scholarly journals Behavior of the Trinomial ArcsB(n,k,r)when0<α<1

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Kaoutar Lamrini Uahabi ◽  
Mohammed Zaoui
Keyword(s):  
Real Number ◽  
Unit Disk ◽  
Complex Number ◽  
Polar Coordinates ◽  
The Family ◽  
Trinomial Equation

We deal with the familyB(n,k,r)of trinomial arcs defined as the set of roots of the trinomial equationzn=αzk+(1−α), wherez=ρeiθis a complex number,nandkare two integers such that0<k<n, andαis a real number between0and1. These arcsB(n,k,r)are continuous arcs inside the unit disk, expressed in polar coordinates(ρ,θ). The question is to prove thatρ(θ)is a decreasing function, for each trinomial arcB(n,k,r).

scholarly journals On the Fekete-Szegö Problem for a Class of Analytic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Zhigang Peng
Keyword(s):  
Power Series ◽  
Real Number ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Upper Bounds ◽  
The Family ◽  
Class Of Functions

Let 𝒜 denote the class of functions which are analytic in the unit disk D={z:|z|<1} and given by the power series f(z)=z+∑n=2∞‍anzn. Let C be the class of convex functions. In this paper, we give the upper bounds of |a3-μa22| for all real number μ and for any f(z) in the family 𝒱={f(z):f∈𝒜, Re(f(z)/g(z))>0 for  some g∈C}.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

Phased Bessel functions

2008 ◽  
Vol 86 (7) ◽  
pp. 863-870 ◽  
Author(s):  
X Hu ◽  
H Wang ◽  
D -S Guo
Keyword(s):  
Real Number ◽  
Number Field ◽  
Complex Number ◽  
Bessel Functions ◽  
Natural Extension ◽  
State Transitions ◽  
Photon State ◽  
Analytical Extension ◽  
Complex Number Field ◽  
Real Number Field

In the study of photon-state transitions, we found a natural extension of the first kind of Bessel functions that extends both the range and domain of the Bessel functions from the real number field to the complex number field. We term the extended Bessel functions as phased Bessel functions. This extension is completely different from the traditional “analytical extension”. The new complex Bessel functions satisfy addition, subtraction, and recurrence theorems in a complex range and a complex domain. These theorems provide short cuts in calculations. The single-phased Bessel functions are generalized to multiple-phased Bessel functions to describe various photon-state transitions.PACS Nos.: 02.30.Gp, 32.80.Rm, 42.50.Hz

scholarly journals On the generalised Ruscheweyh class of analytic functions of complex order

1993 ◽  
Vol 47 (2) ◽  
pp. 247-257 ◽  
Author(s):  
O.P. Ahuja
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Order Complex ◽  
Sharp Estimates ◽  
Complex Order ◽  
Special Cases ◽  
The Family ◽  
Uniform Treatment

The main object of this paper is to identify various classes of analytic functions which are starlike, convex, pre-starlike, Ruscheweyh class of order α, β-spiral-like, β-convex-spiral-like, starlike of complex order, complex of complex order, and others as special cases of a family of analytic functions of complex order in the unit disk. This makes a uniform treatment possible. Finally, we derive sharp estimates for all coefficients of the functions from the family.

Coefficient inequalities for univalent starlike functions

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Milutin Obradović ◽  
Saminathan Ponnusamy
Keyword(s):  
Unit Disk ◽  
Complex Number ◽  
Convex Functions ◽  
Analytic Functions ◽  
Starlike Functions ◽  
Right Hand ◽  
Coefficient Inequalities ◽  
The Right ◽  
Necessary And Sufficient

AbstractLet A be the class of analytic functions in the unit disk $$\mathbb{D}$$ with the normalization f(0) = f′(0) − 1 = 0. In this paper the authors discuss necessary and sufficient coefficient conditions for f ∈ A of the form $$\left( {\frac{z} {{f(z)}}} \right)^\mu = 1 + b_1 z + b_2 z^2 + \ldots$$ to be starlike in $$\mathbb{D}$$ and more generally, starlike of some order β, 0 ≤ β < 1. Here µ is a suitable complex number so that the right hand side expression is analytic in $$\mathbb{D}$$ and the power is chosen to be the principal power. A similar problem for the class of convex functions of order β is open.

scholarly journals GEOMETRIC LIMITS OF MANDELBROT AND JULIA SETS UNDER DEGREE GROWTH

2012 ◽  
Vol 22 (12) ◽  
pp. 1250301 ◽  
Author(s):  
SUZANNE HRUSKA BOYD ◽  
MICHAEL J. SCHULZ
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Julia Sets ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Geometric Limit ◽  
The Family ◽  
Mandelbrot Sets

First, for the family Pn,c(z) = zn + c, we show that the geometric limit of the Mandelbrot sets Mn(P) as n → ∞ exists and is the closed unit disk, and that the geometric limit of the Julia sets J(Pn,c) as n tends to infinity is the unit circle, at least when |c| ≠ 1. Then, we establish similar results for some generalizations of this family; namely, the maps z ↦ zt + c for real t ≥ 2 and the rational maps z ↦ zn + c + a/zn.

scholarly journals On an integral operator for convex univalent functions

1986 ◽  
Vol 34 (2) ◽  
pp. 211-218
Author(s):  
Vinod Kumar ◽  
S. L. Shukla
Keyword(s):  
Integral Operator ◽  
Real Number ◽  
Complex Number ◽  
Univalent Functions ◽  
Corrected Version ◽  
Convex Univalent Functions ◽  
Class Of Functions

Let K (m,M) denote the class of functions regular and satisfying |1 + zf″(z)/f′(z)− m| < M in |z| < 1, where |m−1| < M ≤ m. Recently, R.K. Pandey and G. P. Bhargava have shown that if f ε K (m,M), then the function du also belongs to K (m,M) provided α is a complex number satisfying the inequality |α| ≤ (1−b)/2, where b = (m-1)/M. In this paper we show by a counterexample that their inequality is in general wrong, and prove a corrected version of their result. We show that F ε K (m,M) provided that α is a real number satisfying −φ ≤ α ≤1, φ = (M−|m−1|)/(M + |m−1|), or a complex number satisfying |α| ≤ φ. In both cases the bounds for α are sharp.

scholarly journals Subclasses of Starlike and Convex Functions Associated with the Limaçon Domain

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 942 ◽  
Author(s):  
Vali Soltani Masih ◽  
Stanisława Kanas
Keyword(s):  
Hypothesis Testing ◽  
Unit Disk ◽  
Convex Functions ◽  
Electrical Power ◽  
Low Grade ◽  
Testing Problem ◽  
Starlike And Convex Functions ◽  
Hypothesis Testing Problem ◽  
The Family ◽  
Low Grade Heat

Let ST L ( s ) and CV L ( s ) denote the family of analytic and normalized functions f in the unit disk D : = z : | z | < 1 , such that the quantity z f ′ ( z ) / f ( z ) or 1 + z f ″ ( z ) / f ′ ( z ) respectively are lying in the region bounded by the limaçon ( u − 1 ) 2 + v 2 − s 4 2 = 4 s 2 u − 1 + s 2 2 + v 2 , where 0 < s ≤ 1 / 2 . The limaçon of Pascal is a curve that possesses properties which qualify it for the several applications in mathematics, statistics (hypothesis testing problem) but also in mechanics (fluid processing applications, known limaçon technology is employed to extract electrical power from low-grade heat, etc.). In this paper we present some results concerning the behavior of f on the classes ST L ( s ) or CV L ( s ) . Some appropriate examples are given.

scholarly journals On Extendability of the Principle of Equivalent Utility

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 42
Author(s):  
Małgorzata Chudziak ◽  
Marek Żołdak
Keyword(s):  
Functional Equation ◽  
Utility Function ◽  
Real Number ◽  
Insurance Premium ◽  
Distortion Function ◽  
Utility Model ◽  
Continuous Solutions ◽  
The Family ◽  
Rank Dependent Utility

An insurance premium principle is a way of assigning to every risk a real number, interpreted as a premium for insuring risk. There are several methods of defining the principle. In this paper, we deal with the principle of equivalent utility under the rank-dependent utility model. The principle, generated by utility function and probability distortion function, is based on the assumption of the symmetry between the decisions of accepting and rejecting risk. It is known that the principle of equivalent utility can be uniquely extended from the family of ternary risks. However, the extension from the family of binary risks need not be unique. Therefore, the following problem arises: characterizing those principles that coincide on the family of all binary risks. We reduce the problem thus to the multiplicative Pexider functional equation on a region. Applying the form of continuous solutions of the equation, we solve the problem completely.

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