The support points of several classes of analytic functions with fixed coefficients

Let B denote the set of functions ?(z) that are analytic in the unit disk D and satisfy |?(z)|?1(|z|<1). Let P denote the set of functions p(z) that are analytic in D and satisfy p(0)=1 and Rep(z)>0(|z|<1). Let T denote the set of functions f(z) that are analytic in D, normalized by f(0)=0 and f^′(0)=1 and satisfy that f(z) is real if and only if z is real (|z|<1). In this article we investigate the support points of the subclasses of B, P and T of functions with fixed coefficients.     

Isometries of some F-algebras of holomorphic functions on the upper half plane

Linear isometries of N ^p (D) onto N ^p (D) are described, where N ^p (D), p > 1, is the set of all holomorphic functions f on the upper half plane D = {z ∈ ?: Im z > 0} such that sup _y >0 ∫_? ln ^p (1 + |(x + iy)|) dx < +∞. Our result is an improvement of the results by D.A. Efimov.     

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.     

On multiplicative functions which are additive on sums of primes

Let ${f : \mathbb{N} \to \mathbb{C}}$ be a multiplicative function satisfying f(p ₀) ≠ 0 for at least one prime number p ₀, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p ₁ + p ₂ + . . . + p _k ) = f(p ₁) + f(p ₂) + . . . + f(p _k ) for any prime numbers p ₁, p ₂, . . . ,p _k then f must be the identity f(n) = n for each ${n \in \mathbb{N}}$ . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.     

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.     

Estimates of holomorphic functions in zero-free domains

We study functions f(z) holomorphic in having the property f(z) ≠ 0 for 0 < Im z < 1 and we obtain lower bounds for |f(z)| for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for and I + A(z) is invertible for 0 < Im z < 1 and is unitary for . A. Borichev was partially supported by the ANR project DYNOP.     

Normal families of meromorphic functions with multiple values

Let F be a family of meromorphic functions defined in a domain D, let ψ(?0) be a holomorphic function in D, and k be a positive integer. Suppose that, for every function f∈F, f≠0, f(k)≠0, and all zeros of f(k)−ψ(z) have multiplicities at least (k+2)/k. If, for k=1, ψ has only zeros with multiplicities at most 2, and for k?2, ψ has only simple zeros, then F is normal in D. This improves and generalizes the related results of Gu, Fang and Chang, Yang, Schwick, et al.     

Share fix-points and normal families of holomorphic functions

Let be a family of holomorphic functions in a domain D, let k 2 be a positive integer, and let K be a positive number. In this paper, we prove that: if, for each f and f share fix-points CM, and |f^(k) (z)| K whenever f(z) = z in D, then is normal in D. Some examples are given to show the sharpness of our result.Received: 17 June 2004     

Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P(z)=(1−z)^−b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0<Reb<2. Let where μ is a complex-valued measure on the closed unit disk Such sequences are shown to be multipliers of H^p for 1?p?∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that {μ_n} is a multiplier of H^p for every p>0. When the support of μ is [0,1] we get the multiplier sequence which provides more concrete applications. We show that if the sequences {μ_n} and {ν_n} are related by an asymptotic expansion

     

An improvement of Ozaki’s P-valent conditions

The old result due to[Ozaki,S.:On the theory of multivalent functions Ⅱ.Sci.Rep.Tokyo Bunrika Daigaku Sect.A,45-87(1941)],says that if f(z) = z~p + ∑_(n=p+1~(a_nz~n))~∞ is analytic in a convex domain D and for some real α we have Re{exp(iα)f~((p))(z)} 0 in D,then f(z) is at most p-valent in ED.In this paper,we consider similar problems in the unit disc B = {z ∈ C:|z| 1}.     

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

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.     

Representations of analytic functions as infinite products and their application to numerical computations

Let D be an open disk of radius ≤1 in $\mathbb{C}$ , and let (? _n ) be a sequence of ±1. We prove that for every analytic function $f: D \to \mathbb{C}$ without zeros in D, there exists a unique sequence (α _n ) of complex numbers such that $f(z) = f(0)\prod_{n=1}^{\infty} (1+\epsilon_{n}z^{n})^{\alpha_{n}}$ for every z∈D. From this representation we obtain a numerical method for calculating products of the form ∏ _p prime f(1/p) provided f(0)=1 and f′(0)=0; our method generalizes a well-known method of Pieter Moree. We illustrate this method on a constant of Ramanujan $\pi^{-1/2}\prod _{p~\text{prime}} \sqrt{p^{2}-p}\ln(p/(p-1))$ . From the properties of the exponents α _n , we obtain a proof of the following congruences, which have been the subject of several recent publications motivated by some questions of Arnold: for every n×n integral matrix A, every prime number p, and every positive integer k we have $\operatorname{tr} A^{p^{k}} \equiv\operatorname{tr} A^{p^{k-1}} { \hbox {\rm { (mod\ $p^{k}$) }}}$ .     

Nevanlinna-Pick interpolation by rational functions with a single pole inside the unit disk

We devise an efficient algorithm that, given points z₁,…,z_k in the open unit disk D and a set of complex numbers {f_i,0,f_i,1,…,f_i,ni−1} assigned to each z_i, produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at z_i has f_i,0,f_i,1,…,f_i,ni−1 as its first n_i coefficients.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omit a Function II

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of functions holomorphic on a domain D ? ?, all of whose zeros have multiplicity at least k, where k ?? 2 is an integer. Let h(z) ? 0 and ?? be a meromorphic function on D. Assume that the following two conditions hold for every f ?? F: $$ \begin{gathered} (a)f(z) = 0 \Rightarrow |f^{(k)} (z)| < |h(z)|. \hfill \\ (b)f^{(k)} (z) \ne h(z). \hfill \\ \end{gathered} $$ Then F is normal on D.     

On the asymptotics of the rows of the Padé table of analytic functions with logarithmic branch points

For the functions $ f(z) = \sum\nolimits_{n = 0}^\infty {z^{l_n } } /a_n $ , where l _n and a _n are arithmetic progressions and their Padé approximants π _n,m (z; f), we establish an asymptotics of the decrease of the difference f(z) ? π _n,m (z; f) for the case in which z ∈ D = {z: |z| < 1}, m is fixed, and n → ∞. In particular, we obtain proximate orders of decrease of best uniform rational approximations to the functions ln(1 ? z) and arctan z in the disk D _q = {z: |z| ≤ q < 1}.     

Growth rate of the functions in Bergman type spaces

For 0<p,α<∞, let ‖f‖_p,α be the L^p-norm with respect the weighted measure . We define the weighted Bergman space A_α^p(D) consisting of holomorphic functions f with ‖f‖_p,α<∞. For any σ>0, let A^−σ(D) be the space consisting of holomorphic functions f in D with . If D has C² boundary, then we have the embedding A_α^p(D)⊂A^−(n+α)/p(D). We show that the condition of C²-smoothness of the boundary of D is necessary by giving a counter-example of a convex domain with C^1,λ-smooth boundary for 0<λ<1 which does not satisfy the embedding.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) ? 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator \({f \mapsto f_c}\) maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

Sequences of Exponents in Constructing Strongly Annular Products

This paper gives conditions on the behavior of a sequence of holomorphic functions {f _k (z)} and a strictly increasing sequence of positive integers {m _k } that assures the infinite product Pf_k(z^m_k){\Pi f_k(z^{m_k})} is strongly annular. A constructive proof is given that shows if the sequence {f _k (z)} exhibits certain boundary behavior along with a uniform boundedness condition then a number p > 1 exists such that if {m _k } satisfies m _k+1/m _k ≥ p then the above product is strongly annular.