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Zapiski Nauchnykh Seminarov POMI, 1996, Volume 226, Pages 138–154
(Mi znsl3727)
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This article is cited in 16 scientific papers (total in 16 papers)
Convolution properties of some classes of analytic functions
S. Ponnusamya, Vikramaditya Singhb a Department of Mathematics, University of Helsinki, Finland
b Azad Nagar, KANPUR, India
Abstract:
Let $\mathcal A$ denote the class of functions analytic in $|z|<1$ normalized so that $f(0)=0$ and $f'(0)=1$ and let $\mathcal R(\alpha,\beta)\subset\mathcal A$ be the class of functions $f$ such that
$$
\operatorname{Re}[f'(z)+\alpha zF''(z)]>\beta,\qquad\operatorname{Re}\alpha>0,\quad\beta<1.
$$
We determine conditions so that
(i) $f\in\mathcal R(\alpha_1,\beta_1)$, $g\in\mathcal R(\alpha_2,\beta_2)$ implies $f*g$, convolution of $f$ and $g$, is convex;
(ii) $f\in\mathcal R(0,\beta_1)$, $g\in\mathcal R(0,\beta_2)$ implies $f*g$ is starlike;
(iii) $f\in\mathcal A$ satisfying $f'(z)[f(z)/z]^{\mu-1}\prec1+\lambda z$, $\mu>0$, $0<\lambda<1$ is starlike
and
(iv) $f\in\mathcal A$ satisfying $f'(z)+\alpha zf''(z)\prec1+\delta z$, $\alpha>0$, $\delta>0$ is convex or starlike.
Bibl. 16 titles.
Received: 11.09.1995
Citation:
S. Ponnusamy, Vikramaditya Singh, “Convolution properties of some classes of analytic functions”, Zap. Nauchn. Sem. POMI, 226, 1996, 138–154; J. Math. Sci. (New York), 89:1 (1998), 1008–1020
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https://www.mathnet.ru/eng/znsl3727 https://www.mathnet.ru/eng/znsl/v226/p138
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