scholarly journals Some inequalities for maximum modulus of rational functions

2019 ◽  
Vol 73 (1) ◽  
pp. 33
Author(s):  
Abdullah Mir
Keyword(s):  
Unit Circle ◽  
Rational Function ◽  
Complex Plane ◽  
Maximum Modulus ◽  
Rational Functions ◽  
Restricted Zeros

In this paper, we establish some inequalities for rational functions with prescribed poles and restricted zeros in the sup-norm on the unit circle in the complex plane. Generalizations and refinements of rational function inequalities of Govil, Li, Mohapatra and Rodriguez are obtained.

Some Properties of Rational Functions with Prescribed Poles

1999 ◽  
Vol 42 (4) ◽  
pp. 417-426 ◽  
Author(s):  
Abdul Aziz-Ul-Auzeem ◽  
B. A. Zarger
Keyword(s):  
Unit Circle ◽  
Rational Function ◽  
Rational Functions ◽  
Sharp Result

AbstractLet P(z) be a polynomial of degree not exceeding n and let where |aj| > 1, j = 1, 2,…,n. If the rational function r(z) = P(z)/W(z) does not vanish in |z| < k, then for k = 1 it is known thatwhere B(Z) = W*(z)/W(z) and . In the paper we consider the case when k > 1 and obtain a sharp result. We also show thatwhere , and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

scholarly journals Some inequalities for self-inversive rational functions with prescribed poles

2020 ◽  
Vol 107 (121) ◽  
pp. 109-116
Author(s):  
Abdullah Mir
Keyword(s):  
Unit Circle ◽  
Complex Plane ◽  
Rational Functions ◽  
Polynomial Inequalities

We establish some inequalities for self-inversive rational functions with prescribed poles in the sup-norm on the unit circle in the complex plane. Generalizations of polynomial inequalities of Malik and O?Hara and Rodriguez are obtained for such rational functions

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

The Zero Distribution of Orthogonal Rational Functions on the Unit Circle

1997 ◽  
Vol 49 (4) ◽  
pp. 810-819
Author(s):  
K. Pan
Keyword(s):  
Unit Circle ◽  
Potential Theory ◽  
Rational Functions ◽  
Orthogonal Rational Functions ◽  
Zero Distribution

AbstractRational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied. We use the potential theory to discuss the zeros distribution for the orthogonal rational functions.

Possibility of Uniform Rational Approximation in the Spherical Metric

1976 ◽  
Vol 28 (1) ◽  
pp. 112-115 ◽  
Author(s):  
P. M. Gauthier ◽  
A. Roth ◽  
J. L. Walsh
Keyword(s):  
Compact Subset ◽  
Rational Approximation ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Finite Complex ◽  
Finite Plane ◽  
Extended Plane ◽  
Chordal Metric

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

scholarly journals Hyponormal Toeplitz operators on H2(T) with polynomial symbols

1996 ◽  
Vol 144 ◽  
pp. 179-182 ◽  
Author(s):  
Dahai Yu
Keyword(s):  
Functional Equation ◽  
Hardy Space ◽  
Closed Form ◽  
Unit Circle ◽  
Toeplitz Operator ◽  
Explicit Expression ◽  
Complex Plane ◽  
Toeplitz Operators ◽  
Computing Process

Let T be the unit circle on the complex plane, H2(T) be the usual Hardy space on T, Tø be the Toeplitz operator with symbol Cowen showed that if f1 and f2 are functions in H such that is in Lø, then Tf is hyponormal if and only if for some constant c and some function g in H∞ with Using it, T. Nakazi and K. Takahashi showed that the symbol of hyponormal Toeplitz operator Tø satisfies and and they described the ø solving the functional equation above. Both of their conditions are hard to check, T. Nakazi and K. Takahashi remarked that even “the question about polynomials is still open” [2]. Kehe Zhu gave a computing process by way of Schur’s functions so that we can determine any given polynomial ø such that Tø is hyponormal [3]. Since no closed-form for the general Schur’s function is known, it is still valuable to find an explicit expression for the condition of a polynomial á such that Tø is hyponormal and depends only on the coefficients of ø, here we have one, it is elementary and relatively easy to check. We begin with the most general case and the following Lemma is essential.

scholarly journals Finite-Gap CMV Matrices: Periodic Coordinates and a Magic Formula

2020 ◽  
Author(s):  
Jacob S Christiansen ◽  
Benjamin Eichinger ◽  
Tom VandenBoom
Keyword(s):  
Unit Circle ◽  
Functional Model ◽  
Rational Functions ◽  
Almost Periodic ◽  
Absolutely Continuous ◽  
Magic Formula ◽  
Carathéodory Functions ◽  
Möbius Transforms ◽  
Rotational Phase ◽  
Isospectral Torus

Abstract We prove a bijective unitary correspondence between (1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum ${\textsf{E}}$ and (2) special periodic block-CMV matrices satisfying a Magic Formula. This latter class arises as ${\textsf{E}}$-dependent operator Möbius transforms of certain generating CMV matrices that are periodic up to a rotational phase; for this reason we call them “MCMV.” Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional model in analog to that of GMP matrices. As a corollary of our construction we resolve a conjecture of Simon; namely, that Caratheodory functions associated to such CMV matrices arise as quadratic irrationalities.

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