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In this article, we prove a criterion for quasiconformal extension of the integral transforms of analytic functions defined in the unit disk of the complex plane. Thereafter, we prove a sufficient condition for quasiconformal extensibility of sense-preserving harmonic mappings defined in the unit disk and applying this result, we obtain criteria for the quasiconformal extension of various types of integral transforms that are defined for sense-preserving harmonic mappings. We also study the stability of the quasiconformal extension of sense-preserving harmonic mappings.

Quasiconformal extension of integral transforms of analytic and harmonic mappings