On the inequalities of Turán, Bernstein and Erdős–Lax in quaternionic setting

• Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2021-10, Vol.115 (4), Article 187
• Main Authors: Coroianu, Lucian, Gal, Sorin G.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Inequalities; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Original Paper; Polynomials; Theoretical
• ISSN: 1578-7303; 1579-1505
• DOI: 10.1007/s13398-021-01126-z

In this paper we study the extensions of the classical inequalities of Bernstein, Erdős–Lax and Turán’s inequalities, from complex polynomials to three kinds of quaternionic polynomials, called by us as follows: quaternionic canonical generalized polynomials (shortly (QCG)-polynomials), quaternionic matrix polynomials (shortly (QM)-polynomials) and quaternionic unilateral polynomials (shortly (QU)-polynomials). In the case of the Bernstein and Erdős–Lax inequalities, results in the larger space of quaternionic matrix polynomials that includes the space of quaternionic canonical generalized polynomials (via an embedding) are obtained. In the case of Turán’s inequality, although analogue results to the complex case in some particular large classes of polynomials are obtained, we prove that, in general, the inequality does not hold for none of the three kinds of quaternionic polynomials mentioned above. In the case of quaternionic unilateral polynomials, due to the complicated calculations, the counterexample is produced by performing a MATLAB program.