Unicity of meromorphic functions and their derivatives

In this paper, we prove that if a transcendental meromorphic function f shares two distinct small functions CM with its kth derivative f^(k) (k>1), then f=f^(k). We also resolve the same question for the case k=1. These results generalize a result due to Frank and Weissenborn.     

Construction of bent functions via Niho power functions

A Boolean function with an even number n=2k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f(x)=tr(α₁x^d₁+α₂x^d₂), α₁,α₂,x∈F_2ⁿ, are considered, where the exponents d_i (i=1,2) are of Niho type, i.e. the restriction of x^d_i on F_2^k is linear. We prove for several pairs of (d₁,d₂) that f is a bent function, when α₁ and α₂ fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F_2ⁿ are bijective.     

Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ^* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality. To Reiner Kühnau on his 70th birthday     

Differential polynomials sharing one value

In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k+12. If fⁿ + af^(k) and gⁿ + ag^(k) share b CM and the b-points of fⁿ + af^(k) are not the zeros of f and g, then f and g are either equal or closely related.     

Read-Once Functions Revisited and the Readability Number of a Boolean Function

In this paper, we present the first polynomial time algorithm for recognizing and factoring read-once functions. The algorithm is based on algorithms for cograph recognition and a new efficient method for checking normality. Its correctness is based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal and its co-occurrance graph is P₄-free.We also investigate the problem of factoring certain non-read-once functions. In particular, we show that if the co-occurrence graph of a positive Boolean function f is a tree, then the function is read-twice. We then extend this further proving that if f is normal and its corresponding graph is a partial k-tree, then f is a read 2^k function and a read 2^k formula for F for f can be obtained in polynomial time.     

Log-convexity and log-concavity of hypergeometric-like functions

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a?∑fkk(a)xk, a?∑fkΓ(a+k)xk and a?∑fkxk/k(a). The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.     

Non-L 1 functions with rotation sets of Hausdorff dimension one

Suppose that f: ? → ? is a given measurable function, periodic by 1. For an α ∈ ? put M _n ^α f(x) = 1/n+1 Σ _k=0 ⁿ f(x + kα). Let Γ _f denote the set of those α’s in (0;1) for which M _n ^α f(x) converges for almost every x ∈ ?. We call Γ _f the rotation set of f. We proved earlier that from |Γ _f | > 0 it follows that f is integrable on [0; 1], and hence, by Birkhoff’s Ergodic Theorem all α ∈ [0; 1] belongs to Γ _f . However, Γ _f \? can be dense (even c-dense) for non-L ¹ functions as well. In this paper we show that there are non-L ¹ functions for which Γ _f is of Hausdorff dimension one.     

Sections of Functions and Sobolev-Type Inequalities

We study functions of two variables whose sections by the lines parallel to the coordinate axis satisfy the Lipschitz condition of order 0 < α ≤ 1. We prove that if for a function f the Lip α-norms of these sections belong to the Lorentz space L ^p,1(?) (p = 1/α), then f can be modified on a set of measure zero so as to become bounded and uniformly continuous on ?². For α = 1 this gives an extension of Sobolev’s theorem on continuity of functions of the space W ₁ ^2,2 (?²). We show that the exterior L ^p,1-norm cannot be replaced by a weaker Lorentz L ^p,q -norm with q > 1.     

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.     

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

     

Closure properties of operators on the Ma-Minda type starlike and convex functions

A normalized univalent function f is called Ma-Minda starlike or convex if zf^′(z)/f(z)?φ(z) or 1+zf^″(z)/f^′(z)?φ(z) where φ is a convex univalent function with φ(0)=1. The class of Ma-Minda convex functions is shown to be closed under certain operators that are generalizations of previously studied operators. Analogous inclusion results are also obtained for subclasses of starlike and close-to-convex functions. Connections with various earlier works are made.     

Norm estimates for the Alexander transforms of convex functions of order alpha

The hyperbolic sup norm of the pre-Schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities. We present sharp norm estimates for the Alexander transforms of convex functions of order α, 0?α<1.     

The star function for meromorphic functions of several complex variables

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.     

Riemann integrability and Lebesgue measurability of the composite function

If f is continuous on the interval [a,b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [α,β] and g([α,β])⊂[a,b], then f○g is Riemann integrable (resp. measurable) on [α,β]. A well-known fact, on the other hand, states that f○g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a c²-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f∈V?{0} and g∈W?{0}, f○g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g∈W?{0} there exists a c-dimensional space V of measurable functions such that f○g is not measurable for all f∈V?{0}.     

Entire functions that share a polynomial with their derivatives

Let f be a nonconstant entire function, k and q be positive integers satisfying k>q, and let Q be a polynomial of degree q. This paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial Q is shared by f and f^′ CM, and if f^(k)(z)−Q(z)=0 whenever f(z)−Q(z)=0, then f≡f^′. We give two examples to show that the hypothesis k>q is necessary.     

Coefficient estimates for inverses of starlike functions of positive order

In the present paper, the coefficient estimates are found for the class S∗−1(α) consisting of inverses of functions in the class of univalent starlike functions of order α in . These estimates extend the work of Krzyz, Libera and Zlotkiewicz [J.G. Krzyz, R.J. Libera, E. Zlotkiewicz, Coefficients of inverse of regular starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 33 (10) (1979) 103-109] who found sharp estimates on only first two coefficients for the functions in the class S∗−1(α). The coefficient estimates are also found for the class Σ∗−1(α), consisting of inverses of functions in the class Σ^∗(α) of univalent starlike functions of order α in . The open problem of finding sharp coefficient estimates for functions in the class Σ^∗(α) stands completely settled in the present work by our method developed here.     

On the successive coefficients of close-to-convex functions

Let f∈S, f be a close-to-convex function, f_k(z)=[f(z^k)]^1/k. The relative growth of successive coefficients of f_k(z) is investigated. The sharp estimate of ||c_n+1|−|c_n|| is obtained by using the method of the subordination function.     

The expressive power of voting polynomials

We consider the problem of approximating a Boolean functionf∶{0,1} ⁿ →{0,1} by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP ^A is properly contained in PSPACE ^A . We are also able to prove the old AC⁰ exponential-size lower bounds in a new way. This allows us to prove the new result that an AC⁰ circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.     

A Komlós Theorem for abstract Banach lattices of measurable functions

Consider a Banach function space X(μ) of (classes of) locally integrable functions over a σ-finite measure space (Ω,Σ,μ) with the weak σ-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlós on convergence of Cesàro sums in L¹[0,1] holds also in these spaces; i.e. for every bounded sequence n(fn) in X(μ), there exists a subsequence k(fnk) and a function f∈X(μ) such that for any further subsequence j(hj) of k(fnk), the series converges μ-a.e. to f. In this paper we generalize this result to a more general class of Banach spaces of classes of measurable functions — spaces L¹(ν) of integrable functions with respect to a vector measure ν on a δ-ring — and explore to which point the Fatou property and the Komlós property are equivalent. In particular we prove that this always holds for ideals of spaces L¹(ν) with the weak σ-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlós Theorem.     

Univalent functions without common values

In this paper we establish inequalities involving moduli of derivatives |f′ _k (0)| of functions f _k univalent in the unit disk |z| < 1 having no common values and translating zero into a point on the segment [−1, 1], k = 1, …, n. We estimate f _k by means of Schwarzian derivatives.