Classical families of univalent functions in the Hornich space

In this paper the simple structure between some convex sets in the Banach spaceH introduced by Hornich is used to determine the extreme points of the familiesK() of convex functions of order andV(k) of functions with bounded boundary rotationk. For close-to-convex functions of order ,]0,1[, a partial result is given. The results forK() andV(k) agree with those that hold for the closed convex hulls of the same families with respect to the usual linear structure and the topology of locally uniform convergence. However, in this case, fork]2,4[ the question of determining the extreme points of V(k) is still open.     

多项式函数的极值点与拐点判别及个数公式

本文讨论了多项式函数(x-a)~n,(x-a)~ng(x)(n∈N,n1,a∈R)和∏ki=1(x-a_i)~(n_i)(n_i∈N,n_i0,a_i∈R)的极值点和拐点,并给出了函数∏ki=1(x-a~i)~(n_i)(n_i∈N,n_i0,a_i∈R)所有极值点和拐点的个数公式.     

Classes of analytic functions subordinate to convex functions and extreme points

In this paper we prove two sufficient conditions for an analytic function f to be an extreme point of the set of functions subordinate to a given convex mapping F when the image of the unit disk under F is a convex domain other than a half-plane, a strip or an infinite wedge.     

Barycenters of extreme points in the cone of non-negative entire functions

In the cone of non-negative (on the real axis) entire functions, a certain simple functionf(x) is shown not to be the barycenter of any finite number of extreme functions. This is in contradistinction to S. Karlin's result that every non-negativepolynomial is the barycenter oftwo extreme ones.     

Convex hull and extreme points of two families of univalent functions

Closed convex hull and extreme points are obtained for the classR (α) of univalent functions ? inU = {z:¦z ¦< 1}. satisfying inU the conditions ?(0) = 0, ?′(0) = 1Re {f(z)/z} >α, 0 ≤ α < 1. We also obtain the closed convex hull for the classR (α, β) of univalent functionsg inU, satisfying inU the conditionsg (0) = 0,g (0) = 1 and Re{g(z)/F(z)} > β, 0 ? β< 1, wheref ∈ R (α). Integral representations are given for the hulls of these two classes in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example the upper bounds are determined for the coefficients of a function subordinate to some function in R(α) when 1/2 ≤ α < 1.

     

On the dirichlet problem of extreme points for non-continuous functions

Let X be a compact convex set and f be a bounded function defined on the set ext X of extreme points of X. We present a necessary and sufficient condition ensuring that f can be extended to a strongly affine Baire-α function. This generalizes a result of E. M. Alfsen from [2]. We also consider extensions of vector-valued mappings, thus generalizing another result of E. M. Alfsen.     

Banach spaces of locally Schlicht functions with the Hornich operations

Let \(S_ \propto ( \propto \geqq 0)\) be the set of normalized (see (1.2)) functions f holomorphic in D:|z|<1 with \(f''(z)/f'(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and let be the set of normalized (see (1.6)) functions f meromorphic in D with the Schwarzian derivative \(\left\{ {f,z} \right\} = 0((1 - \left| z \right|^2 )^{ - \propto } )\) . We shall show that some topological properties of \(S_ \propto\) and , and of subsets of them, follow from those of the weighted H^∞ space \(H_ \propto ^\infty\) , consisting of functions f holomorphic in D with \(f(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\) , and those of subsets of \(H_ \propto ^\infty\) . The set S₁ is denoted by X in [3] and [4].     

Fixed points for operators in a space of continuous functions and applications

This paper investigates the fixed points for self-maps of a closed set in a space of abstract continuous functions. Our main results essentially extend the Banach contracting mapping principle. An application to integro-differential equations is given.

     

Higher order monotonic (multi-) sequences and their extreme points

Functions on the half-line which are non-negative and decreasing of a higher order have a long tradition. When normalized they form a simplex whose extreme points are well-known. For functions on ${\mathbb{N}_{0} = \{0, 1, 2, . . .\}}$ the situation is different. Since an n-monotone sequence is in general not the restriction of an n-monotone function on ${\mathbb{R}_{+}}$ (apart from n = 1 and n = 2), it is not even clear at the beginning if the normalized n-monotone sequences form a simplex. We will show in this paper that this is actually true, and we determine their extreme points. A corresponding result will also be proved for multi-sequences. The main ingredient in the proof will be a relatively new characterization of so-called survival functions of probability measures on (subsets of) ${\mathbb{R}^n}$ , in this case on ${\mathbb{N}^{n}_{0}}$ .