Abstract

In this paper we obtain sharp coefficient bounds for functions analytic in the unit disc $U$ and belonging to the class $R(b,M)$, $b\neq 0$ is a complex number. Also, we maximize $| a_{3}-\mu a_{2}^{2}| $ over the class $R(b,M)$ and obtain distortion theorem for functions in this class.

Keywords: Analytic functions; complex order; starlike functions; bounded functions.

MSC: 30C45

Abstract

In this paper, a new algorithm for the numerical solution of semidifferential equations with constant coefficients and fractional derivative defined in the Caputo sense is presented. The algorithm is obtained by using the spline collocation method. Moreover, a new technique for calculating the fractional derivative of the spline polynomial is derived. Numerical examples are also presented to test and illustrate the method.

Keywords: Fractional derivative; semidifferential equation; numerical solution; spline space.

MSC: 26A33, 65D07

Abstract

In this paper, we introduce a new general integral operator defined by Hadamard product. Some properties involving this operator on a class of functions of complex order are determined . Furthermore, we obtained new sufficient conditions for this operator to be univalent in the open unit disc. Finally, we prove several subordination results involving starlike and convex functions of complex order. Several corollaries and consequences of the main results are also considered.

Keywords: Analytic and univalent functions; starlike and convex functions of complex order; Hadamard product; subordination; integral operator.

MSC: 30C45

Abstract

The purpose of this paper is to give necessary and sufficient condition of modified three-step iteration scheme with errors to converge to common fixed points for a pair of quasi-nonexpansive and asymptotically quasi-nonexpansive mappings in Banach spaces. The results presented in this paper generalize, improve and unify the corresponding results in several papers.

Keywords: Quasi-nonexpansive mappings, asymptotically quasi nonexpansive mapping; common fixed points; modified three-step iteration scheme with errors with respect to a pair of mappings; strong convergence; Banach space.

MSC: 47H09, 47H10

Abstract

In this paper we introduce and study the new properties $(ab)$, $(gab)$, $(aw)$ and $(gaw)$ as a continuation of our previous article [4], where we introduced the two properties $(b)$ and $(gb)$. Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space $X$, then $T$ possesses property $(gb)$ if and only if $T$ possesses property $(gab)$ and $\tx{\rm ind}(T-\lambda I)=0$ for all $\lambda\in\sigma_a(T)\setminus\sigma_{SBF_+^-}(T)$; where $\sigma_{SBF_+^-}(T)$ is the essential semi-B-Fredholm spectrum of $T$ and $\sigma_a(T)$ is the approximate spectrum of $T$. We prove also that $T$ possesses property $(gaw)$ if and only if $T$ possesses property $(gab)$ and $E_a(T)=\Pi_a(T)$.

Keywords: Property $(ab)$; property $(gab)$; property $(aw)$; property $(gaw)$.

MSC: 47A53, 47A10, 47A11

Abstract

We consider a local algebra $A$ (in the sense of André Weil), a smooth paracompact manifold $M$ and the manifold $M^{A}$ of infinitely near points on $M$ of kind $A$. In this paper, we define and study the notions of $A$-Jacobi structures on $M^{A}$.

Keywords: Infinitely near point; local algebra, Lie-Rinehart algebra, Jacobi algebra.

MSC: 58A20, 58A32, 11F50

Abstract

In the paper, we prove two uniqueness theorems concerning nonlinear differential polynomials, one of which generalizes a recent result of A. Banerjee, and the other supplements a recent result of I. Lahiri and P. Sahoo.

Keywords: Meromorphic function; uniqueness; differential polynomials.

MSC: 30D35

Abstract

In this paper, we obtain that the space $\Cal{W}$ of orthonormal wavelets enjoys the complete invariance property with respect to homeomorphisms. Further, it is obtained that the cylinder, the cone and the suspension of $\Cal{W}$ possess the complete invariance property. Certain subspaces of $\Cal{W}$ are also considered in this connection.

Keywords: Wavelets; MSF Wavelets; MRA Wavelets; CIP; CIPH; dimension function.

MSC: 42C15, 42C40, 54H25