Complex-Order \(q\)-Bi-Bazilevič-Type Bi-Univalent Functions Associated with a \(q\)-Poisson Convolution Operator and Leaf-Like Domains

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DOI: https://doi.org/10.28924/2291-8639-24-2026-300
Published: Sep 21, 2026
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Abdullah Alsoboh, Yousef Al-Qudah, Ala Amourah, Nashat Ali Almasria, Faisal Al-Sharqi, Maryam K. Rasheed, Sarah Jawad Shoja, Complex-Order \(q\)-Bi-Bazilevič-Type Bi-Univalent Functions Associated with a \(q\)-Poisson Convolution Operator and Leaf-Like Domains, Int. J. Anal. Appl., 24 (2026), 300.

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Abdullah Alsoboh, Yousef Al-Qudah, Ala Amourah, Nashat Ali Almasria, Faisal Al-Sharqi, Maryam K. Rasheed, Sarah Jawad Shoja

Abstract

A new subclass of bi-univalent functions of complex order is formulated by combining a \(q\)-Poisson convolution operator with a Ma–Minda subordination associated with a leaf-shaped image domain. The defining conditions are imposed simultaneously on a function and its inverse and involve a weighted interaction between a Bazilevič-type expression and the corresponding \(q\)-derivative term. This construction produces the class \(\mathsf{QPL}_{\Sigma}^{{r},{q}} ({\epsilon},\pounds,{\gamma};{\Omega}),\) whose parameters provide considerable flexibility in controlling its analytic and geometric structure. The non-emptiness of the proposed family is verified, and several relevant specializations are identified, including leaf-like \(q\)-Poisson bi-starlike and derivative-type subclasses. By expanding the subordinating functions and comparing the resulting coefficient identities for the function and its inverse, upper bounds for the initial Taylor–Maclaurin coefficients \(\left|{a}_{2}\right|\) and \(\left|{a}_{3}\right|\) are established. A Fekete–Szegö inequality for \( \left|{a}_{3}-\varkappa{a}_{2}^{2}\right|, ~ \varkappa\in\mathbb{C},\) is also derived under appropriate nondegeneracy restrictions on the parameters. The results exhibit how the \(q\)-Poisson distribution, complex-order Bazilevič structure, and leaf-like geometry interact in the coefficient theory of bi-univalent functions.

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