Recently Milovanovic et al. [Bulletin T.CLIII de l?Aca?mie serbe des sciences arts - 2020.] proved that if P(z) ? Pn with no zeros in |z| < k, k 1, then, |P?(z)| ?P?/2 [n {n(k 1/k + 1) 2/k 1 (|c0| kn|cn|/|c0| kn|cn|)} |P(z)|2 ?P?2], = where c0 c1z ... cnzn is a polynomial of degree n. In this paper, we obtain some results concerning the class polynomials having s?fold zero at origin. These n...

A 2n-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux transformations is established for this Lax integrable system. The Vandermonde and generalized Cauchy determinant formulas lead to a description for deriving expl...

A typical and effective way to construct a higher dimensional integrable equation is to extend the Lax pair for a (1 + 1) dimensional equation known as integrable to higher dimensions. Here we construct an alternative modified KdV equation in (2+1) dimensions by the higherdimensional extension of a Lax pair. And it is shown that this higher dimensional modified KdV equation passes the Painlevé ...

We construct noncommutative extension of U(N) principal chiral model with Wess-Zumino term and obtain an infinite set of local and non-local conserved quantities for the model using iterative procedure of Brezin et.al [17]. We also present the equivalent description as Lax formalism of the model. We expand the fields perturbatively and derive zerothand first-order equations of motion, zero-curv...

Using the symmetry reductions of the self-dual Yang-Mills (SDYM) equations in (2+2) dimensions, we introduce new integrable equations which are nonautonomous versions of the chiral model in (2 + 1) dimensions, generalized nonlinear Schrödinger, Korteweg-de Vries, Toda lattice, Garnier and Euler-Arnold equations. The Lax pairs for all of these equations are derived by the symmetry reductions of ...

Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form. Nonlinear equations arising from the compatibility condition of the Lax pairs in the matrix case include, in particular, Nahm equations, Volterra, Bogoyavlenskii and...

In [8], I. M. Krichever invented the space of matrices parametrizing the cotangent bundle of moduli space of stable vector bundles over a compact Riemann surface, which is named as the Hitchin system after the investigation [5]. We study a necessary and sufficient condition for the linearity of flows on the space of Krichever-Lax matrices in a Lax representation in terms of cohomological classe...