Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Injectivity of sections of convex harmonic mappings and convolution
  theorems   In the article the authors 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 $h(0)=0=h'(0)-1$ 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 $\mathcal{P}_H^0(\alpha)$ 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
$n\geq 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\geq2$
and $n\neq 3$. 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 is shown to be convex
in a smaller disk.

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The categories of this paper are: physics
18
The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin