A function f = u + iv defined in the domain D ⊂ C is harmonic in D if u, v are real harmonic. Such functions can be represented as f = h+ ḡ where h, g are analytic in D. In this paper the class of harmonic functions constructed by the Hadamard product in the unit disk, and properties of some of its subclasses are examined.

Correspondence: [email protected] Department of Mathematics, Dezhou University, Dezhou, 253023 Shandong, China Abstract Using a estimate on the Perron root of the nonnegative matrix in terms of paths in the associated directed graph, two new upper bounds for the Hadamard product of matrices are proposed. These bounds improve some existing results and this is shown by numerical examples. MS...

In this paper, we define a generalized class of starlike functions with negative coefficients and obtain coefficient estimates, distortion bounds, closure theorems and extreme points. Further we obtain modified Hadamard product, radii of close-to-convex, starlikeness and convexity for functions belonging to this class. Furthermore neighborhood results are discussed.

The present paper systematically investigates a new class of functions involving certain fractional derivative operators. Characterization and distortion theorems, and other interesting properties of this class of functions are studied. Further, the modified Hadamard product of several functions belonging to this class are also investigated.

By means of the Hadamard product, the present paper introduces new classes, Σ t, * a (α, β, ρ) and Σ t a (α, β, ρ) of Hurwitz-Lerch-Zeta function in the punctured disk U * = {z : 0 < |z| < 1}. In addition, the study investigates a number of inclusion relationships, properties and derives some interesting properties depending on some integral properties.

Using the ζ functional equation and the Hadamard product, an analytical expression for the sum of the reciprocal of the ζ zeros is established. We then demonstrate that on the critical line, |ζ| is convex, and that in the region 0 < <(s) ≤ 0.5, |ζ| has a negative slope. In each case, analytical formulae are established, and numerical examples are presented to validate these formulae.

Recently, K.M.R. Audenaert (2010), and R.A. Horn and F. Zhang (2010) proved inequalities between the spectral radius of Hadamard products of finite nonnegative matrices and the spectral radius of their ordinary matrix product. We will prove these inequalities in such a way that they extend to infinite nonnegative matrices A and B that define bounded operators on the classical sequence spaces lp.

Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce a class Qp(a, c;h) of analytic and multivalent functions in the open unit disk. An inclusion relation and a convolution property for the class Qp(a, c;h) are presented. Some integral-preserving properties are also given.