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In this paper, we establish a Jackson q-difference analogue of Jack's lemma and its higher-order extension by Miller and Mocanu, which serve as cornerstone tools in geometric function theory and differential subordination. Applying this foundation, we establish q-counterparts of Nunokawa's boundary theorem and the Miller-Mocanu subordination lemma, and deduce applications to circular enclosure bounds. All obtained results naturally recover their classical counterparts in the limiting case as q → 1⁻.
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Ali et al. (2026) studied this question.
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