EXTREME POINTS OF INTEGRAL FAMILIES OF ANALYTIC FUNCTIONS

Extreme Points of Vector Functions

Several previous investigations have appeared in the literature which discuss the nature of the set T in »-dimensional space spanned by the vectors ifsdui, ■ ■ ■ , fsdp.n) where 5 ranges over all measurable sets. It was first shown by Liapounov that xi ui, • • ■ , un are atomless measures, then the set F is convex and closed. Extensions of this result were achieved by D. Blackwell [l] and Dvore...

Fixed Points of Analytic Functions

Integral Means of Certain Analytic Functions

We introduce the analytic functions f (z) = z + ∞ ∑ k=n+1 akz k ( n ∈ N) and p(z) = z + m ∑ s=1 bsj−s+1zsj−s+1 (j ≥ n + 1;n ∈ N) in the open unit disk U. By means of the subordination theorem of J.E. Littlewood, we shall investigate the integral means with coefficients inequalities of analytic functions f(z) and p(z). Some applications of the integral mean of f(z) are considered. 2000 Mathemati...

Integral Specialization of Families of Rational Functions

Suppose C is an algebraic curve, f is a rational function on C defined over Q, and A is a fractional ideal of Q. If f is not equivalent to a polynomial, then Siegel’s theorem gives a necessary condition for the set C(Q)∩f−1(A) to be infinite: C is of genus 0 and the fiber f−1(∞) consists of two conjugate quadratic real points. We consider a converse. Let P be a parameter space for a smooth fami...

Extreme and Exposed Points of Spaces of Integral Polynomials

We show that if E is a real Banach space such that E′ has the approximation property and such that `1 6↪→ ⊗̂ n,s, E then the set of extreme points of the unit ball of PI(E) is equal to {±φn : φ ∈ E′, ‖φ‖ = 1}. Under the additional assumption that E′ has a countable norming set we see that the set of exposed points of the unit ball of PI(E) is also equal to {±φn : φ ∈ E′, ‖φ‖ = 1}.