Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li; Saminathan Ponnusamy

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 331-350
  • ISSN: 0011-4642

Abstract

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We consider the class ℋ 0 of sense-preserving harmonic functions f = h + g ¯ defined in the unit disk | z | < 1 and normalized so that h ( 0 ) = 0 = h ' ( 0 ) - 1 and g ( 0 ) = 0 = g ' ( 0 ) , where h and g are analytic in the unit disk. In the first part of the article we present two classes 𝒫 H 0 ( α ) and 𝒢 H 0 ( β ) of functions from ℋ 0 and show that if f ∈ 𝒫 H 0 ( α ) and F ∈ 𝒢 H 0 ( β ) , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums) s n , n ( f ) ( z ) = s n ( h ) ( z ) + s n ( g ) ( z ) ¯ , where f = h + g ¯ ∈ ℋ 0 , s n ( h ) and s n ( g ) denote the n -th partial sums of h and g , respectively. We prove, among others, that if f = h + g ¯ ∈ ℋ 0 is a univalent harmonic convex mapping, then s n , n ( f ) is univalent and close-to-convex in the disk | z | < 1 / 4 for n ≥ 2 , and s n , n ( f ) is also convex in the disk | z | < 1 / 4 for n ≥ 2 and n ≠ 3 . Moreover, we show that the section s 3 , 3 ( f ) of f ∈ 𝒞 H 0 is not convex in the disk | z | < 1 / 4 but it is convex in a smaller disk.

How to cite

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Li, Liulan, and Ponnusamy, Saminathan. "Injectivity of sections of convex harmonic mappings and convolution theorems." Czechoslovak Mathematical Journal 66.2 (2016): 331-350. <http://eudml.org/doc/280103>.

@article{Li2016,
abstract = {We consider the class $\{\mathcal \{H\}\}_0$ of sense-preserving harmonic functions $f=h+\overline\{g\}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h^\{\prime \}(0)-1$ and $g(0)=0=g^\{\prime \}(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal \{P\}_H^0(\alpha )$ and $\mathcal \{G\}_H^0(\beta )$ of functions from $\{\mathcal \{H\}\}_0$ and show that if $f\in \mathcal \{P\}_H^0(\alpha )$ and $F\in \mathcal \{G\}_H^0(\beta )$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha $ and $\beta $ are satisfied. In the second part we study the harmonic sections (partial sums) \[ s\_\{n, n\}(f)(z)=s\_n(h)(z)+\overline\{s\_n(g)(z)\}, \] where $f=h+\overline\{g\}\in \{\mathcal \{H\}\}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline\{g\}\in \{\mathcal \{H\}\}_0$ is a univalent harmonic convex mapping, then $s_\{n, n\}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\ge 2$, and $s_\{n, n\}(f)$ is also convex in the disk $|z|< 1/4$ for $n\ge 2$ and $n\ne 3$. Moreover, we show that the section $s_\{3,3\}(f)$ of $f\in \{\mathcal \{C\}\}_H^0$ is not convex in the disk $|z|<1/4$ but it is convex in a smaller disk.},
author = {Li, Liulan, Ponnusamy, Saminathan},
journal = {Czechoslovak Mathematical Journal},
keywords = {harmonic mapping; partial sum; univalent mapping; convex mapping; starlike mapping; close-to-convex mapping; harmonic convolution; direction convexity preserving map; close-to-convex; convex; starlike and univalent harmonic mappings; partial sum},
language = {eng},
number = {2},
pages = {331-350},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Injectivity of sections of convex harmonic mappings and convolution theorems},
url = {http://eudml.org/doc/280103},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Li, Liulan
AU - Ponnusamy, Saminathan
TI - Injectivity of sections of convex harmonic mappings and convolution theorems
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 331
EP - 350
AB - We consider the class ${\mathcal {H}}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h^{\prime }(0)-1$ and $g(0)=0=g^{\prime }(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal {P}_H^0(\alpha )$ and $\mathcal {G}_H^0(\beta )$ of functions from ${\mathcal {H}}_0$ and show that if $f\in \mathcal {P}_H^0(\alpha )$ and $F\in \mathcal {G}_H^0(\beta )$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha $ and $\beta $ are satisfied. In the second part we study the harmonic sections (partial sums) \[ s_{n, n}(f)(z)=s_n(h)(z)+\overline{s_n(g)(z)}, \] where $f=h+\overline{g}\in {\mathcal {H}}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline{g}\in {\mathcal {H}}_0$ is a univalent harmonic convex mapping, then $s_{n, n}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\ge 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\ge 2$ and $n\ne 3$. Moreover, we show that the section $s_{3,3}(f)$ of $f\in {\mathcal {C}}_H^0$ is not convex in the disk $|z|<1/4$ but it is convex in a smaller disk.
LA - eng
KW - harmonic mapping; partial sum; univalent mapping; convex mapping; starlike mapping; close-to-convex mapping; harmonic convolution; direction convexity preserving map; close-to-convex; convex; starlike and univalent harmonic mappings; partial sum
UR - http://eudml.org/doc/280103
ER -

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