Differential superordinations and sandwich-type results. (English) Zbl 1316.30024
Joshi, Santosh (ed.) et al., Current topics in pure and computational complex analysis. Most of the selected papers based on the presentations at the international workshop on complex analysis and its applications, Sangli, India, June 11–15, 2012. New Delhi: Birkhäuser/Springer (ISBN 978-81-322-2112-8/hbk; 978-81-322-2113-5/ebook). Trends in Mathematics, 109-146 (2014).
From the introduction: Let \(\Omega\subset\mathbb{C}\), let \(p\) be analytic in the unit disc \(\mathrm{U}=\left\{z\in\mathbb{C}:|z|<1\right\}\), and let \(\psi(r,s,t;z):\mathbb{C}^3{\times}\mathrm{U}\to\mathbb{C}\). In a series of articles, S. S. Miller, P. T. Mocanu and many others have determined properties of functions {\(\psi\)} that satisfy the differential subordination (i.e., the differential inclusion)
\[ \big\{\psi\big(p(z),zp'(z), z^2p''(z); z\big)\,:\, z\in U\big\}\subset \Omega. \]
Reversely, let us consider the dual problem of determining properties of functions \(\psi\) that satisfy the differential superordination
\[ \Omega\subset \big\{\psi\big(p(z),zp'(z), z^2p''(z); z\big)\,:\, z\in U\big\}. \]
Since many of these kind of results can be expressed in terms of subordination and superordination, we will give the required definitions, and note that these results have been first presented in [S. S. Miller and P. T. Mocanu, J. Math. Anal. Appl. 329, No. 1, 327–335 (2007; Zbl 1138.30009)].
For the entire collection see [Zbl 1305.30003].
\[ \big\{\psi\big(p(z),zp'(z), z^2p''(z); z\big)\,:\, z\in U\big\}\subset \Omega. \]
Reversely, let us consider the dual problem of determining properties of functions \(\psi\) that satisfy the differential superordination
\[ \Omega\subset \big\{\psi\big(p(z),zp'(z), z^2p''(z); z\big)\,:\, z\in U\big\}. \]
Since many of these kind of results can be expressed in terms of subordination and superordination, we will give the required definitions, and note that these results have been first presented in [S. S. Miller and P. T. Mocanu, J. Math. Anal. Appl. 329, No. 1, 327–335 (2007; Zbl 1138.30009)].
For the entire collection see [Zbl 1305.30003].
MSC:
| 30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |
Citations:
Zbl 1138.30009References:
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