Elementary approximations for zeros of Bessel functions

Let j_vk, y_vk and c_vk denote the kth positive zeros of the Bessel functions J_v(x), Y_v(x) and of the general cylinder function C_v(x) = cos αJ_v(x)?sin αY_v(x), 0 ? α < π, respectively. In this paper we extend to c_vk, k = 2, 3,..., some linear inequalities presently known only for j_vk. In the case of the zeros y_vk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero j_v1 we compare the linear enequalities given in [9] with some other known inequalities.     

Sharpened forms of the generalized Schwarz inequality on the boundary

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function f(z) = 1 + c_pz^p + c_{p + 1}z^{p + 1}…, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account c_{p + 1} and zeros of the function f(z) ? 1. The sharpness of these inequalities is also proved.     

A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices

We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip-Simon (Ann. Math. 158 (2003) 253) of their result characterizing the spectral measures of Jacobi matrices, J, with J−J₀ Hilbert-Schmidt. We prove a nonlocal version of Case and step-by-step sum rules.     

A new class of generalized close-to-starlike functions defined by the Srivastava-Attiya operator

In this paper, we consider a new class CS*s,b of generalized close-to-starlike functions, which is defined by means of the Srivastava-Attiya operator Js,b involving the Hurwitz-Lerch Zeta function Φ(z, s, a). Basic results such as inclusion relations, coefficient inequalities and other interesting properties of this class are investigated. Relevant connections of some of the results presented here with those that were obtained in earlier investigations are pointed out briefly.     

Darboux transformations of Jacobi matrices and Padé approximation

Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_C=UL is a monic generalized Jacobi matrix associated with the function F_C(λ)=λF(λ)+1. It turns out that the Christoffel transformation J_C of a bounded monic Jacobi matrix J can be unbounded. This phenomenon is shown to be related to the effect of accumulating at ∞ of the poles of the Padé approximants of the function F_C although F_C is holomorphic at ∞. The case of the UL-factorization of J is considered as well.     

Über Bruwirsche Reihen mit nicht negativen Koeffizienten

LetK be the space of entire functions $$f(z) = \sum\limits_{n = 0}^\infty {a_n (1 + z/n)^n + a_\infty \exp (z),z \in \mathbb{C}} ,$$ withf(0)=1 and nonnegative coefficientsa _n. Then it is shown thatK is a Choquetsimplex and can be characterized in three ways: by its set of extrem points, by differential inequalities or as closed convex hull of a set of normed polynomials having only negative zeros and with harmonic mean of these zeros less or equal ?n, n being the degree of the polynomial. Additional properties ofK are derived by using the above characterizations.     

Minkowski's successive minima and the zeros of a convexity-function

For a convex bodyK?E ^d letV _i (K),i=0, 1,...,d denote its normalized quermassintegrals. We show that there are various inequalities between the zeros of the polynomial W(?μ K) = ∑ _i=0 ^d V_i(K)(?μ)ⁱ and Minkowski's successive minima in geometry of numbers.     

Inequalities and properties of some generalized Orlicz classes and spaces

We discuss and complement the knowledge about generalized Orlicz classes $ \tilde X_\Phi $ and Orlicz spaces X _Φ obtained by replacing the space L ¹ in the classical construction by an arbitrary Banach function space X. Our main aim is to focus on the task to study inequalities in such spaces. We prove a number of new inequalities and also natural generalizations of some classical ones (e.g., Minkowski’s, Hölder’s and Young’s inequalities). Moreover, a number of other basic facts for further study of inequalities and function spaces are included.     

On determinants of Toeplitz-Hessenberg matrices arising in power series

The determinant, J_n, of [a_{i ? j + 1}]_{n, n} with a_{i ? j + 1} = 0 for j ? i > 1 is obtained explicitly, in terms of the zeros of an associated polynomial, as the solution of a difference equation. These determinants, which appear often under various guises, provide the coefficients of the reciprocal series of unit formal power series. The paper concludes with some examples including an application yielding an asymptotic formula for the Bernoulli numbers.     

Generalized iterative process and associated regularization for J-Pseudomonotone mixed variational inequalities

A generalized iterative process for solving mixed variational inequalities with J-Pseudomonotone operators in uniformly smooth Banach spaces is proposed and its weak convergence is established. An application to the stationary filtration problem is given. For such variational inequalities, a generalized iterative regularization method is constructed and its weak convergence under the assumption that the iterative parameter may vary from step to step is analyzed. Our results extend and generalize the corresponding theorems of [A.M. Saddeek, S.A. Ahmed, Convergence analysis of iterative methods for some variational inequalities with J-Pseudomonotone operators in uniformly smooth Banach spaces, Appl. Sci. Comput., accepted for publication, A.M. Saddeek, S.A. Ahmed, On the convergence of some iteration processes for J-Pseudomonotone mixed variational inequalities in uniformly smooth Banach spaces, Math. Comput. Modell., 46(3-4) (2007) 557-572].     

On a conjecture regarding the extrema of Bessel functions and its generalization

It was conjectured by Á. Elbert in J. Comput. Appl. Math. 133 (2001) 65-83 that, given two consecutive real zeros of a Bessel function of order ν, j_ν,κ and j_ν,κ+1, the zero of the derivative between such two zeros j_ν,κ′ satisfies . We prove that this inequality holds for any Bessel function of any real order. In addition to these lower bounds, upper bounds are obtained. In this way we bracket the zeros of the derivative. It is discussed how similar relations can be obtained for other special functions which are solutions of a second order ODE; in particular, the case of the zeros of is considered.     

On some criteria for completely regular growth of entire functions of exponential type

This paper sharpens the author’s previous results concerning the completely regular growth of an entire function of exponential type all of whose zeros are simple, forming a sequence Λ = {λ_k} _k=1 ^∞ . For a function with real zeros, we write the growth regularity conditions (on the real axis and on the entire plane) in terms of lower bounds only for the absolute value of the derivative at the points λ_k. We also obtain an analog of Krein’s theorem concerning the functions whose inverse can be expanded in the corresponding series of simple fractions.     

The parabolic map f1/e(z)=(1/e)e

We consider the exponential maps ?_λ : ? → ? defined by the formula ?_λ (z) = λe^z, λ(0,1/e]. Let J_r(?_λ) be the subset of the Julia set consisting of points that do not escape to infinity under forward iterates of ?. Our main result is that the function λh_λ :=HD(J_r(?_λ),)), λ(0, 1/e], is continuons at the point 1/e. As a preparation for this result we deal with the map ?₁/e itself. We prove that the h₁/e-dimensional Hausdorff measure of J_r(?₁/e) is positive and finite on each horizontal strip, and that the h₁/e-dimensional packing measure of J_r(?_λ) is locally infinite at each point of J_r(?_λ). Our main technical devices are formed by the, associated with ?_λ, maps F_λ defined on some strip P of height 2π and also associated with them tonformal measures.     

Extremes of conditioned elliptical random vectors

Let {X_n,n?1} be iid elliptical random vectors in R^d,d≥2 and let I,J be two non-empty disjoint index sets. Denote by X_n,I,X_n,J the subvectors of X_n with indices in I,J, respectively. For any a∈R^d such that a_J is in the support of X_1,J the conditional random sample X_n,I|X_n,J=a_J,n≥1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X₁ has distribution function in the max-domain of attraction of a univariate extreme value distribution.     

（0,1;0）-INTERPOLATION ON INFINITE INTERVAL （-∞, ＋∞）

In this paper, we study the explicit representation and convergence of （0, 1;0）-interpolation on infisite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values areprescribed at two set of points namely the zeros of Hn（x） and H′n （x） and the first derivatives at the zerosof H′n（x）.     

Second Order Linear Differential Polynomials and Real Meromorphic Functions

The main result determines all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles, while f′′ + a ₁ f′ + a ₀ f has finitely many non-real zeros, where a ₁ and a ₀ are real rational functions which satisfy a ₁(∞) = 0 and a ₀(x) ≥ 0 for all real x with |x| sufficiently large. This is accomplished by refining some earlier results on the zeros in a neighbourhood of infinity of meromorphic functions and second order linear differential polynomials. Examples are provided illustrating the results.     

A sufficient condition for a polynomial to be stable

A real polynomial is called Hurwitz (stable) if all its zeros have negative real parts. For a given n∈N we find the smallest possible constant dn>0 such that if the coefficients of F(z)=a₀+a₁z+?+anzn are positive and satisfy the inequalities akak+1>dnak−1ak+2 for k=1,2,…,n−2, then F(z) is Hurwitz.     

Uniform asymptotics and zeros of a system of orthogonal polynomials defined via a difference equation

We study a class of sieved Pollaczek polynomials defined by a second-order difference equation (three-term recurrence relation). The measure of orthogonality is determined by using the Markov theorem and the Perron-Stieltjes inversion formula, and is shown consisting of an absolutely continuous part and a discrete part with infinitely many mass points. Uniform asymptotic approximations of these polynomials for large degree n are derived at a turning point αn and a critical point βn, involving respectively the Airy function Ai, and . Darboux's method, the method of steepest descents, and various uniform asymptotic techniques such as cubic transformations are used to derive the results. Asymptotic formulas for the least zeros, the largest zeros, and the zeros on both sides of βn are also obtained. Several numerical examples are provided to compare the approximate zeros with the true values.