same fixed points, namely, E_f(z)=E_f′(z), and if f^(k)(z)=z whenever f(z)=z, then f≡f′.     

Growth conditions for entire functions with only bounded Fatou components

Let f be a transcendental entire function of order less than 1/2. Denote the maximum and minimum modulus of f by M(r, f) = max{|f(z)|: |z| = r} and m(r, f) = min{|f(z)|: |z| = r}. We obtain a minimum modulus condition satisfied by many f of order zero that implies all Fatou components are bounded. A special case of our result is that if

Representing measures for R(X) and their Green's functions

Let X be a compact subset of the complex plane with a nonempty interior, R(X) the uniform closure in C(X) of the rational functions with poles off X, and m a representing measure on ∂X for the functional on R(X) of evaluation at a point a in int X. Let N² be the space of functions f in L²(m) satisfying ∝ fdm = ∝ fKdm = 0 for all h in R(X), and let T be the operator on N² of multiplication by z followed by projection onto N². The spectral properties of T are investigated and shown to depend in part on the behavior of the so-called Green's function of m. In case m is the harmonic measure on ∂X for a the latter function is the classical Green's function for int X with singularity at a. Special attention is paid to the case where X is the closure of a finitely connected Jordan domain whose boundary curves are analytic. In that context, new proofs are given of Beurling's invariant subspace theorem and of Forelli's theorem on extreme points in the unit ball of the Hardy space H¹(m).     

A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

Fried and MacRae (Math. Ann. 180, 220?C226 (1969)) proved that for univariate polynomials ${p,q, f, g \in \mathbb{K}[t]}$ ( ${\mathbb{K}}$ a field) with p, q nonconstant, p(x) ? q(y) divides f(x) ? g(y) in ${\mathbb{K}[x,y]}$ if and only if there is ${h \in \mathbb{K}[t]}$ such that f?=?h(p(t)) and g?=?h(q(t)). Schicho (Arch. Math. 65, 239?C243 (1995)) proved this theorem from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In the present note, we give a new proof of one of these generalizations. The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions f, g from ${\mathbb{C}}$ to ${\mathbb{C}}$ : if both the composition ${f \circ g}$ and g are polynomial functions, then f has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. This provides a way to prove a generalization of a result by Carlitz (Acta Sci. Math. (Szeged) 24, 196?C203 (1963)) that describes those univariate polynomials over finite fields that induce bijective functions on all of their finite extensions.     

Some relations among multiplicative and <Emphasis Type="Italic">q</Emphasis>-additive functions

We give all solutions of the equation f(n) = g(n) + h(n) for every n ∈ ?, where f is a completely multiplicative, g is a 2-additive, and h is a 3-additive function. We also determine all completely multiplicative functions f and all q-additive functions g for which f(n) = g ²(n) for every n ∈ ?.     

Normal families and value distribution in connection with composite functions

We prove a value distribution result which has several interesting corollaries. Let k∈N, let α∈C and let f be a transcendental entire function with order less than 1/2. Then for every nonconstant entire function g, we have that (f○g)^(k)−α has infinitely many zeros. This result also holds when k=1, for every transcendental entire function g. We also prove the following result for normal families. Let k∈N, let f be a transcendental entire function with ρ(f)<1/k, and let a₀,…,ak−1,a be analytic functions in a domain Ω. Then the family of analytic functions g such that

     

On the value distribution theory of elliptic functions

The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr ² (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , having coefficientsa(z) satisfyingT(r, a)=o(r ²) asr→∞, will the meromorphic functionh _Q (z)=Q(z, ?(z),...,?⁽ⁿ⁾(z)) either be identically zero or satisfyN(r, 1/h _Q )=o(r ²) asr→∞? In fact, we answer this question for rational functionsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z).     

Optimally uniform approximation by meromorphic functions in the strip

The paper studies the uniform approximation problem of functions f, which are continuous in a closed strip S _h and holomorphic in its interior. Such functions are approximated on S _h by meromorphic functions g, the growth of which is estimated in the terms of the Nevanlinna characteristic T (r, g) and depends on the growth of f in the strip and the differential properties of f on the boundary of the strip. Also, the possible location of the poles of g in the complex plane is studied.     

Arithmetic properties of certain functions in several variables

If T is an n × n matrix with nonnegative integral entries, we define a transformation T: Cⁿ → Cⁿ by w = Tz where

$\text{W}{}_1\text{=}\text{∏}\text{j=1}\text{n}\text{z}{}_{\mathrm{j}}{}^{\mathrm{t}}\text{ij}\text{(1?i?n).}$

We consider functions f(z) of n complex variables which satisfy a functional equation giving f(Tz) as a rational function of ¹f(z) and we obtain conditions under which such a function f(z) takes transcendental values at algebraic points.