In this paper, we find Fekete-Szegö bounds for a generalized class M q (γ, φ). Also, we discuss some remarkable results.

A classical theorem of Fekete and Szegő [4] says that if E is a compact set in the complex plane, stable under complex conjugation and having logarithmic capacity γ(E) ≥ 1, then every neighborhood of E contains infinitely many conjugate sets of algebraic integers. Raphael Robinson [5] refined this, showing that if E is contained in the real line, then every neighborhood of E contains infinitely...

An optimal algorithm is presented for computing the smallest set of linear inequalities that define the integer hull of a possibly unbounded two-dimensional convex polygon R. Input to the algorithm is a set of linear inequalities defining R, and the integer hull computed is the convex hull of the integer points of R. It is proven that the integer hull has at most O(n logAmax) inequalities, wher...

Let be the class of analytic functions in the unit disk that are normalized with f (0)= f ′(0)− 1 = 0 and let −1 ≤ B < A ≤ 1. In this paper we study the class Gλ,α = { f ∈ : |(1− α + αz f (z)/ f ′(z))/z f ′(z)/ f (z)− (1− α)| < λ, z ∈ }, 0 ≤ α ≤ 1, and give sharp sufficient conditions that embed it into the classes S∗[A,B] = { f ∈ : z f ′(z)/ f (z) ≺ (1 +Az)/(1 + Bz)} and K(δ) = { f ∈ : 1 + z f...

Let A and B be n n positive semidefinite Hermitian matrices, let c and/ be real numbers, let o denote the Hadamard product of matrices, and let Ak denote any k )< k principal submatrix of A. The following trace and eigenvalue inequalities are shown: tr(AoB) <_tr(AoBa), c_<0or_> 1, tr(AoB)a_>tr(AaoBa), 0_a_ 1, A1/a(A o Ba) <_ Al/(Az o B), a <_ /,a O, Al/a[(Aa)k] <_ A1/[(A)k], a <_/,a/ 0. The equ...

which are analytic in the open unit disk U {z : z ∈ C and |z| < 1} and S denote the subclass ofA that are univalent in U. A function f z inA is said to be in class S∗ of starlike functions of order zero in U, if R zf ′ z /f z > 0 for z ∈ U. Let K denote the class of all functions f ∈ A that are convex. Further, f is convex if and only if zf ′ z is star-like. A function f ∈ A is said to be close...

Suppose that K ⊂ IR is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c1, c2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, Fn ⊂ K, c1 n ≤ min b∈Fn b 6=a dist(a, b) ≤ c2 n , ∀a ∈ Fn where dist(a, b) is a natural distance on K that will be described in the text. §

In this paper we present a procedure for the numerical estimation of the Fekete points of a wide variety of compact sets in IR. We understand the problem of the Fekete points in terms of the identification of nearly equilibrium configurations for a potential energy that depends on the relative position of N particles. The compact sets for which our procedure has been designed can be described b...

In this note we present inequalities relating the successive minima of a o-symmetric convex body and the successive inner and outer radii of the body. These inequalities build a bridge between known inequalities involving only either the successive minima or the successive radii.