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Hardy-Type Inequalities for the Jacobi Weight with Applications

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Abstract

We prove some new Hardy-type inequalities for the Jacobi weight function. The resulting inequalities contain additional terms with the weight functions characteristic of Poincaré–Friedrichs inequalities. One of the constants in the inequality is unimprovable. We apply the inequalities to extending the available classes of univalent analytic functions in simply-connected domains and find univalence conditions in terms of estimates for the Schwartz derivative of an analytic function on the unit disk, the exterior of the unit disk, and the right half-plane.

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Acknowledgment

The author is grateful to Professor Avkhadiev for the valuable suggestions that helped improve this article.

Funding

The author was supported by the Russian Science Foundation (Grant no. 18–11–00115).

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Authors and Affiliations

  1. Kazan (Volga Region) Federal University, Kazan, Russia

    R. G. Nasibullin

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  1. R. G. Nasibullin
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Correspondence to R. G. Nasibullin.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 6, pp. 1313–1333. https://doi.org/10.33048/smzh.2022.63.612

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Nasibullin, R.G. Hardy-Type Inequalities for the Jacobi Weight with Applications. Sib Math J 63, 1121–1139 (2022). https://doi.org/10.1134/S003744662206012X

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  • DOI: https://doi.org/10.1134/S003744662206012X

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