Abstract
Seminal works of Hardy and Littlewood on the growth of analytic functions contain the comparison of the integral means \(M_p(r,f)\), \(M_p(r,f')\), \(M_q(r,f)\). For a complex-valued harmonic function f in the unit disk, using the notation \(\vert \nabla f \vert =(\vert f_z{\vert }^2+\vert f_{\bar{z}}{\vert ^2})^{1/2}\) we explore the relation between \(M_p(r,f)\) and \(M_p(r,\nabla f)\). We show that if \(\vert \nabla f \vert \) grows sufficiently slowly, then f is continuous on the closed unit disk and the boundary function satisfies a Lipschitz condition. We also prove that for \(1 \le p < q \le \infty \), it is possible to give an estimate on the growth of \(M_q(r,f)\) whenever the growth of \(M_p(r,f)\) is known. We notably obtain Baernstein type inequalities for the major geometric subclasses of univalent harmonic mappings such as convex, starlike, close-to-convex, and convex in one direction functions. Some of these results are sharp. A growth estimate and a coefficient bound for the whole class of univalent harmonic mappings are given as well.
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Acknowledgements
The second author thanks Prof. S. Ponnusamy and Prof. Y. Abu-Muhanna for their suggestion to work on Baernstein type theorems for univalent harmonic mappings. The authors thank the anonymous referee for careful reading of the paper and constructive suggestions that improved the presentation of the article.
Funding
The first author thanks Indian Institute of Technology Ropar for providing financial support for his PhD research. The second author is partly supported by the ISIRD Project (F.No. 9-308/2018/IITRPR/4092) of Indian Institute of Technology Ropar.
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Das, S., Sairam Kaliraj, A. Growth of Harmonic Mappings and Baernstein Type Inequalities. Potential Anal 60, 1121–1137 (2024). https://doi.org/10.1007/s11118-023-10081-w
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DOI: https://doi.org/10.1007/s11118-023-10081-w
Keywords
- Integral means
- Hardy spaces
- Univalent functions
- Harmonic functions
- Growth problems
- Baernstein theorem
- Convex
- Starlike
- Close-to-convex