Some properties of two analytic functions associated with complex polynomials

Two basic analytic functions α(z) and β(z) defined in domains depending on the location of the zeros of a complex polynomial P(z) are given by $\text{P′}\text{P}\text{=}\text{n}\text{(z ? α)}$ and P = (z ? β)ⁿ. These functions are studied with respect to their growth and their Laurent expansion coefficients. Applications to the location of zeros of complex polynomials are indicated.     

Growth and zeros of meromorphic solution of some linear difference equations

In this paper, we study growth and zeros of linear difference equations

Pn(z)f(z+n)+?+P₁(z)f(z+1)+P₀(z)f(z)=F(z)     

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.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Lemniscates and inequalities for the logarithmic capacities of continua

It is shown that if P(z) = z ⁿ + ? is a polynomial with connected lemniscate E(P) = {z: ¦P(z)¦ ≤ 1} and m critical points, then, for any n? m+1 points on the lemniscate E(P), there exists a continuum γ ? E(P) of logarithmic capacity cap γ ≤ 2^?1/n which contains these points and all zeros and critical points of the polynomial. As corollaries, estimates for continua of minimum capacity containing given points are obtained.     

Composite meromorphic functions and normal families

In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and \({\mathcal{F}}\) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair \({f(z),g(z)\in \mathcal{F}}\) and one of the following conditions holds: (1) P(z) ? α(z ₀) has at least three distinct zeros for any \({z_{0}\in D}\); (2) There exists \({z_{0}\in D}\) such that P(z) ? α(z ₀) has at most two distinct zeros and α(z) is nonconstant. Assume that β ₀ is a zero of P(z) ? α(z ₀) with multiplicity p and that the multiplicities l and k of zeros of f(z) ? β ₀ and α(z) ? α(z ₀) at z ₀, respectively, satisfy k ≠ lp, for all \({f(z)\in\mathcal{F}}\). Then \({\mathcal{F}}\) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.     

On the Distribution of the Derivative Zeros of a Complex Polynomial

We consider the class S(n) of all complex polynomials of degree n > 1 having all their zeros in the closed unit disk ē. By S(n,β) we denote the subclass of p ? S(n) vanishing in the prescribed point β ? ē. For an arbitrary point α ? C and p ? S(n,β) let |p| _α be the distance of α and the set of zeros of P'. Then there exists some P ? S(n,β) with maximal |P|α. We give an estimation for the number of zeros of P on |z| = 1$ resp. P' on $ |z-α| = |P| _α .     

Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

Entire functions that share a small function with its derivative

In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n?2). This result is a generalization of several previous results.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

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.     

On the divergence of polynomial interpolation

Consider a triangular interpolation scheme on a continuous piecewise C¹ curve of the complex plane, and let Γ be the closure of this triangular scheme. Given a meromorphic function f with no singularities on Γ, we are interested in the region of convergence of the sequence of interpolating polynomials to the function f. In particular, we focus on the case in which Γ is not fully contained in the interior of the region of convergence defined by the standard logarithmic potential. Let us call Γ_out the subset of Γ outside of the convergence region.In the paper we show that the sequence of interpolating polynomials, {P_n}_n, is divergent on all the points of Γ_out, except on a set of zero Lebesgue measure. Moreover, the structure of the set of divergence is also discussed: the subset of values z for which there exists a partial sequence of {P_n(z)}_n that converges to f(z) has zero Hausdorff dimension (so it also has zero Lebesgue measure), while the subset of values for which all the partials are divergent has full Lebesgue measure.The classical Runge example is also considered. In this case we show that, for all z in the part of the interval (−5,5) outside the region of convergence, the sequence {P_n(z)}_n is divergent.     

Uniqueness and value distribution of differences of entire functions

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of fn(z)(λfm(z+c)+μfm(z))−a, where f is an entire function, n, m are two integers such that n?m>0, and λ, μ are non-zero complex numbers. We also obtain a uniqueness result in the case where shifts of two entire functions share a small function.     

Comparison of the best uniform approximations of analytic functions in the disk and on its boundary

Denote by C _A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ? _n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R _n (f) (R _n (f) _A ) the best uniform approximation of a function f ∈ C _A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ? _n . The following equality is proved for any n ≥ 1: sup{R _n (f) _A /R _n (f): f ∈ C _A ? ? _n } = 2. We also consider a similar problem of comparing the best approximations of functions in C _A by polynomials and trigonometric polynomials.     

Differential operators and entire functions with simple real zeros

Let ? and f be functions in the Laguerre-Pólya class. Write ?(z)=e−αz²?₁(z) and f(z)=e−βz²f₁(z), where ?₁ and f₁ have genus 0 or 1 and α,β?0. If αβ<1/4 and ? has infinitely many zeros, then ?(D)f(z) has only simple real zeros, where D denotes differentiation.     

On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function

In this paper we study the distribution of zeros of each entire function of the sequence , which approaches the Riemann zeta function for Rez<−1, and is closely related to the solutions of the functional equations f(z)+f(2z)+?+f(nz)=0. We determine the density of the zeros of Gn(z) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of Gn(z) inside certain rectangles in the critical strip.     

Entire solutions of certain type of differential equations II

We analyze the transcendental entire solutions of the following type of nonlinear differential equations: fn(z)+P(f)=p₁eα₁z+p₂eα₂z in the complex plane, where p₁, p₂ and α₁, α₂ are nonzero constants, and P(f) denotes a differential polynomial in f of degree at most n−1 with small functions of f as the coefficients.     

On the equivalence of the Euler-Pommier operators in spaces of analytic functions

In the space A (θ) of all one-valued functions f(z) analytic in an arbitrary region G ? ? (0 ∈ G) with the topology of compact convergence, we establish necessary and sufficient conditions for the equivalence of the operators L ₁=α _n z ⁿ Δ ⁿ + ... + α₁ zΔ+α₀ E and L ₂= z ⁿ a _n (z)Δ ⁿ + ... + za ₁(z)Δ+a ₀(z)E, where δ: (Δ?)(z)=(f(z)-?(0))/z is the Pommier operator in A(G), n ∈ ?, α _n ∈ ?, a _k (z) ∈ A(G), 0≤k≤n, and the following condition is satisfied: Σ _j=s ^n?1 α _j+1 ∈ 0, s=0,1,...,n?1. We also prove that the operators z ^s+1Δ+β(z)E, β(z) ∈ A _R , s ∈ ?, and z ^s+1 are equivalent in the spaces A _R, 0?R?-∞, if and only if β(z) = 0.     

On Value Distribution of Δ~nf

In this paper, we mainly study zeros and poles of the forward differences Δnf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.