On The Constant ?

ON SOME CLASSES OF BI-UNIVALENT FUNCTIONS

Consider a family of integral operators and a related family of differential operators, both defined on a class of analytic functions holomorphic in the unit disk, distortion properties of the real part are derived from a general aspect. 1. Integral and differential operators. Let F denote the class of analytic functions which are holomorphic in the unit disk E = {| z |< 1}. Let F nad G be its subclasses consisting of f ∈ F normalized by f(0) = f ′(0) − 1 = 0 and f(0) = 1, respectively. In previous papers [3, 6] we have observed an integral operator L(a) defined on F , which is represented by L(a)f(z) = a ∫

Radius of convexity of partial sums of functions in the close-to-convex family

Let $\Cal P$ denote the well-known class of functions of the form $p(z)=1+q_1z+\ldots$ holomorphic in the unit disc $\bold D$ and fulfilling the conditions $Rep(z)>0$ in $\bold D$. Let $0\leq b<1, b<B, 0<\alpha <1$, be fixed real numbers and $zbold F$ a given measurable subset of the unit circle $\bold T$ of Lebesgue measure $2\pi\alpha$. For each $r \in (-\pi,\pi)$, denote by $\bold F_r=\{\xi\in \bold T; e^{-iT}\xi \in \bold F\}$ the set arising by rotation of $\bold F$ through the angle $\tau$. Denote by $\Cal P(B,b,\alpha;\bold F)$ the class of functions $p\in \Cal P$ satisfying the following conditions: there exists $\tau \in (-\pi,\pi)$ such that Rep(^{i\theta})\geq b$ a.e. on $\bold T\ \bold F_r$. In the paper the properties of the class $\Cal P(B,b,\alpha;\bold F)$ for different values of the parameters $B, b, \alpha$ and measurable sets $\bold F$ are examined. This article belongs to the series of papers ([4], [5], [6]) where different classes of functions defined by conditions on the circle $\bold T$ were studied. The results of papers [5], [6] are generalized.

State space methods for solving tangential interpolation problems for matrix functions in a given class have been developed extensively. These are interpolation problems in which the values in certain directions of the interpolant matrix functions, and several of their derivatives, are prescribed at given points in the complex plane. For the most part, the matrix functions under consideration are rational functions or polynomials (perhaps with symmetries); however, other important classes of matrix functions have been studied as well, notably, the Nevanlinna classes on the unit disk and the upper half-plane, and the Hardy class H/sub 2/ on the unit disk. This development is motivated by many important applications in systems and control. We present some recent results and open problems in this area.

In this paper we study the distribution of values of harmonic functions in non-Euclidean disks. We introduce the notion of a P′-sequence, which enables us to characterize the class of normal harmonic functions defined in the unit disk. Moreover, we obtain sufficient conditions for the existence of such sequences and give examples which show that these conditions are essential in the stated theorems.

In order to investigate the effective circular image analysis method, a class of Haar functions defined on the unit disk, named as disk Haar Functions (DHFs), is introduced in this paper. Compared with traditional bivariate tensor product orthogonal functions (TPOFs), DHFs have great advantages in handling the circular images. Experimental results demonstrate that DHFs is more suitable for representing images on circular region than TPOFs. 

Interpolation of analytic functions of moderate growth in the unit disc and zeros of solutions of a linear differential equation

There are many classes of analytic functions in the unit disc $U$. We consider about the special classes $S^*_\lambda(A,B,\alpha,\beta)$ and $C^*_\lambda(A,B,\alpha,\beta)$( $-1\le A &lt; B \le 1$, $0 &lt; B \le 1$, $0 \le \alpha &lt; 1$ and $0 &lt; \beta \le 1$) of analytic functions in the unit disc $U$. And the purpose of this paper is to study the classes $S^*_\lambda(A,B,\alpha,\beta)$ and $C^*_\lambda(A,B,\alpha,\beta)$. We prove some distortion theorems and some coefficient estimates for these classes $S^*_\lambda(A,B,\alpha,\beta)$ and $C^*_\lambda(A,B,\alpha,\beta)$.

We continue the study of the so‐called Hausdorff–Zhu operators. In this paper, we study the behavior of the singular values of such operators. We prove the general fact that the sequence of singular numbers tends to zero, as well as we prove power‐type estimates for such behavior under additional conditions on the kernel of the operator. We give application of these results to the boundedness of Hausdorff–Zhu operators in general classes of analytic functions in the unit disc and also in some special classes of analytic functions defined in terms of conditions on Taylor or Fourier coefficients.

We study the relation between the growth of a subharmonic function in the half space and the size of its asymptotic set. A function f defined on a domain D has an asymptotic value b∈[−∞, ∞] at a∈∂D if there exists a path γ in D ending at a such that u(p) tends to b as p tends to a along γ. The set of all points on ∂D at which f has an asymptotic value b is denoted by A(f, b). G. R. MacLane [10, 11] studied the class of analytic functions in the unit disk having asymptotic values at a dense subset of the unit circle. Hornblower [8, 9] studied the analogous class for subharmonic functions. Many theorems have since been proved having the following character: for a function f of a given growth, if A(f, +∞) is a small set then f has nice boundary behavior on a large set. See [1, 3–7] and the references therein. For α>0, let Mα be the class of subharmonic functions u in R + n + 1 ≡{(x, y):x∈ℝn, y>0} satisfying the growth condition u(x, y) ⩽ C(u)y−α for 0 < y < 1 for some constant C(u) depending on u. Denote by F(u) the Fatou set of u, which consists of points on ∂ R + n + 1 where u has finite vertical limits. For β>0, denote by Hβ the β-dimensional Hausdorff content. The following theorem is due to Barth and Rippon [1], Fernández, Heinonen and Llorente [5], and Gardiner [6].

Let $\mathcal{H}$ denote the class of analytic functions in the unit disc on the complex plane $\mathbf{C}$. Let $\mathcal{E}$ be a subclass of $\mathcal{H}$. If the operator $I:\mathcal{E}\rightarrow \mathcal{H}$ satisfies \begin{equation*} f(z)\prec g(z) \Rightarrow I[f](z)\prec I[g](z) \end{equation*} for all $f,g\in\mathcal{E}$, then it is called subordination-preserving operator on the class $\mathcal{E}$. In this work we consider the convexity of the Bernardi operator. We prove also that the Bernardi is the subordination-preserving operator on the class of starlike functions. The applications of main results are also presented.

Some extremal bounds for subclasses of univalent functions

Let denote the class of functions that are harmonic univalent and sense-preserv- ing in the unit disk , where . In this paper, we introduce the class of functions which are harmonic in . A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class if , where and . Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class , are obtained.

Generalizations of Hadamard Products of Functions with Negative Coefficients