Coefficients Of Such Functions Research Articles

Sobolev spaces revisited

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$ n, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$.

Coefficient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial

In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of n−th(n≥3) Taylor–Maclaurin coefficients an is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established.

Coefficient estimates for m-fold symmetric bi-subordinate functions

A function is said to be bi-univalent in the open unit disk U if both the function and its inverse map are univalent in U. By the same token, a function is said to be bisubordinate in U if both the function and its inverse map are subordinate to a given function in U. In this paper, we consider the m-fold symmtric transform of such functions and use their Faber polynomial expansions to find upper bounds for their n-th (n ≥ 3) coefficients subject to a given gap series condition. We also determine bounds for the first two coefficients of such functions with no restrictions imposed.

On harmonic univalent mappings with nonzero pole

In this article we consider the class AH(p) of all sense preserving harmonic functions f in the open unit disc D having a simple pole at z=p∈(0,1) with the normalizations f(0)=fz(0)−1=0. We first derive a sufficient condition for univalence of such functions. Next we consider the class SH0(p) which consists of all functions belonging to AH(p) and are univalent with the additional normalization fz¯(0)=0. We study the class SH0(p) in the geometric function theoretic viewpoint. As a byproduct of our investigation, we see that consideration of nonzero pole yields nontrivial lower bounds for the Taylor coefficients of such functions. We also prove a growth result for a non-vanishing subclass associated with SH0(p).

Faber Polynomial Coefficient Estimates for Subclass of Analytic Bi-Bazilevic Functions Defined by Differential Operator

In this work, an explicit formula for a class of Bi-Bazilevic univalent functions involving differential operator is given, as well as the determination of upper bounds for the general Taylor-Maclaurin coefficient of a functions belong to this class, are established Faber polynomials are used as a coordinated system to study the geometry of the manifold of coefficients for these functions. Also determining bounds for the first two coefficients of such functions.
 In certain cases, our initial estimates improve some of the coefficient bounds and link them to earlier thoughtful results that are published earlier.

Faber Polynomial Coefficient Estimates for Subclass of Analytic Bi-Bazilevic Functions Defined by Differential Operator

In this work, an explicit formula for a class of Bi-Bazilevic univalent functions involving differential operator is given, as well as the determination of upper bounds for the general Taylor-Maclaurin coefficient of a functions belong to this class, are established Faber polynomials are used as a coordinated system to study the geometry of the manifold of coefficients for these functions. Also determining bounds for the first two coefficients of such functions.
 In certain cases, our initial estimates improve some of the coefficient bounds and link them to earlier thoughtful results that are published earlier.

Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination

In the present paper, we introduce and investigate a new class of analytic and bi-univalent functions f (z) in the open unit disk U. For this purpose, we make use of a linear combination of the following three functions: f(z)/z, f'(z) and z f''(z) for a function belonging to the normalized univalent function class S. By applying the technique involving the Faber polynomials, we determine estimates for the general Taylor-Maclaurin coefficient of functions belonging to the analytic and bi-univalent function class which we have introduced here. We also demonstrate the not-too-obvious behaviour of the first two Taylor-Maclaurin coefficients of such functions.

Faber polynomial coefficients of bi-subordinate functions

A function is said to be bi-univalent in the open unit disk D if both the function and its inverse map are univalent in D. By the same token, a function is said to be bi-subordinate in D if both the function and its inverse map are subordinate to certain given function in D. The behavior of the coefficients of such functions are unpredictable and unknown. In this paper, we use the Faber polynomial expansions to find upper bounds for the n-th (n≥3) coefficients of classes of bi-subordinate functions subject to a gap series condition as well as determining bounds for the first two coefficients of such functions.

JACK–LAURENT SYMMETRIC FUNCTIONS FOR SPECIAL VALUES OF PARAMETERS

AbstractWe consider the Jack–Laurent symmetric functions for special values of parametersp0=n+k−1m, wherekis not rational andmandnare natural numbers. In general, the coefficients of such functions may have poles at these values ofp0. The action of the corresponding algebra of quantum Calogero–Moser integrals$\mathcal{D}$(k,p0) on the space of Laurent symmetric functions defines the decomposition into generalised eigenspaces. We construct a basis in each generalised eigenspace as certain linear combinations of the Jack–Laurent symmetric functions, which are regular atp0=n+k−1m, and describe the action of$\mathcal{D}$(k,p0) in these eigenspaces.

COEFFICIENT DISCS AND GENERALIZED CENTRAL FUNCTIONS FOR THE CLASS OF CONCAVE SCHLICHT FUNCTIONS

We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle <TEX>${\pi}{\alpha}$</TEX>, <TEX>${\alpha}{\in}(1,2]$</TEX>, at infinity. We derive the exact interval for the variability of the real Taylor coefficients of these functions and we prove that the corresponding complex Taylor coefficients of such functions are contained in certain discs lying in the right half plane. In addition, we also determine generalized central functions for the aforesaid class of functions.

Unpredictability of the coefficients of m-fold symmetric bi-starlike functions

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

Determination of order of magnitude of multiple Fourier coefficients of functions of bounded ϕ-variation having lacunary Fourier series using Jensen's inequality

For a Lebesgue integrable complex-valued function f defined over the m -dimensional torus Tm := [0,2π)m , let f (n) denote the Fourier coefficient of f , where n = (n(1), . . . ,n(m)) ∈ Zm . Recently, in one of our papers [to appear in Mathematical Inequalities & Applications], we have defined the notion of bounded φ -variation for a complex-valued function on a rectangle [a1,b1 ]× . . .× [am,bm ] and studied the order of magnitude of Fourier coefficients of such functions on [0,2π]m . In this paper, the order of magnitude of Fourier coefficients of a function of bounded φ -variation from [0,2π]m to C and having lacunary Fourier series with certain gaps is studied and a generalization of our earlier result (Theorem in [Acta Sci. Math. (Szeged), 78, (2012), 97–109]) is proved. Interestingly, the Jensen’s inequality for integrals is used to prove the main result. Mathematics subject classification (2010): 42B05, 26B30, 26D15.

An application of Jensen's inequality in determining the order of magnitude of multiple Fourier coefficients of functions of bounded φ-variation

For a Lebesgue integrable complex-valued function f defined over the n -dimensional torus Tn := [0,2π)n , n∈N , let f (k) denote the Fourier coefficient of f , where k = (k1, . . . ,kn) ∈ Zn . The Riemann-Lebesgue lemma shows that f (k) = o(1) as |k| → 0 for any f ∈ L1(Tn) . However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive results are due to V. Fulop and F. Moricz for functions of bounded variation, and due to B. L. Ghodadra for functions of bounded p -variation. In this paper, defining the notion of bounded φ -variation for a function from [0,2π]n to C in two different ways, we prove that this is the case for Fourier coefficients of such functions also. Interestingly, in proving our main results we use the famous Jensen’s inequality for integrals. Our new results with φ(x) = xp (p 1) gives our earlier results [Acta Math. Hungar, 128 (4) (2010), 328–343]. Mathematics subject classification (2010): 42B05, 26D15, 26B30.

Coefficient Estimates for Certain Classes of Bi-Univalent Functions

A function analytic in the open unit disk is said to be bi-univalent in if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions.

Order of magnitude of multiple Fourier coefficients of functions of bounded p-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus \( \mathbb{T}^n \):= [0, 2π)n, let \( \hat f \)(k) denote the Fourier coefficient of f, where k = (k1, … kn) ∈ ℤn. In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π]n to ℜ in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Moricz [5] and Fulop and Moricz [3].

On the Fourier tails of bounded functions over the discrete cube

In this paper we consider bounded real-valued functions over the discrete cube, f: {−1, 1}n → [−1, 1]. Such functions arise naturally in theoretical computer science, combinatorics, and the theory of social choice. It is often interesting to understand when these functions essentially depend on few coordinates. Our main result is a dichotomy that includes a lower bound on how fast the Fourier coefficients of such functions can decay: we show that $$\sum\limits_{|S| > k} {\hat f(S)^2 \geqslant exp( - O(k^2 logk))} ,$$ unless f depends essentially only on 2 O(k) coordinates. We also show, perhaps surprisingly, that this result is sharp up to the log k factor. p ]The same type of result has already been proven (and shown useful) for Boolean functions [Bou02, KS]. The proof of these results relies on the Booleanity of the functions, and does not generalize to all bounded functions. In this work we handle all bounded functions, at the price of a much faster tail decay. As already mentioned, this rate of decay is shown to be both roughly necessary and sufficient. p ]Our proof incorporates the use of the noise operator with a random noise rate and some extremal properties of the Chebyshev polynomials.

Stability analysis of a strictly efficient solution of a vector problem of Boolean programming in the metric l 1

We consider a vector (multicriteria) problem of Boolean programming where sub-criteria are projections of linear functions onto R+. We give a bound for variation of coefficients of such functions in the metric l1 which preserves strict efficiency of the solution.

Bounds for the taylor coefficients of locally univalent bloch functions

For a holomorphic function F defined on the unit disk D let . F is called a Bloch function if We describe a method to get inequalities for real functionals varying on the class of locally univalent Bloch functions F with M(F)≤1. As an application inequalities for the moduli of the Taylor coefficients of such functions are derived.

Algebraic Families of Analytic Functions, I

The aim of this paper is to consider analytic functions, arising in connection with algebraic differential equations, from the following point of view: Taylor coefficients of such functions usually turn out to be polynomials (or rational functions) in the parameters of the problem. Following pioneering work of Bautin [1], we relate analytic properties of the functions considered with algebraic properties of various ideals, generated by their Taylor coefficients. Bautin's approach has been successfully applied to the study of the Poincare mapping of the planar polynomial systems in [4, 9, 27], and to the study of the period function of polynomial Hamiltonian systems in [5, 6, 12, 13]. However, the study of the Poincare mapping of the polynomial differential equations (closely related to the second part of Hilbert 16th problem) is, probably, the most difficult among a wide variety of problems, to which Bautin's approach can be applied. In particular, counting zeroes of solutions of algebraic differential equations is an important and mostly open problem. To our best knowledge, the first application of Bautin's method to this problem was given in [26]. In the present paper we systematically apply this method in the following two situations: to algebraic functions and to solutions of the first order polynomial differential equations (higher order linear equations are also considered in Section 5). article no. DE963250

Coefficient Multipliers of Bloch Functions

The class $\mathcal {B}$ of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by $c/(1 - r)$ on $|z| \leqslant r$. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from ${l^p}$ into $\mathcal {B}$ and from $\mathcal {B}$ into ${l^p}$. We find the second Köthe dual of $\mathcal {B}$ and show its relevance to the multiplier problem. We identify all power series $\sum {a_n}{z^n}$ such that $\sum {w_n}{a_n}{z^n}$ is a Bloch function for every choice of the bounded sequence $\{ {w_n}\}$. Analogous problems for ${H^p}$ spaces are discussed briefly.