Let f be analytic in D = {z: |z| < 1} with f(0) = f′(0)−1=0. For γ > 0, the largest α (γ) and β(γ) are found such that . The results solve the inclusion problem for convex and starlike functions defined in a sector.

Let A denote the set of all functions analytic in U = {z: |;z| < 1} equipped with the topology of unifrom convergence on compact subsets of U. For F ∈ A define Let s(F) and s(F) denote the closed convex hull of s(F) and the set of extreme points of , respectively. Let R denote the class of all F ∈ A such that = {Fx}: |x| = 1} where Fx = F(xz).

We prove that |An| ≤ |AMN| for all positive integers M and N, and for . We also prove that if , then F is a univelaent halfplane mapping.

Logarithmic coefficient bounds for some univalent functions are given in this paper.

For a sequence of polynomials (Pn) orthonormal on the interval [−1, 1], we consider the sequence of transforms (gn) of the series given by . We establish necessary and sufficient conditions on the matrix (bnk) for the sequence (gn) to converge uniformly on compact subsets of the interior of an appropriate ellipse to a function holomorphic on that interior.

Pommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

Let f(x) be a monic polynomial of degree n with complex coefficients, which factors as f(x) = g(x)h(x), where g and h are monic. Let be the maximum of on the unit circle. We prove that , where β = M(P0) = 1 38135 …, where P0 is the polynomial P0(x, y) = 1 + x + y and δ = M(P1) = 1 79162…, where P1(x, y) = 1 + x + y - xy, and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.

Let S(p) be the family of holomorphic functions f defined on the unit disk D, normalized by f(0) = f1(0) – 1 = 0 and univalent in every hyperbolic disk of radius p. Let C(p) be the subfamily consisting of those functions which are convex univalent in every hyperbolic disk of radius p. For p = ∞ these become the classical families S and C of normalized univalent and convex functions, respectively. These families are linearly invariant in the sense of Pommerenke; a natural problem is to calculate the order of these linearly invariant families. More precisely, we give a geometrie proof that C(p) is the universal linearly invariant family of all normalized locally schlicht functions of order at most coth(2p). This gives a purely geometric interpretation for the order of a linearly invariant family. In a related matter, we characterize those locally schlicht functions which map each hyperbolically k-convex subset of D onto a euclidean convex set. Finally, we give upper and lower bounds on the order of the linearly invariant family S(p) and prove that this class is not equal to the universal linearly invariant family of any order.

Let be the class of normalized univalent functions in the unit disk. For f ∈ let Sf be the set of all star center points of f. Let 0 = where is the interior of Sf. The influence that the size of the set has on the Taylor coefficients of a function f ∈ 0 is examined, and estimates of these coefficients depending only on , as well as other results, are obtained.

We investigate a family consisting of functions whose convolution with is starlike of order α 0 ≤ α < 1. We determine extreme points, inclusion relations, and show how this family acts under various linear operators.

In this note the L2-angle between two concentric rings and between the ring and the exterior of the disc in the complex plane are calculated. In the second part we prove that the L2-angles between domains A and B and between A × C and B × C are equal. We give also some examples of discontinuity of the L2-angle between domains.

We introduce the class of Harnack domains in which a Harnack type inequality holds for positive harmonic functions with bounds given in terms of the distance to the domain's boundary. We give conditions connecting Harnack domains with several different complete metrics. We characterize the simply connected plane domains which are Harnack and discuss associated topics. We extend classical results to Harnack domains and give applications concerning the rate of growth of various functions defined in Harnack domains. We present a perhaps new characterization for quasidisks.

For α≥0 and β≥0 we denote by K (α, β) the Kaplan classes of functions f analytic and non-zero in the open unit disk U = {z: |z| < 1} such that f ∈ K(α, β), if, and only if, for θ1 < θ2 < θ1 + 2π and 0 < r < 1,

Where

In this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

Convergent iterative sequences are constructed for the polynomials fm = z + zm, m ≧ 2, with initial point the lemniscate {z: |fm (z)| ≦1}. In the particular case m = 2 convergent iterative sequences are constructed also for f-1m, (z) with an arbitrary initial point. The method is based on a certain variational principle which allows reducing the problem to the well known situation of an analytic function mapping a simply connected domain into a proper subset of itself and possessing a fixed point in the domain.

Let S(m, M) be the set of functions regular and satisfying │zf′(z)/f(z) – m│< M in │z│ <1, where│m –│ <M;≦ m; and let S*(p) be the set of starlike functions of order p, 0≦ p <1. In this paper we obtain integral operators which map S(m, M) into S(mM) and S* (p) × S(mM) into S*(p). Our results improve and generalize many recent results.

In this paper the authors obtain a series transformation of concentric circles into concentric rectangles.

A function

analytic and univalent in U = {z: |z| < 1} is said to be starlike there, if f(U) is f starshaped with respect to the origin, that is, if w ε f(U) implies tw ε f(U) for 0 ≤t ≤ 1. We denote by S* the class of all such functions. The Koebe function; k(z) = z(l – z)-2, z ε U, maps U onto the complex plane minus a slit along the I negative real axis from - ¼ to ∞, and thus belongs to the class S*. Recently Leung [4] has shown that, if

then, for f ε S*,

for every p > 0.

The classes of prestarlike functions Rα, α ≧ – 1, were studied recently by St. Ruscheweyh. The author generalizes and extends these classes. In particular the author obtains the radius of Ra+1 for the class Rα, α ≧ –1.