bluuebunny/crossref_metadata_2025_split huggingface dataset:
Brittle fracture in PVC pipe material
Authors: McGarry Frederick J. , Mandell John F. , Hsueh‐Lee Lidia 
Published: June 2010

AbstractMethods to measure the plane strain fracture toughness (K1c) of PVC pipe material, valid according to linear elastic fracture mechanics (LEFM), have been developed. As in metals practice, the crack tip plastic zone must be small compared to initial crack length, section thickness, and other specimen dimensions to achieve validity. Other conditions of test rate and temperature also must be fulfilled. K1c values vary from 2.8 to 4.0 MN m−3/2. Quantitative micromechanical analyses of the plastic flow regions on the fracture surfaces account for only a fraction of the measured toughness. Other deformation mechanisms (crazing) below but near the surfaces have been observed; apparently they absorb the balance of the work required to drive the stable crack. Under catastrophic, rapid crack propagation, only a single craze exists ahead of the crack tip; the multiple crazes disappear. Thus the dynamic fracture toughness may be several factors less than the static or stable value.
Diffusion in polymer‐solvent systems
Authors: Duda J. Larry , Vrentas James S. 
Published: June 2010

AbstractRecent advances in the area of molecular diffusion in polymer‐solvent systems will be reviewed. Alfrey's classification scheme for the diffusion of solvent molecules in polymers will be used as a starting point. It will be shown that the various regions on Alfrey's diagram are distinguished by the ratio of two characteristic times, a characteristic relaxation time for the polymer‐solvent system and a characteristic diffusion time. Fickian diffusion is realized when this characteristic ratio is a small number, and anomalous behavior occurs when these two characteristic times are the same order of magnitude. Alfrey's “null” region corresponds to those conditions where this dimensionless ratio is much greater than one. The different manifestations of anomalous diffusion with polymer‐solvent systems will be discussed, including case II transport as defined by Alfrey, Gurnee, and Lloyd. Finally, diffusion in polymer‐solvent systems above the glass transition temperature where classical diffusion theory is applicable will be considered. Emphasis will be placed on the demonstration of the ability of theoretical methods based on free volume concepts to predict the concentration and temperature dependencies of diffusion coefficients. The extension of this free volume theory to describe diffusion in glassy polymers will also be demonstrated. Experimental results for several polymer‐solvent systems which exhibit different types of transport behavior will be used to illustrate the correlative and predictive capabilities of these recently developed theories.
Some contributions of turner alfrey to fabrication of polymers
Authors: Schrenk W. J. 
Published: June 2010

AbstractEnhancement of polymer properties by controlling supramolecular structure during fabrication was of major interest to Turner Alfrey. Supramolecular structure, i.e., the geometric arrangement of macromolecules in a specimen, is established during mechanical processing and can have a profound effect on properties. Several examples of Alfrey's contributions to fabrication of polymers are reviewed which demonstrate the Alfrey method of analyzing complicated fabrication processes.
Pump‐assisted plasticating extrusion*
Authors: McKelvey James M. 
Published: June 2010

AbstractWithin the past several years considerable interest has developed in the practice of retrofitting of plasticating extruders with gear pumps. The primary motivation for this is to reduce surges and fluctuations in the extrusion rate, thereby improving product quality and reducing waste. It is shown that the effectiveness of a gear pump for this purpose is directly related to its volumetric efficiency. The volumetric efficiency depends upon the pump dimensions, the flow behavior of the polymer, and the operating variables of temperature, pressure, and pump speed. Experimental data on a variety of thermoplastic materials has been obtained using a pump with gear wheels of 3.6 cm diameter. These data have been correlated with an empirical equation. Some tentative conclusions are drawn concerning the sizing of gear pumps for specific applications.
Gelation by additive polymerization in two dimensions
Authors: Rushton Andrew F. , Family Fereydoon , Herrmann Hans J. 
Published: June 2010

AbstractKinetics of gelation by additive polymerization on a two‐dimensional lattice were investigated with a Monte Carlo simulation. We have determined the critical component γ and the ratio R of the critical amplitudes of the weight‐average degree of polymerization (“susceptibility”) to be γ = 3.8 ± 0.5 and R = 140 ± 45. These results show that this model is neither in the universality class of standard percolation (γ ± 43/18 ± 2.38; R = 219 ± 25) nor the Flory‐Stockmayer model (γ = 1, R = 1).
Kinetics of aggregation and gelation
Authors: Stanley H. Eugene , Family Fereydoon , Gould Harvey 
Published: June 2010

AbstractWe define six kinetic growth models that have witnessed an explosion of recent activity:
Cancer Growth (Eden)
Colloid Growth (Langer‐Müller‐Krumbhaar; Witten‐Sander)
Breakdown (Sawada et al.)
Crystallization (Rikvold)
Invasion Percolation (Schlumberger group)
Addition Polymerization (Manneville‐de Seze; Herrmann et al.)
We also describe briefly some of the approaches used to study these models, with emphasis on the re normalization group approach being developed by us.
Polymer reptation in semidilute solution
Authors: Schaefer D. W. 
Published: June 2010

AbstractThe self‐diffusion constant and solution viscosity are calculated for polymers in semidilute solution using the reptation concept. A space‐filling constraint (rather than binary contacts) is used to construct the tube in the reptation model. In good solvents, scaling predictions are recovered, but in marginal systems unexpected effective power‐law exponents are predicted for the concentration dependence of the self‐diffusion constant, relative viscosity, and reptation time.
Multifunktionale Gebäudehülle und Raum im Raum – Die Cité du Design in St. Etienne/Frankreich
Authors: Sobek Werner , Winterstetter Thomas 
Published: June 2010

AbstractDie nach den Plänen des deutsch‐französischen Architekten Finn Geipel errichtete Cité du Design in Saint‐Etienne soll sich zum führenden Zentrum für Design entwickeln und als solches wesentliche Impulse zum Strukturwandel in dieser zentralfranzösischen Region geben. Die Cité du Design bietet Raum für verschiedene Institutionen, Initiativen und Unternehmen aus dem Bereich des Design sowie der Informationstechnologie. Das Bauvorhaben besteht aus der sogenannten Platine, einem Aussichtsturm und den umgebauten Gebäuden einer ehemaligen Waffenmanufaktur (Bild 1). Die Platine spielt die zentrale Rolle im Gesamtprojekt: sie bietet wichtige Funktionen wie Ausstellungsräume, Säle für Veranstaltungen und eine Bibliothek. Der Aussichtsturm hatte als erstes fertig gestelltes Gebäude eine wesentliche kommunikationsstrategische Aufgabe zu erfüllen. Der vorliegende Aufsatz widmet sich den neu errichteten Gebäuden, d. h. der Platine und dem Aussichtsturm, die sich durch ihr filigran gestaltetes Stahltragwerk auszeichnen.Multifunctional building envelope and space‐and‐space – The Cité du Design in St. Etienne/France. The Cité du Design in Saint‐Etienne, built according to the German‐French architect Finn Geipel's designs, is planned to be developed as a leading centre of design and to drive further the structural change in this central French region. The Cité du Design offers space for various institutions, initiatives and companies from the field of design as well as information technology. The project consists of the so‐called Platine, an observation tower and renovated buildings of a former weapon factory. The Platine is the main part of the project and offers important functions such as showrooms, event halls, and a library. The observation tower, the first completed building, had to fulfil an essential task in communication strategy. The following essay presents the newly set up buildings, i. e. the Platine and the observation tower, both of which are characterised by their filigree steel structure design.
Radiologic and Pathological Correlation of Staging of Rectal Cancer with 3 Tesla Magnetic Resonance Imaging
Authors: Chatterjee Parangama , Eapen Anu , Perakath Benjamin , Singh Ashish 
Published: June 2010

Purpose To assess the sensitivity and specificity of 3 Tesla magnetic resonance imaging (MRI) in the prediction of extramural spread and metastatic adenopathy in rectal carcinoma. Materials and Methods This was a prospective cohort study that included forty consecutive patients with rectal carcinoma from the Department of Colorectal Surgery. Three Tesla (3T) MRI was performed on these patients after a 4-hour fast and cleansing water enema. TI-weighted and T2-weighted images were obtained with high-resolution images T2-weighted sequences through the pelvis. Sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV) of 3T MRI for prediction of metastatic adenopathy and extramural spread were calculated. The TNM staging based on MRI was compared with histopathology of the resected specimen (taken as the criterion standard). Results In our study, sensitivity, specificity, PPV, and NPV of 3T MRI for prediction of metastatic adenopathy were 100%, 78.3%, 77.3%, and 100%, respectively. Sensitivity, specificity, PPV, and NPV of 3T MRI for prediction of extramural tumour spread were 100% and 20%, 89.7% and 100%, respectively (ie, prediction of stages T3 and above). Conclusion MRI allows accurate measurement of the depth of extramural tumour spread. In the assessment of metastatic adenopathy, however, MRI has a low specificity. This study shows that MRI is unlikely to miss any significant parameter in staging of rectal carcinoma. However, it has a tendency to overstage extramural spread of tumour.
Home‐Based Life‐Saving Skills in Liberia: Acquisition and Retention of Skills and Knowledge
Authors: Lori Jody R. , Majszak Christina M. , Martyn Kristy K. 
Published: June 2010

Introduction: In 2006, Home‐Based Life‐Saving Skills was introduced in three Liberian counties, in partnership with Africare‐Liberia and the Liberian Ministry of Health and with funding by the United States Agency for International Development.Methods: Traditional midwives and trained traditional midwives (N = 412) underwent pre‐ and immediate posttesting on four topics. Three hundred eighty‐nine (94%) of the original participants also completed 1‐year follow‐up posttesting.Results: Mean scores significantly improved between pre‐ and immediate posttests and knowledge improvement was stable at 1 year. Correct responses on individual steps for each of four topics significantly increased over the course of training. The data demonstrate a major change in the knowledge base of the traditional and trained traditional midwives acting as village guides.Discussion: Home‐Based Life‐Saving Skills is a viable means by which to improve community knowledge and decrease maternal and neonatal morbidity and mortality.
Plankton of the central Great Barrier Reef: abundance, production and trophodynamic roles
Authors: Sorokin Yu.I. , Sorokin P.Yu. 
Published: June 2010

The abundance, composition and metabolic activity of plankton were assessed in the Tribulation zone of the central Great Barrier Reef (16°–17°S). Wet phytoplankton biomass ranged in shallow reef waters from 30 to 70 mg m−3, and from 60 to 270 mg m−3in the deep lagoon and in the estuarine areas which are dominated by pico- and nano-algae. Wet bacterioplankton biomass varied from 70 to 290 mg m−3. Wet meroplankton biomass was less than 10 mg m−3. Wet daytime mesozooplankton biomass ranged from 100 to 300 mg m−3in the deep lagoon. In the estuarine area, it reached 400 to 1300 mg m−3and in the shallow inner lagoon of the Low Isles ring reef it varied from 10 to 30 mg m−3. Zooplankton density increased at night and was 3 to 5 fold greater in the deep lagoon, for about 2 orders of magnitude greater over the reef shallows and up to 3 orders of magnitude greater in mangrove habitats, due to the emergence of demersal components from the benthos. The biomass of zooplankton hidden in the benthic substrates during the day reached 10 to 40 g m−2. Pelagic primary production in the deep lagoon varied between 0.2 and 0.5 g C m−2 d−1. A calculation of the energy balance suggests that the basic energy source for heterotrophic plankton production in the deep lagoon is the organic matter exported from surrounding reef benthic communities and from mangroves. The trophic status of coral reef pelagic ecosystem might range from mesotrophic to eutrophic.
Diversity of Indo-AustralianPlakortis(Demospongiae: Plakinidae), with description of four new species
Authors: Muricy Guilherme 
Published: June 2010

A collection of 32 specimens of the genusPlakortis(Demospongiae: Plakinidae) from Australia and the Indo-Pacific is here examined. Six species are described, four of which are new to science.Plakortis litahas microrhabds, an irregular tangential ectosomal reticulation, a confused choanosomal skeleton and irregular diods and triods.Plakortis quasiamphiasterhas quasiamphiasters (spined diods and triods), spheres and a skeleton with a distinct subectosomal region.Plakortis communissp. nov. has diods and a tangential ectosomal reticulation with circular meshes; triods and spheres may be present or absent.Plakortis bergquistaesp. nov. has diods in two size-classes, the larger one up to 330–356 µm long, and large triods (actines up to 75–121 µm long).Plakortis fromontaesp. nov. has large diods (up to 220 µm long), rare triods and a double ectosomal reticulation.Plakortis hooperisp. nov. has diods, triods, microrhabds and spheres, with a confused ectosomal skeleton, a thinly encrusting shape and cream colour. Detailed descriptions of external morphology, skeletal arrangement of the ectosome and spicules in SEM are essential for the taxonomy ofPlakortis. The number of valid species in the genus is raised from 15 to 19. A key to Indo-Australian species ofPlakortisis given.
A distortion theorem for analytic functions
Authors: Robertson M. S. 
Published: June 2010

Let

f
(
z
)

f(z)

be a function analytic in the disk

E
{
z
:

|

z

|

>
1
}

E\{ z:|z| > 1\}

and for some real number

n
>
0

n > 0

let

|

f
(
z
)

|

≦

(
1
−

|

z

|

2

)

−
n

,
z
∈
E

|f(z)| \leqq {(1 - |z{|^2})^{ - n}},z \in E

. In this paper it is shown that
\[

|

f
′

(
z
)

|

≦

(
n
+
1
)

n
+
1

n
n

[

1
−

(

n

n
+
1

)

2
n

(
1
−

|

z

|

2

)

2
n

|

f
(
z
)

|

2

]

÷

(
1
−

|

z

|

2

)

n
+
1

,

|f’(z)| \leqq \frac {{{{(n + 1)}^{n + 1}}}}{{{n^n}}}\left [ {1 - {{\left ( {\frac {n}{{n + 1}}} \right )}^{2n}}{{(1 - |z{|^2})}^{2n}}|f(z){|^2}} \right ] \div {(1 - |z{|^2})^{n + 1}},

\]

z
∈
E

z \in E

. In the special case

n
=
1

n = 1

there is a constant

K
,
3
≦
K
≦
4

K,3 \leqq K \leqq 4

, so that
\[

f
′

(
z
)

|

+

|

f
(
z
)

|

2

≦
K

(
1
−

|

z

|

2

)

−
2

.

f’(z)| + |f(z){|^2} \leqq K{(1 - |z{|^2})^{ - 2}}.

\]
This result has application in univalent function theory.
Note on the Kloosterman sum
Authors: Williams Kenneth S. 
Published: June 2010

The Kloosterman sum
\[

∑

x
=
0
;
(
x
,
p
)
=
1

p
α

−
1

exp
⁡
(
2
π
i
n
(
x
+

x
¯

)

/

p
α

)
,

\sum \limits _{x = 0;(x,p) = 1}^{{p^\alpha } - 1} {\exp (2\pi in(x + \bar x)/{p^\alpha }),}

\]
where p is an odd prime,

α
≧
2

\alpha \geqq 2

and

(
n
,
p
)
=
1

(n,p) = 1

, is evaluated in a very short direct way.
Coefficients for the area theorem
Authors: Goodman A. W. 
Published: June 2010

Let

f
(
z
)
=

∑

n
=
1

∞

a
n

z
n

f(z) = \sum \nolimits _{n = 1}^\infty {{a_n}{z^n}}

, and set

G
(
z
)
=
f

(

z

−
p

)

−

/

1
p

=

∑

n
=
0

∞

g

n
p
−
1

z

1
−
n
p

G(z) = f{({z^{ - p}})^{ - /1p}} = \sum \nolimits _{n = 0}^\infty {{g_{np - 1}}{z^{1 - np}}}

. This paper finds an explicit formula for

g

n
p
−
1

{g_{np - 1}}

in terms of the

a
n

{a_n}

. Such a formula (apparently previously unknown) may be very useful in the theory of univalent functions.
Homotopy types of the deleted product of unions of two simplexes
Authors: Whitley W. T. 
Published: June 2010

If X is a space, let

X
∗

=
X
×
X
−
D

{X^\ast } = X \times X - D

, where D is the diagonal. If f is a map on X to a space Y, let

X
f
∗

=
{
(
x
,
y
)
∈

X
∗

|

f
(
x
)
≠
f
(
y
)
}

X_f^\ast = \{ (x,y) \in {X^\ast }|f(x) \ne f(y)\}

. In this paper we continue our investigation, begun in [6], of the homotopy types of

X
∗

{X^\ast }

and

X
f
∗

X_f^\ast

, and of a question due to Brahana [1, p. 236], as to when the homotopy types of

X
∗

{X^\ast }

and

X
f
∗

X_f^\ast

are the same. If X is the union of two nondisjoint simplexes, and if f is a simplicial map on X, we are able, using results and techniques developed in [6], to express the homotopy types of

X
∗

{X^\ast }

and

X
f
∗

X_f^\ast

in terms of spheres, and then to determine when the homotopy types of these spaces are the same.
On the behavior of solutions of sublinear second order differential equations
Authors: Teufel Hugo 
Published: June 2010

The equations in question generalize
\[

x
+
a
(
t
)

|

x

|

γ

sgn
⁡
x
=
0
,

0
>
γ
>
1
,

a
(
t
)
≧
0.

x + a(t)|x{|^\gamma }\operatorname {sgn} x = 0,\quad 0 > \gamma > 1,\quad a(t) \geqq 0.

\]
. A comparison theorem and a uniqueness theorem for initial value problems are proved. Boundary value problems are studied. Oscillation is discussed via comparison.
Differentiable semigroups with a neighborhood of the identity homeomorphic to a Banach space
Authors: Neuberger J. W. 
Published: June 2010

Suppose S is a topological semigroup with identity e, H is a Banach space and there is a homeomorphism g from an open subset of S containing e onto H. Theorem. Suppose that

V
(
x
,
y
)
=
g
(

g

−
1

(
x
)

g

−
1

(
y
)
)

V(x,y) = g({g^{ - 1}}(x){g^{ - 1}}(y))

for all x, y in H for which the second expression is defined. If V is

C

(
2
)

{C^{(2)}}

then there is an open subset D of S containing e so that if q is in D, then for some number t and some one-parameter subgroup f of S,

q
=
f
(
t
)

q = f(t)

.
A random Trotter product formula
Authors: Kurtz Thomas G. 
Published: June 2010

Let

X
(
t
)

X(t)

be a pure jump process with state space S and let

ξ
0

,

ξ
1

,

ξ
2

,
⋯

{\xi _0},{\xi _1},{\xi _2}, \cdots

be the succession of states visited by

X
(
t
)
,

Δ
0

Δ
1

⋯

X(t),{\Delta _0}{\Delta _1} \cdots

the sojourn times in each state,

N
(
t
)

N(t)

the number of transitions before t and

Δ
t

=
t
−

∑

k
=
0

N
(
t
)
−
1

Δ
k

{\Delta _t} = t - \sum \nolimits _{k = 0}^{N(t) - 1} {{\Delta _k}}

. For each

x
∈
S

x \in S

let

T
x

(
t
)

{T_x}(t)

be an operator semigroup on a Banach space L. Define

T
λ

(
t
,
w
)
=

T

ξ
0

(
(
1

/

λ
)

Δ
0

)

T

ξ
1

(
(
1

/

λ
)

Δ
1

)
⋯

T

ξ

N
(
λ
t
)

(
(
1

/

λ
)

Δ

λ
t

)

{T_\lambda }(t,w) = {T_{{\xi _0}}}((1/\lambda ){\Delta _0}){T_{{\xi _1}}}((1/\lambda ){\Delta _1}) \cdots T_{\xi _{N(\lambda t)}}((1/\lambda ){\Delta _{\lambda t}})

. Conditions are given under which

T
λ

(
t
,
w
)

{T_\lambda }(t,w)

will converge almost surely (or in probability) to a semigroup of operators as

λ
→
∞

\lambda \to \infty

. With

S
=
{
1
,
2
}

S = \{ 1,2\}

and
\[

X
(
t
)
=
1
,

=
2
,

2
n
≦
t
>
2
n
+
1
,

2
n
+
1
≦
t
>
2
n
+
2
,

\begin {array}{*{20}{c}} \hfill {X(t) = 1,\quad } \\ \hfill { = 2,\quad } \\ \end {array} \begin {array}{*{20}{c}} {2n \leqq t > 2n + 1,} \hfill \\ {2n + 1 \leqq t > 2n + 2,} \hfill \\ \end {array}

\]

n
=
0
,
1
,
2
,
⋯

n = 0,1,2, \cdots

the result is just the “Trotter product formula".
Colength of derivation ideals
Authors: Kramer Kenneth 
Published: June 2010

In this paper,

D
D

is a derivation acting on the formal power series ring

K
[
[

t
1

,
⋯
,

t
r

]
]

K[[{t_1}, \cdots ,{t_r}]]

over a field

K
K

of characteristic

p
≠
0

p \ne 0

. We conjecture that the colengths of the ideals (

(

D

p
n

t
1

,
⋯
,

D

p
n

t
r

)

({D^{{p^n}}}{t_1}, \cdots ,{D^{{p^n}}}{t_r})

and

(

D

p

n
−
1

t
1

,
⋯
,

D

p

n
−
1

t
r

)

({D^{{p^{n - 1}}}}{t_1}, \cdots ,{D^{{p^{n - 1}}}}{t_r})

are congruent modulo

p
n

{p^n}

, provided they are finite. We give a proof for the case

r
=
1

r = 1

and any

n
≧
1

n \geqq 1

, and for the case

n
=
1

n = 1

and any

r
≧
1

r \geqq 1

.
Fixed points of certain self maps on an interval
Authors: Wong Chi Song 
Published: June 2010

Let T be a self map on a bounded interval [a, b] with

a
,
b
∈
T
(
[
a
,
b
]
)

a,b \in T([a,b])

. Suppose that for any x, y in [a, b],
\[

|

T
(
x
)
−
T
(
y
)

|

≦

1
2

(

|

x
−
T
(
x
)

|

+

|

y
−
T
(
y
)

|

)
.

|T(x) - T(y)| \leqq \tfrac {1}{2}(|x - T(x)| + |y - T(y)|).

\]
It is shown without the continuity of T that the midpoint of [a, b] is a fixed point of T. A nontrivial example is given.
Fixed points by a new iteration method
Authors: Ishikawa Shiro 
Published: June 2010

The following result is shown. If

T
T

is a lipschitzian pseudo-contractive map of a compact convex subset

E
E

of a Hilbert space into itself and

x
1

{x_1}

is any point in

E
E

, then a certain mean value sequence defined by

x

n
+
1

=

α
n

T
[

β
n

T

x
n

+
(
1
−

β
n

)

x
n

]
+
(
1
−

α
n

)

x
n

{x_{n + 1}} = {\alpha _n}T[{\beta _n}T{x_n} + (1 - {\beta _n}){x_n}] + (1 - {\alpha _n}){x_n}

converges strongly to a fixed point of

T
T

, where

{

α
n

}

\{ {\alpha _n}\}

and

{

β
n

}

\{ {\beta _n}\}

are sequences of positive numbers that satisfy some conditions.
Coefficients associated with the expansion of certain products
Authors: Roselle D. P. 
Published: June 2010

Simple combinatorial interpretations are given for the coefficients of the polynomials

H
n

(
x
,
y
)

{H_n}(x,y)

and

G
n

(
x
,
y
)

{G_n}(x,y)

defined by

∏
(
1
+

x
n

y
m

t
)
=
Σ

G
n

(
x
,
y
)

t
n

/

(
x

)
n

(
y

)
n

\prod (1 + {x^n}{y^m}t) = \Sigma {G_n}(x,y){t^n}/{(x)_n}{(y)_n}

and

∏

(
1
−

x
n

y
m

t

)

−
1

=
Σ

H
n

(
x
,
y
)

t
n

/

(
x

)
n

(
y

)
n

\prod {(1 - {x^n}{y^m}t)^{ - 1}} = \Sigma {H_n}(x,y){t^n}/{(x)_n}{(y)_n}

.
On a subclass of spiral-like functions
Authors: Silvia E. M. 
Published: June 2010

Let

α
≧
0
,
0
≦
β
>
1
,

|

λ

|

>
π

/

2

\alpha \geqq 0,0 \leqq \beta > 1,|\lambda | > \pi /2

and suppose that

f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}

is holomorphic in

U
=
{
z
:

|

z

|

>
1
}

U = \{ z:|z| > 1\}

. If
\[

Re
⁡

[

e

i
λ

z

f
′

(
z
)

f
(
z
)

+
α

(

z
f
(
z
)

f
′

(
z
)

+
1
−

z

f
′

(
z
)

f
(
z
)

)

]

>
β
cos
⁡
λ

\operatorname {Re} \left [ {{e^{i\lambda }}\frac {{zf’(z)}}{{f(z)}} + \alpha \left ( {\frac {{zf(z)}}{{f’(z)}} + 1 - \frac {{zf’(z)}}{{f(z)}}} \right )} \right ] > \beta \cos \lambda

\]
for

z
∈
U

z \in U

, then

f
(
z
)

f(z)

is said to be

α
\alpha

-

λ
\lambda

-spiral-like of order

β
\beta

and we write

f
(
z
)
∈

S
α
λ

(
β
)

f(z) \in S_\alpha ^\lambda (\beta )

. The author shows that for each

α
≧
0
,
α

\alpha \geqq 0, \alpha

-

λ
\lambda

-spiral-like functions of order

β
\beta

are

λ
\lambda

-spiral-like of order

β
\beta

. The following representation theorem is obtained: The function

f
(
z
)
∈

S
α
λ

(
β
)
(
α
>
0
,
0
≦
β
>
1
,

|

λ

|

>
π

/

2
)

f(z) \in S_\alpha ^\lambda (\beta )(\alpha > 0,0 \leqq \beta > 1,|\lambda | > \pi /2)

, if and only if there exists a function

F
(
ζ
)
λ

F(\zeta )\lambda

-spiral-like of order

β
\beta

such that
\[

F
(
z
)
=

[

(

e

i
λ

/

α
)

∫
0
z

F

(
ζ
)

e

i
λ

/

α

ζ

−
1

d
ζ

]

α

e

−
i
λ

.

F(z) = {\left [ {({e^{i\lambda }}/\alpha )\int _0^z {F{{(\zeta )}^{{e^{i\lambda }}/\alpha }}{\zeta ^{ - 1}}d\zeta } } \right ]^{\alpha {e^{ - i\lambda }}}}.

\]
A distortion theorem for

log
⁡

|

f
(
z
)

/

z

|

\log |f(z)/z|

and a rotation theorem for

arg
⁡
f
(
z
)

/

z

\arg f(z)/z

are also proved for functions

f
(
z
)
∈

S
0
λ

(
β
)

f(z) \in S_0^\lambda (\beta )

.