Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 2. P. 38 – 47. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-2-38-47
Research Article
Full text in English
MSC 46L05, 30C10

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C^{\ast}-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture

K.M. Krishna^{\ast}

School of Mathematics and Natural Sciences, Chanakya University Global Campus, NH-648, Haraluru Village, Devanahalli Taluk, Bengaluru North District, Karnataka State, 562110, India

Abstract. We develop C^{\ast}-algebraic analogues of two central problems in complex analysis: the Smale Mean Value Conjecture and the Dubinin-Sugawa Dual Mean Value Conjecture. Building on Smale’s classical mean value theorem and its dual formulation, we extend these conjectures to the setting of commutative C^{\ast}-algebras. The proposed framework defines algebraic differentiation for polynomials over C^{\ast}-algebras and formulates both standard and strong versions of the conjectures. We further introduce higher-order and dynamical extensions, including a C^{\ast}-algebraic analogue of the Miles-Leighton-Pilgrim dynamics conjecture. Rigorous verification is provided for degree two polynomials, establishing the validity of these conjectures in this foundational case. The results highlight the deep interplay between operator-algebraic structures and analytic inequalities, offering new perspectives on long-standing problems in polynomial theory. This work opens pathways for generalizations to Banach and von Neumann algebras, while emphasizing the challenges posed by noncommutative settings.

Key words: C^{\ast}-algebra, Smale mean value theorem, Smale mean value conjecture, Dubinin-Sugawa dual mean value theorem, Dubinin-Sugawa dual mean value conjecture.

Received: 25.06.2026; Revised: 21.07.2026; Accepted: 08.08.2026; First online: 14.09.2026

For citation. Krishna K.M. C^{\ast}-algebraic Smale mean value conjecture and Dubinin-Sugawa dual mean value conjecture. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 2, 38-47. EDN: NMXQAM. https://doi.org/10.26117/2079-6641-2026-54-2-38-47.

Funding. The study was conducted without the support of foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for the submission of the final version of the article for publication.

^{\ast}Correspondence: E-mail: kmaheshak@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Krishna K.M., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

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Information about the author

Krishna Mahesh – PhD (Phys. & Math.), Assistant Professor, School of Mathematics and Natural Sciences, Chanakya University Global Campus, Haraluru, India, ORCID 0000-0003-4872-8634.

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