scholarly journals THE SHARP BOUND FOR THE DEFORMATION OF A DISC UNDER A HYPERBOLICALLY CONVEX MAP

2006 ◽  
Vol 93 (2) ◽  
pp. 395-417 ◽  
Author(s):  
ROGER W. BARNARD ◽  
LEAH COLE ◽  
KENT PEARCE ◽  
G. BROCK WILLIAMS
Keyword(s):  
Extremal Function ◽  
Schwarzian Derivative ◽  
Special Function ◽  
Function Theory ◽  
Sharp Bound ◽  
Spherical Case ◽  
Euclidean Case ◽  
Step Down ◽  
The Extremal Function

We complete the determination of how far convex maps can deform discs in each of the three classical geometries. The euclidean case was settled by Nehari in 1976, and the spherical case by Mejía and Pommerenke in 2000. We find the sharp bound on the Schwarzian derivative of a hyperbolically convex function and thus complete the hyperbolic case. This problem was first posed by Ma and Minda in a series of papers published in the 1980s. Mejía and Pommerenke then produced partial results and a conjecture as to the extremal function in 2000. Their function maps onto a domain bounded by two proper geodesic sides, a ‘hyperbolic strip’. Applying a generalization of the Julia variation and a critical Step Down Lemma, we show that there is an extremal function mapping onto a domain with at most two geodesic sides. We then verify using special function theory that, among the remaining candidates, the two-sided domain of Mejía and Pommerenke is in fact extremal. This correlates nicely with the euclidean and spherically convex cases in which the extremal is known to be a map onto a two-sided ‘strip’.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

scholarly journals Some estimates for extremal decomposition of the complex plane

2018 ◽  
Vol 32 ◽  
pp. 42-47
Author(s):  
Iryna Denega
Keyword(s):  
Complex Variable ◽  
Open Problem ◽  
Function Theory ◽  
Extremal Problems ◽  
Geometric Function Theory ◽  
Geometric Function ◽  
Inner Radius ◽  
Overlapping Domains ◽  
Disjoint Domains

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. In this paper, we consider the well-known problem of maximum of the functional \(r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right)\), where \(B_{0}\),..., \(B_{n}\) are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \( a_0=0 \), \(|a_{k}|=1\), \(k=\overline{1,n}\) are different points of the circle, \(\gamma\in (0, n]\), and \(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to the point \( a \). This problem was posed as an open problem in the Dubinin paper in 1994. Till now, this problem has not been solved, though some partial solutions are available. In the paper an estimate for the inner radius of the domain that contains the point zero is found. The main result of the paper generalizes the analogous results of [1, 2] to the case of an arbitrary arrangement of systems of points on \(\overline{\mathbb{C}}\).

scholarly journals A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

scholarly journals The extremal function for partial bipartite tilings

2012 ◽  
Vol 33 (5) ◽  
pp. 807-815 ◽  
Author(s):  
Codruţ Grosu ◽  
Jan Hladký
Keyword(s):  
Extremal Function ◽  
The Extremal Function

Unpredictability of the coefficients of m-fold symmetric bi-starlike functions

2014 ◽  
Vol 25 (07) ◽  
pp. 1450064 ◽  
Author(s):  
Samaneh G. Hamidi ◽  
Jay M. Jahangiri
Keyword(s):  
Extremal Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Square Root ◽  
Koebe Function ◽  
The Extremal Function

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

scholarly journals Convolutions of prestarlike functions

1983 ◽  
Vol 6 (1) ◽  
pp. 59-68 ◽  
Author(s):  
O. P. Ahuja ◽  
H. Silverman
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Extreme Points ◽  
Distortion Theorems ◽  
The Extremal Function ◽  
Class Of Functions

The convolution of two functionsf(z)=∑n=0∞anznandg(z)=∑n=0∞bnzndefined as(f∗g)(z)=∑n=0∞anbnzn. Forf(z)=z−∑n=2∞anznandg(z)=z/(1−z)2(1−γ), the extremal function for the class of functions starlike of orderγ, we investigate functionsh, whereh(z)=(f∗g)(z), which satisfy the inequality|(zh′/h)−1|/|(zh′/h)+(1-2α)|<β,0≤α<1,0<β≤1for allzin the unit disk. Such functionsfare said to beγ-prestarlike of orderαand typeβ. We characterize this family in terms of its coefficients, and then determine extreme points, distortion theorems, and radii of univalence, starlikeness, and convexity. All results are sharp.

scholarly journals Note on Schiffer’s Variation in the Class of Univalent Functions in the Unit Disc

1968 ◽  
Vol 32 ◽  
pp. 273-276
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Univalent Functions ◽  
The Real ◽  
Maximum Value ◽  
The Extremal Function

Let S denote the class of univalent functions f(z) in the unit disc D: | z | < 1 with the following expansion: (1) f(z) = z + a2z2 + a3z3 + · · · · anzn + · ··.We denote by fn(z) the extremal function in S which gives the maximum value of the real part of an and by Dn the image of D under w = fn(z).

DETERMINATION OF DYNAMIC RESPONSE COMPUTATION OF CONSTANT VOLTAGE PULSE CONVERTER MICROCHIPS

2021 ◽  
Vol 2021 (3-4) ◽  
pp. 68-76
Author(s):  
Vitaliy Zotin
Keyword(s):  
Experimental Determination ◽  
Frequency Domain ◽  
Frequency Analysis ◽  
Operational Amplifier ◽  
Voltage Pulse ◽  
Pass Filter ◽  
Analysis Scheme ◽  
External Frequency ◽  
Step Down

Work objective is the determination of data on the frequency-domain behavior of components in the composition of the microcircuit and within the microcircuit as a whole, intended for using in DC voltage pulse converters. Research methods: simulation modeling. Research results and novelty: for microcircuits with an external frequency weighing network a frequency analysis scheme using an auxiliary operational amplifier and an active low-pass filter of the second order is proposed. Using a simulator in the LT-spice environment, the amplitude and phase frequency response data of the microchips of the step-down and step-up converters have been obtained. The possibility of technical implementation of the proposed scheme for experimental determination of frequency-domain behavior including final control of the parameters of the appropriate microchips has been noted. For microchips having built-in frequency weighing circuit, the possibility of determining frequency-domain characteristics during the formation of a test signal in an external control circuit has been confirmed. It is found that the use of this method for microchips with increased dynamic properties is problematic. Conclusion: the frequency analysis scheme using an auxiliary operational amplifier is suitable for both step-down and step-up DC/DC converters with an external frequency weighing network. This scheme can be recommended for experimental determination of frequency-domain behavior, including final control of the parameters of the appropriate microchips.

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