Some Lp Inequalities for Polynomials Not Vanishing inside a Circle

If P(z) is a polynomial of degree n having no zeros in |z|<1, then it is known that, for all α,β∈𝒞 with |α|≤1, |β|≤1, R>r≥1, and p>0, P(Rz)-αP(rz)+β{((R+1)/(r+1))n-|α|}P(rz)p≤([Rn-αrn+β{((R+1)/(r+1))n-|α|}]z+[1-α+β{((R+1)/(r+1))n-α}]p/1+zp)P(z)p. In this paper, we will prove a result which not only generalizes the above inequality but also generalize and refines the various results pertaining to the Lp norm of P(z)∀p>0. We will also prove a result which extends and refines a result of Boas Jr. and Rahman (1962). Also we will see that our results lead to some striking conclusions giving refinements and generalizations of other well-known results.

Hamburger and Stieltjes Moment Problems for Operators

Solutions to some operator-valued, unidimensional, Hamburger and Stieltjes moment problems in this paper are given. Necessary and sufficient conditions on some sequences of bounded operators being Hamburger, respectively, Stieltjes operator-valued moment sequences are obtained. The determinateness of the operator-valued Hamburger and Stieltjes moment sequence is studied.

On the Sumudu Transform and Its Extension to a Class of Boehmians

Boehmians are used for all objects obtained by an algebraic construction similar to that of the field of quotients. In literature, several integral transforms have been extended to various Boehmian spaces but a few to the space of strong Boehmians. As shown in the work of Al-Omari (2013), this work describes certain spaces of Boehmians. The Sumudu transform is therefore established and it is one-one and continuous in the space of Boehmians. The inverse transform is given and some results are also discussed.

Approximation by T-Transformation of Double Walsh-Fourier Series to Multivariable Functions

We study the Walsh series expansion of multivariate functions in Lp (1≤p≤∞) and, in particular, in Lip(α,p). The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996), Nagy (2012). Our results can be applied to many summability methods, including the Nörlund summability and weighted summability.

Contiguous Function Relations for k-Hypergeometric Functions

In this research work, our aim is to determine the contiguous function relations for k-hypergeometric functions with one parameter corresponding to Gauss fifteen contiguous function relations for hypergeometric functions and also obtain contiguous function relations for two parameters. Throughout in this research paper, we find out the contiguous function relations for both the cases in terms of a new parameter k>0. Obviously if k→1, then the contiguous function relations for k-hypergeometric functions are Gauss contiguous function relations.

Meromorphic Parabolic Starlike Functions Associated with q-Hypergeometric Series

We introduce a new class of meromorphic parabolic starlike functions with a fixed point defined in the punctured unit disk Δ*≔z∈ℂ:0<z<1 involving the q-hypergeometric functions. We obtained coefficient inequalities, growth and distortion inequalities, and closure results for functions f∈ℳmlλ,β,γ. We further established some results concerning convolution and the partial sums.

An Alternate Proof of the De Branges Theorem on Canonical Systems

The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on ℂ. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.

Global Existence of Solution for Cauchy Problem of Two-Dimensional Boussinesq-Type Equation

In this paper, we consider the Cauchy problem of two-dimensional Boussinesq-type equations utt-Δu-Δutt+Δ2u=Δfu. Under the assumptions that fu is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence of global weak solution.

A Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Space

We prove the existence of a weak solution for the degenerate nonlinear elliptic Dirichlet boundary-value problem Lu-μug1+hu,∇ug2=f in Ω, u=0 on ∂Ω, in a suitable weighted Sobolev space, where Ω⊂ℝn is a bounded domain and h is a continuous bounded nonlinearity.

A Formula for the Numerical Range of Elementary Operators

Let BH be the algebra of bounded linear operators on a complex Hilbert space H. For k-tuples of elements of BH, A=A1,…,Ak and B=B1,…,Bk, let RA,B denote the elementary operator on BH defined by RA,BX=∑i=1kAiXBi. In this paper, we prove the following formula for the numerical range of RA,B: VRA,B,BBH=[∪U∈UHW(∑i=1kUAiU*Bi)-]-, where UH is the set of unitary operators.