Spirallike Functions Research Articles

The generalized Roper–Suffridge extension operator for some biholomorphic mappings

Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C . Let Ω n , p 1 , p 2 , … , p n = { z = ( z 1 , z 2 , … , z n ) ′ ∈ C n : ∑ j = 1 n | z j | p j < 1 } , where 1 ⩽ p 1 ⩽ 2 (or 0 < p 1 ⩽ 2 ), p j ⩾ 1 , j = 2 , … , n , are real numbers. In this paper, we prove that Φ n , β 2 , γ 2 , … , β n , γ n ( f ) ( z ) = ( f ( z 1 ) , ( f ( z 1 ) z 1 ) β 2 ( f ′ ( z 1 ) ) γ 2 z 2 , … , ( f ( z 1 ) z 1 ) β n ( f ′ ( z 1 ) ) γ n z n ) ′ preserves spirallikeness of type β (or starlikeness of order α) on Ω n , p 1 , p 2 , … , p n , where β j ∈ [ 0 , 1 ] , γ j ∈ [ 0 , 1 p j ] , and β j + γ j ⩽ 1 , p j is the same as above, we choose the branches such that ( f ( z 1 ) z 1 ) β j | z 1 = 0 = 1 , ( f ′ ( z 1 ) ) γ j | z 1 = 0 = 1 , j = 2 , … , n .

Angle distortion theorems for starlike and spirallike functions with respect to a boundary point

We exhibit angle bounds for starlike and spirallike functions with respect to a boundary point. As an application, we obtain a covering theorem for functions convex in one direction.

Strongly starlike and spirallike functions

The aim of this paper is to present a new method of proof of an analytic characterization of strongly starlike functions of order $(\alpha ,\beta )$. The relation between strong starlikeness and spirallikeness of the same order is discussed in detail. Som

On some results for λ-Spirallike and λ-Robertson functions of complex order

We give some results of various kinds concerning ?-Spirallike functions of complex order and ?-Robertson functions of complex order in the unit disc U = (z: (z) &lt; 1). They represent extensions and generalizations of many previous results. We mainly used the subordination method.

Spiral-like functions with respect to a boundary point

In this chapter, we introduce the class of spirallike functions with respect to a boundary point. It turns out that, in fact, each such function is a complex power of a starlike function with respect to a boundary point. So, what we studied in the previous chapter forms the base for our present considerations. We start by looking at the images of spirallike functions with respect to a boundary point, i.e., spirallike domains, the boundary of which contains the origin.

The norm estimates of pre-schwarzian derivatives of spiral-like functions

For a constant , a normalized analytic function on the unit disk is said to be β-spiral-like if for any point z in the unit disk. In this paper, for such a function f, we shall present the optimal estimate of the norm of f″/f′.

A subordination theorem for spirallike functions

We prove a subordination relation for a subclass of the class ofλ-spirallike functions.

SUBCLASSES OF SPIRALLIKE FUNCTIONS DEFINED BY RUSCHEW EYH DERIVATIVES

&#x0D; &#x0D; &#x0D; An attempt is being made perhaps for the first time, to relate the spirallike functions with the concept of Ruscheweyh derivatives. A new subclass of spirallike functions is introduced with the help of Ruscheweyh derivatives. &#x0D; &#x0D; &#x0D;

ON SPIRALLIKE INTEGRAL OPERATORS

In this paper the integral operators \[ F(z)=\left[\frac{\beta+\gamma}{z^\gamma}\int_0^z [f(t)]^\beta t^{\gamma-1} dt\right]^{1/\beta}\] for $f(z) \in S^\alpha(\lambda, a, b)$ are studied. $S^\alpha(\lambda, a, b)$ as a subclass of the class of all spirallike functions was introduced and studied by the authors. It is shown that $F(z)$ is also in $S^\alpha(\lambda, a, b)$, whenever $f(z)$ is in $S^\alpha(\lambda, a, b)$, under certain restrictions.

Compactness of resolvent operators generated by a class of composition semigroups on Hp

Consider a semigroup of composition operators $T_tf=f\circ \phi_t$, $t\geq 0$, acting on the standard Hardy space $H^p$ $(1\leq p<\infty)$ of the unit disk $D$. Assuming that the $\phi_t$ have a common fixed point at 0, a unique univalent function $h$ can be found such that $h(\phi_t(z))=e^{ct}h(z)$, $z\in D$, $t\geq 0$, where $c$ is a constant which is related to the infinitesimal generator $A$ of $\{T_t\}$. In this paper the compactness of the resolvent operator $R(\lambda,A)$ is studied using the function $h$. It is shown that $R(\lambda,A)$ is compact if and only if $h$ lies in $H^q$ for each $q<\infty$. In the cases where $R(\lambda,A)$ is not compact it is shown that the spectrum of $R(\lambda,A)$ contains a nontrivial disk. A condition under which compactness of $R(\lambda,A)$ implies compactness of the operators in $\{T_t\}$ is also obtained. The methods used are function-theoretic. In particular, the notion of a spiral-like function plays a key role. (Less)

On a class of functions unifying the classes of Paatero, Robertson and others

We study a class of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of order β, spirallike functions of order β and functions for which zf′ is spirallike of order β, functions of boundary rotation utmost kπ, α‐convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the class Vk of functions of bounded boundary rotation, yield some representation theorems and a variational principle for our class. A consequence of these basic theorems is a theorem for this class which unifies some earlier results concerning the radii of convexity of functions in the class of Moulis and those concerning the radii of starlikeness of functions in the classes Uk of Pinchuk and U2(β) of Robertson etc. By applying an estimate of Moulis concerning functions in , we obtain an inequality in the class which will contain an estimate for the Schwarzian derivative of functions in the class and in particular the estimate of Moulis for the Schwarzian of functions in .

Extremal properties of some spirallike functions

Extremal properties of some spirallike functions

Some Results on Spiral-Like Functions

We begin with the following definitions.Let f(z) be regular near z=0 and let f(0)=0, f'(0)≠0. Let α and λ be two real numbers such that |α|&lt;π/2 and 0≤λ1. Then Rα,λ is the largest value of r such that the following conditions are satisfied for |z|&lt;r

Boundary Behavior of Starlike Functions

For a starlike function f, we impose a geometric condition on the image of the open unit disc by the mapping $w = zf’(z)/f(z)$ to insure that f be one-to-one on the closed unit disc. Applications are given to certain classes of univalent functions, including spiral-like functions.

ON A SUBCLASS OF SPIRAL-LIKE FUNCTIONS

ON A SUBCLASS OF SPIRAL-LIKE FUNCTIONS

Boundary behavior of starlike functions

For a starlike function f, we impose a geometric condition on the image of the open unit disc by the mapping w = z f ′ ( z ) / f ( z ) w = zf’(z)/f(z) to insure that f be one-to-one on the closed unit disc. Applications are given to certain classes of univalent functions, including spiral-like functions.

The Hardy class of a spiral-like function.

The Hardy class of a spiral-like function.

The Hardy Class of a Spiral-Like Function and its Derivative

The Hardy Class of a Spiral-Like Function and its Derivative

The Hardy class of a spiral-like function and its derivative

A determination is made of the Hardy classes to which a spiral-like univalent function and its derivative belong. An estimate for the size of the Taylor coefficients is deduced.