This paper discusses the behavior of the second-order modes (Hankel singular values) of linear discrete-time systems under bounded-real transformations, where the transformations are given by arbitrary transfer functions with magnitude bounded by unity. Our main result reveals that the values of the second-order modes are decreased under any of the above-mentioned transformations. This result i...

This paper discusses the behavior of the second-order modes (Hankel singular values) of linear continuous-time systems under typical frequency transformations, such as lowpass-lowpass, lowpasshighpass, lowpass-bandpass, and lowpass-bandstop transformations. Our main result establishes the fact that the second-order modes are invariant under any of these typical frequency transformations. This m...

Abstract Sharp bounds are given for the second Hankel determinant of logarithmic coefficients strongly starlike and convex functions.

The main purpose of this paper is to obtain an upper bound for the second Hankel determinant functions belonging a subclass bi-univalent in open unit disk complex plane. Furthermore, presented results work improve or generalize recent works other authors.

This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions, including 4-term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal pol...

In this paper, bounds are established for the second Hankel determinant of logarithmic coefficients normalised analytic functions satisfying certain differential inequality.

Classically, formal orthogonal polynomials are studied with respect to a linear functional, which gives rise to a moment matrix with a Hankel structure. Moreover, in most situations, the moment matrix is supposed to be strongly regular. This implies a number of algebraic properties which are well known, like for example the existence of a three-term recurrence relation (characterised by a tridi...