scholarly journals Estimates for the maximal modulus of rational functions with prescribed poles

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1511-1517
Author(s):  
Gradimir Milovanovic ◽  
Abdullah Mir ◽  
Abrar Ahmad
Keyword(s):  
Rational Function ◽  
Rational Functions ◽  
Schwarz Lemma ◽  
Sharp Estimates ◽  
Regular Functions ◽  
Maximal Modulus ◽  
The Schwarz Lemma

In this paper, we obtain certain sharp estimates for the maximal modulus of a rational function with prescribed poles. The proofs of the obtained results are based on the new version of the Schwarz lemma for regular functions which was suggested by Osserman. The obtained results produce several inequalities for polynomials as well.

SHARP BOUNDS OF SOME COEFFICIENT FUNCTIONALS OVER THE CLASS OF FUNCTIONS CONVEX IN THE DIRECTION OF THE IMAGINARY AXIS

2019 ◽  
Vol 100 (1) ◽  
pp. 86-96 ◽  
Author(s):  
NAK EUN CHO ◽  
BOGUMIŁA KOWALCZYK ◽  
ADAM LECKO
Keyword(s):  
Analytic Functions ◽  
Imaginary Axis ◽  
Schwarz Lemma ◽  
Hankel Determinant ◽  
Sharp Estimates ◽  
The Third ◽  
Carathéodory Functions ◽  
Unit Polydisk ◽  
Class Of Functions ◽  
The Schwarz Lemma

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.

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 .

Rational Interpolation of the Exponential Function

1995 ◽  
Vol 47 (6) ◽  
pp. 1121-1147 ◽  
Author(s):  
L. Baratchart ◽  
E. B. Saff ◽  
F. Wielonsky
Keyword(s):  
Rational Function ◽  
Exponential Function ◽  
Complex Plane ◽  
Rational Interpolation ◽  
Real Interval ◽  
Compact Set ◽  
Rational Approximants ◽  
Sharp Estimates ◽  
Image Position ◽  
Uniformly Bounded

AbstractLet m, n be nonnegative integers and B(m+n) be a set of m + n + 1 real interpolation points (not necessarily distinct). Let Rm,n = Pm,n/Qm.n be the unique rational function with deg Pm,n ≤ m, deg Qm,n ≤ n, that interpolates ex in the points of B(m+n). If m = mv, n = nv with mv + nv → ∞, and mv / nv → λ as v → ∞, and the sets B(m+n) are uniformly bounded, we show that locally uniformly in the complex plane C, where the normalization Qm,n(0) = 1 has been imposed. Moreover, for any compact set K ⊂ C we obtain sharp estimates for the error |ez — Rm,n(z)| when z ∈ K. These results generalize properties of the classical Padé approximants. Our convergence theorems also apply to best (real) Lp rational approximants to ex on a finite real interval.

Sign in / Sign up

Export Citation Format

Share Document