scholarly journals Extreme points of subordination and weak subordination families of harmonic mappings

2011 ◽  
Vol 61 (1) ◽  
pp. 145-155 ◽  
Author(s):  
Jinjing Qiao ◽  
Xiantao Wang
Keyword(s):  
Extreme Points ◽  
Harmonic Mappings

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

2020 ◽  
pp. 455-469
Author(s):  
Deepali Khurana ◽  
Raj Kumar ◽  
Sibel Yalcin
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Harmonic Mappings ◽  
Distortion Bounds ◽  
Radius Of Convexity ◽  
Univalent Harmonic Mappings ◽  
Harmonic Starlike ◽  
Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

On a subclass of p-harmonic mappings

2013 ◽  
Vol 44 (3) ◽  
pp. 313-325 ◽  
Author(s):  
Saurabh Porwal ◽  
Kaushal Kishore Dixit
Keyword(s):  
Extreme Points ◽  
Harmonic Mapping ◽  
Harmonic Mappings

The purpose of the present paper is to introduce two new classes $HS_p(\alpha)$ and $HC_p(\alpha)$ of $p$-harmonic mappings together with their corresponding subclasses $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$. We prove that the mappings in $HS_p(\alpha)$ and $HC_p(\alpha)$ are univalent and sense-preserving in $U$ and obtain extreme points of $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$, $HS_p(\alpha)\cap T_p$ and $HC_p(\alpha)\cap T_p$ are determined, where $T_p$ denotes the set of $p$-harmonic mapping with non negative coefficients. Finally, we establish the existence of the neighborhoods of mappings in $HC_p(\alpha)$. Relevant connections of the results presented here with various known results are briefly indicated.

scholarly journals Subclasses of Harmonic Mappings Defined by Convolution

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Santosh B. Joshi ◽  
Girish D. Shelake
Keyword(s):  
Convex Combination ◽  
Univalent Functions ◽  
Extreme Points ◽  
Harmonic Mappings ◽  
Coefficient Bounds ◽  
Integral Convolution ◽  
Distortion Bounds ◽  
Special Cases

Two new subclasses of harmonic univalent functions defined by using convolution and integral convolution are introduced. These subclasses generate several known and new subclasses of harmonic univalent functions as special cases and provide a unified treatment in the study of these classes. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are also determined.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

Extreme Points and Majorization: Economic Applications

2020 ◽  
Author(s):  
Andreas Kleiner ◽  
Benny Moldovanu ◽  
Philipp Strack
Keyword(s):  
Extreme Points ◽  
Economic Applications

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

scholarly journals Investigation of ancestral alleles in the Bovinae subfamily

BMC Genomics ◽  
2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Maulana M. Naji ◽  
Yuri T. Utsunomiya ◽  
Johann Sölkner ◽  
Benjamin D. Rosen ◽  
Gábor Mészáros
Keyword(s):  
Extreme Points ◽  
Genetic Mutation ◽  
Zebu Cattle ◽  
Ancestral State ◽  
Initial State ◽  
High Count ◽  
Environmental Pressures ◽  
Ancestral States ◽  
Productive Traits ◽  
Species Specific

Abstract Background In evolutionary theory, divergence and speciation can arise from long periods of reproductive isolation, genetic mutation, selection and environmental adaptation. After divergence, alleles can either persist in their initial state (ancestral allele - AA), co-exist or be replaced by a mutated state (derived alleles -DA). In this study, we aligned whole genome sequences of individuals from the Bovinae subfamily to the cattle reference genome (ARS.UCD-1.2) for defining ancestral alleles necessary for selection signatures study. Results Accommodating independent divergent of each lineage from the initial ancestral state, AA were defined based on fixed alleles on at least two groups of yak, bison and gayal-gaur-banteng resulting in ~ 32.4 million variants. Using non-overlapping scanning windows of 10 Kb, we counted the AA observed within taurine and zebu cattle. We focused on the extreme points, regions with top 0. 1% (high count) and regions without any occurrence of AA (null count). High count regions preserved gene functions from ancestral states that are still beneficial in the current condition, while null counts regions were linked to mutated ones. For both cattle, high count regions were associated with basal lipid metabolism, essential for survival of various environmental pressures. Mutated regions were associated to productive traits in taurine, i.e. higher metabolism, cell development and behaviors and in immune response domain for zebu. Conclusions Our findings suggest that retaining and losing AA in some regions are varied and made it species-specific with possibility of overlapping as it depends on the selective pressure they had to experience.

On subclasses of harmonic mappings involving Frasin operator

2021 ◽  
Author(s):  
Muhammad G. Khan ◽  
Bakhtiar Ahmad ◽  
Zabidin Salleh ◽  
Iing Lukman
Keyword(s):  
Harmonic Mappings

Determination of catastrophic sets of a tubular chemical reactor by two-parameter continuation method

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Marek Berezowski
Keyword(s):  
Continuation Method ◽  
Extreme Points ◽  
Chemical Reactor ◽  
Stationary States ◽  
Parameter Continuation Method ◽  
It Systems ◽  
Parametric Continuation ◽  
Parameter Continuation ◽  
Two Parameter

AbstractThe work relates to development and presentation a two-parameter continuation method for determining catastrophic sets of stationary states of a tubular chemical reactor with mass recycle. The catastrophic set is a set of extreme points occurring in the bifurcation diagrams of the reactor. There are many large IT systems that use the parametric continuation method. The most popular is AUTO’97. However, its use is sometimes not convenient. The method developed in this work allows to eliminate the necessity to use huge IT systems from the calculations. Unlike these systems, it can be inserted into the program as a short subroutine. In addition, this method eliminates time-consuming iterations from the calculations.

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