Hadamard product

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

We studied convolution properties of spirallike, starlike and convex unctions, and some special cases of the main results are also pointed out. Further, we also obtained inclusion and convolution properties for some new subclasses on p-valent functions defined by using the Dziok–Srivastava operator.

We studied convolution properties of spirallike, starlike and convex unctions, and some special cases of the main results are also pointed out. Further, we also obtained inclusion and convolution properties for some new subclasses on p-valent functions defined by using the Dziok–Srivastava operator.

Two subclasses P m n and P 0 m n of certain analytic functions having pos- itive real part in the open unit disk U are introduced. In the present paper, several properties of the subclass P m n of analytic functions with real part greater than m n are derived. For p(z) 2 P m n and 0, the neighborhood N (p(z)) of p(z) is defined. For P m n , P0 m n , and N (p(z)), we prove that if p(z) 2 P 0 m n , then N (p(z)) P m n .

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

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices 1 N (Dn ◦Xn)(Dn ◦Xn)∗, studied in Girko 2001. Here, Xn = (xij) is an n×N random matrix consisting of independent complex standardized random variables, Dn = (dij), n × N , has nonnegative entries, and ◦ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of Xn and Dn which are different from those in Girko (2001), which include a Lindeberg condition on the entries of Dn ◦Xn, as well as a bound on the average of the rows and columns of Dn ◦ Dn. The present paper separates the assumptions needed on Xn and Dn. It assumes a Lindeberg condition on the entries of Xn, along with a tigntness-like condition on the entries of Dn,

ABSTRACT. For functions belonging to each of the subclasses M ∗ (α) and N ∗ (α) of normalized analytic functions in the open unit disk U, which are investigated in this paper when α> 1, the authors derive several subordination results involving the Hadamard product (or convolution) of the associated functions. A number of interesting consequences of some of these subordination results are also discussed.

Abstract. In this paper, we introduce a new class of functions which are analytic and univalent with negative coefficients defined by using certain fractional operators descibed in the Caputo sense. Characterization property, the results on modified Hadamard product and integral transforms are discussed. Further, distortion theorem and radii of starlikeness and convexity are also determined here.

We introduce the subclass Tj(n,m,, ) of analytic functions with negative coecients defined by Salagean operators Dn and Dn+m. In this paper we give some properties of functions in the class Tj(n,m,, ) and obtain numerous sharp results including (for example) coecient es- timates, distortion theorems, closure theorems and modified Hadamard products of several functions belonging to the class Tj(n,m,,