scholarly journals On the Ninth Coefficient of the Inverse of a Convex Function

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 706
Author(s):  
Toshiyuki Sugawa
Keyword(s):  
Convex Function ◽  
Unit Disk ◽  
Complex Plane ◽  
Univalent Function ◽  
Inverse Function ◽  
The Other ◽  
Other Hand ◽  
Convex Univalent Function

We consider the inverse function z=g(w)=w+b2w2+⋯ of a normalized convex univalent function w=f(z)=z+a2z2+⋯ on the unit disk in the complex plane. So far, it is known that |bn|≤1 for n=2,3,⋯,8. On the other hand, the inequality |bn|≤1 is not valid for n=10. It is conjectured that |b9|≤1. The present paper offers the estimate |b9|<1.617.

scholarly journals The asymptotic expansion of integral functions defined by Taylor series

1940 ◽  
Vol 238 (795) ◽  
pp. 423-451 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Plane ◽  
Taylor Series ◽  
Asymptotic Expansions ◽  
Integral Function ◽  
The Other ◽  
General Integral ◽  
Other Hand

1.1. Considerable attention has been devoted to the behaviour of the general integral function for large values of the variable, and many important theorems have been proved in this field. On the other hand, the behaviour of a large number of particular integral functions has been studied in detail and their asymptotic expansions for certain regions of the plane obtained. There is, however, a substantial gap between the two theories. For example, much of the most interesting work on the general integral function deals with the distribution of its zeroes and other values; but many of the asymptotic expansions obtained for particular functions are not valid in the regions in which these functions have zeroes. In this paper and its sequels I propose to study several fairly wide classes of functions defined by Taylor series; from the properties of the coefficients I deduce asymptotic expansions of the function defined by the series. For the sort of functions I consider we can usually divide the whole complex plane, with the exception of certain “ barrier regions” , into a number of regions R , in each of which the function is given asymptotically by an equation of the form

ON THE FREQUENT UNIVERSALITY OF UNIVERSAL TAYLOR SERIES IN THE COMPLEX PLANE

2016 ◽  
Vol 59 (1) ◽  
pp. 109-117 ◽  
Author(s):  
A. MOUZE ◽  
V. MUNNIER
Keyword(s):  
Complex Plane ◽  
Taylor Series ◽  
The Other ◽  
Other Hand ◽  
Universal Taylor Series

AbstractWe prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.

scholarly journals A Construction of Meromorphic Functions with Prescribed Asymptotic Behavior

1971 ◽  
Vol 41 ◽  
pp. 75-87 ◽  
Author(s):  
J.L. Stebbins
Keyword(s):  
Asymptotic Behavior ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
The Other ◽  
Asymptotic Values ◽  
Other Hand ◽  
Extended Complex ◽  
The Given ◽  
Prescribed Asymptotic Behavior ◽  
Extended Complex Plane

Although there are several constructions of meromorphic functions with prescribed asymptotic sets [e.g., 5,6], it is usually difficult to determine or prescribe the nature of the asymptotic paths used in these constructions. On the other hand, there are several other constructions of meromorphic functions with prescribed asymptotic paths [e.g., 1, 10, 12], but the extent of the asymptotic values for these functions cannot always be restricted to the values approached along the given paths. Gross [3] has accomplished both results by prescribing paths for every value in the extended complex plane.

scholarly journals A problem on the Riesz-Dunford operator calculus and convex univalent functions

1983 ◽  
Vol 24 (2) ◽  
pp. 129-130 ◽  
Author(s):  
J. S. Hwang
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Univalent Function ◽  
Convex Set ◽  
Univalent Functions ◽  
Operator Calculus ◽  
Convex Univalent Functions ◽  
Ky Fan ◽  
Convex Univalent Function

In his paper [3], Ky Fan asked whether if f is a convex univalent function in the unit disk, with f(0) = 0 and f'(0) = 1, then is it true that the set of f(A) is a convex set of operators, when A runs through all proper contractions on a Hilbert space? We answer this question in the negative.

scholarly journals On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

2020 ◽  
Author(s):  
Toshiyuki Sugawa ◽  
Li-Mei Wang
Keyword(s):  
Extremal Problem ◽  
Unit Disk ◽  
Complex Plane ◽  
Inverse Function ◽  
Starlike Function ◽  
Extremal Functions ◽  
Positive Order ◽  
Sharp Estimates ◽  
The Third ◽  
Straightforward Proof

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0&lt;\alpha&lt;1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

scholarly journals Some Geometric Properties of a Hyperbolic Univalent Function

2021 ◽  
pp. 2022-2028
Author(s):  
Sabah S. Al-Azawee ◽  
Shatha S. Alhily
Keyword(s):  
Lower Bound ◽  
Unit Disk ◽  
Univalent Function ◽  
Convex Region ◽  
Connected Subset ◽  
Growth Properties ◽  
Convexity Properties ◽  
Disk Image ◽  
Convexity Estimation ◽  
Convex Univalent Function

In this paper, we analyze several aspects of a hyperbolic univalent function related to convexity properties, by assuming  to be the univalent holomorphic function maps of the unit disk  onto the hyperbolic convex region  ( is an open connected subset of). This assumption leads to the coverage of some of the findings that are started by seeking a convex univalent function distortion property to provide an approximation of the inequality and confirm the form of the lower bound for . A further result was reached by combining the distortion and growth properties for increasing inequality  . From the last result, we wanted to demonstrate the effect of the unit disk image on the condition of convexity estimation by proving the two inequalities of  , and   .

scholarly journals Hankel Determinants for Univalent Functions Related to the Exponential Function

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1211
Author(s):  
Paweł Zaprawa
Keyword(s):  
Unit Disk ◽  
Exponential Function ◽  
Univalent Functions ◽  
Main Idea ◽  
The Other ◽  
Hankel Determinants ◽  
Other Hand ◽  
Almost All

Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f ′ ( 0 ) − 1 = 0 and z f ′ ( z ) f ( z ) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Life Beyond 1MeV

1980 ◽  
Vol 38 ◽  
pp. 30-33
Author(s):  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
Past Experience ◽  
The Other ◽  
Good Case ◽  
Other Hand ◽  
The U.S

Life beyond 1MeV – like life after 40 – is not too different unless one takes advantage of past experience and is receptive to new opportunities. At first glance, the returns on performing electron microscopy at voltages greater than 1MeV diminish rather rapidly as the curves which describe the well-known advantages of HVEM often tend towards saturation. However, in a country with a significant HVEM capability, a good case can be made for investing in instruments with a range of maximum accelerating voltages. In this regard, the 1.5MeV KRATOS HVEM being installed in Berkeley will complement the other 650KeV, 1MeV, and 1.2MeV instruments currently operating in the U.S. One other consideration suggests that 1.5MeV is an optimum voltage machine – Its additional advantages may be purchased for not much more than a 1MeV instrument. On the other hand, the 3MeV HVEM's which seem to be operated at 2MeV maximum, are much more expensive.

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