• Title/Summary/Keyword: harmonic convolution

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Some Properties of Harmonic Functions Defined by Convolution

  • Dixit, Kaushal Kishor;Porwal, Saurabh
    • Kyungpook Mathematical Journal
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    • v.49 no.4
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    • pp.751-761
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    • 2009
  • In this paper, we introduce and study a comprehensive family of harmonic univalent functions which contains various well-known classes of harmonic univalent functions as well as many new ones. Also, we improve some results obtained by Frasin [3] and obtain coefficient bounds, distortion bounds and extreme points, convolution conditions and convex combination are also determined for functions in this family. It is worth mentioning that many of our results are either extensions or new approaches to those corresponding previously known results.

A GENERALIZED CLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH AL-OBOUDI OPERATOR INVOLVING CONVOLUTION

  • Sangle, N.D.;Metkari, A.N.;Joshi, S.B.
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.5
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    • pp.887-902
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    • 2021
  • In this paper, we have introduced a generalized class SiH (m, n, 𝛾, 𝜙, 𝜓; 𝛼), i ∈ {0, 1} of harmonic univalent functions in unit disc 𝕌, a sufficient coefficient condition for the normalized harmonic function in this class is obtained. It is also shown that this coefficient condition is necessary for its subclass 𝒯 SiH (m, n, 𝛾, 𝜙, 𝜓; 𝛼). We further obtained extreme points, bounds and a covering result for the class 𝒯 SiH (m, n, 𝛾, 𝜙, 𝜓; 𝛼). Also, show that this class is closed under convolution and convex combination. While proving our results, certain conditions related to the coefficients of 𝜙 and 𝜓 are considered, which lead to various well-known results.

ON A NEW CLASS OF SALAGEAN-TYPE HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SUBORDINATION

  • Altinkaya, Sahsene;Cakmak, Serkan;Yalcin, Sibel
    • Honam Mathematical Journal
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    • v.40 no.3
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    • pp.433-446
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    • 2018
  • In this present investigation, we introduce a new class of harmonic univalent functions of the form $f=h+{\bar{g}}$ in the open unit disk ${\Delta}$. We get basic properties, like, necessary and sufficient convolution conditions, distortion bounds, compactness and extreme points for these classes of functions.

SOME INCLUSION RELATIONS OF CERTAIN SUBCLASSES OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH GENERALIZED DISTRIBUTION SERIES

  • Magesh, Nanjundan;Porwal, Saurabh;Themangani, Rajavadivelu
    • Communications of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.843-854
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    • 2020
  • The purpose of this present paper is to obtain inclusion relations between various subclasses of harmonic univalent functions by using the convolution operator associated with generalized distribution series. To be more precise, we obtain such inclusions with harmonic starlike and harmonic convex mappings in the plane.

CONSTRUCTION OF SUBCLASSES OF UNIVALENT HARMONIC MAPPINGS

  • Nagpal, Sumit;Ravichandran, V.
    • Journal of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.567-592
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    • 2014
  • Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk are widely studied. A new methodology is employed to construct subclasses of univalent harmonic mappings from a given subfamily of univalent analytic functions. The notions of harmonic Alexander operator and harmonic Libera operator are introduced and their properties are investigated.

STARLIKENESS OF MULTIVALENT MEROMORPHIC HARMONIC FUNCTIONS

  • Murugusundaramoorthy, G.
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.4
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    • pp.553-564
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    • 2003
  • We give sufficient coefficient conditions for starlikeness of a class of complex-valued multivalent meromorphic harmonic and orientation preserving functions in outside of the unit disc. These coefficient conditions are also shown to be necessary if the coefficients of the analytic part of the harmonic functions are positive and the coefficients of the co-analytic part of the harmonic functions are negative. We then determine the extreme points, distortion bounds, convolution and convex combination conditions for these functions.

A NOTE ON CONVEXITY OF CONVOLUTIONS OF HARMONIC MAPPINGS

  • JIANG, YUE-PING;RASILA, ANTTI;SUN, YONG
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.6
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    • pp.1925-1935
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    • 2015
  • In this paper, we study right half-plane harmonic mappings $f_0$ and f, where $f_0$ is fIxed and f is such that its dilatation of a conformal automorphism of the unit disk. We obtain a sufficient condition for the convolution of such mappings to be convex in the direction of the real axis. The result of the paper is a generalization of the result of by Li and Ponnusamy [11], which itself originates from a problem posed by Dorff et al. in [7].