Inequalities For Rational Functions Research Articles

Inequalities for Maximum Modulus of Rational Functions with Prescribed Poles

In this paper we prove some results concerning the rational functions with prescribed poles and restricted zeros. These results in fact generalize or strengthen some known inequalities for rational functions with prescribed poles and in turn produce new results besides the refinements of some known polynomial inequalities. Our method of proof may be useful for proving other inequalities for polynomials and rational functions.

Some Inequalities Involving the Derivative of Rational Functions

In this paper, we establish some inequalities involving the modulus of the derivative of rational functions with prescribed poles and restricted zeros. The obtained results generalize some known inequalities for rational functions. Moreover, our results also contain certain known polynomial inequalities.

Inequalities for Rational Functions with Prescribed Poles

Inequalities for Rational Functions with Prescribed Poles

Extremal problems in geometric function theory

This survey is devoted to a number of achievements in the theory of extremal problems in geometric function theory. The approaches to the solution of problems under consideration and the methods used are based on conformal isomorphisms and on the theory of univalent functions developed since the beginning of the 20th century. Results on integral means of conformal mappings of a disc are presented and, in particular, Dolzenko's inequality for rational functions is extended to arbitrary domains with rectifiable boundaries. Investigations in the field of Bohr-type inequalities are described. An emphasis is made on integral inequalities of Hardy and Rellich type, in which the analytic properties of inequalities are intertwined with geometric characteristics of the boundaries of domains. Results related to the solution of the Vuorinen problem on the behaviour of conformal moduli under unlimited dilations of the plane are presented. Formulae for the variation of Robin capacity are obtained. One-parameter families of rational and elliptic functions whose critical values vary in accordance with a prescribed law are characterized. The last results on Smale's conjecture and Smale's dual conjecture are described. Bibliography: 149 titles.

INEQUALITIES PERTAINING TO RATIONAL FUNCTIONS WITH PRESCRIBED POLES

Let \(\Re_n\) be the set of all rational functions of the type \(r(z) = p(z)/w(z),\) where \(p(z)\) is a polynomial of degree at most \(n\) and \(w(z) = \prod_{j=1}^{n}(z-a_j)\), \(|a_j|>1\) for \(1\leq j\leq n\). In this paper, we set up some results for rational functions with fixed poles and restricted zeros. The obtained results bring forth generalizations and refinements of some known inequalities for rational functions and in turn produce generalizations and refinements of some polynomial inequalities as well.

A Kreiss Type Inequality for Rational Functions on General Domains

A Kreiss Type Inequality for Rational Functions on General Domains

Refinement of some Bernstein type inequalities for rational functions

Refinement of some Bernstein type inequalities for rational functions

Generalizations of some inequalities for rational functions with prescribed poles and restricted zeros

Generalizations of some inequalities for rational functions with prescribed poles and restricted zeros

Rational difference equations with parameter, boundedness and periodic solutions

The boundedness of solutions of rational difference equations that contain a parameter is investigated. For this purpose, we prove and use a basic inequality for rational functions. The existence of periodic solutions is also treated. Some examples are given to illustrate our results.

Inequalities for complex rational functions

UDC 517.5 For the rational function having all its zeros in it is known thatwhere is a positive integer, and In this paper, we improve the above mentioned inequality for the rational function with all zeros in and a zero of order at the origin. Our main results refine and generalize some known rational inequalities.

Inequalities for the Derivative of Rational Functions with Prescribed Poles

In this paper, we consider a class of rational functions rsz of degree mn where sz is a polynomial of degree m and establish some inequalities for rational functions with prescribed poles which generalize and refine the result of I. Qasim and A. Liman.

Some Inequalities for Rational Functions with Fixed Poles

By using lemmas of Dubinin and Osserman some results for rational functions with fixed poles and restricted zeros are proved. The obtained results strengthen some known results for rational functions and, in turn, produce refinements of some polynomial inequalities as well.

Some inequalities for maximum modulus of rational functions

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.

Inequalities concerning rational functions with prescribed poles

In this paper, we establish some inequalities for rational functions with prescribed poles which generalize and refine the results of Li, Mohapatra and Rodriguez, Aziz and Shah, Qasim and Liman and others. A result of O’Hara and Rodriguez for self-inversive polynomials is also extended to rational functions by one of our result as a special case.

Inequalities For rational functions with prescribed poles

Let Rn be the set of all rational function of the type r(z) = p(z) / W(z), where p(z) be a polynomial of degree not exceeding n and let where |aj |>1, j = 1, 2, … , n. If r, s = ∊ Rn and s(z) n zeros in |z| ≤ 1, then Xin Li [Proc. Amer. Math. Soc.,Vol. 139, No. 5 (2011), pp. 1659–1665] proved thatWhere |r(z)| ≤ | s(z) for |z| =1 and |β| ≤ 1.In this paper, we consider a class of rational function r(z) with restricted zeros and present certain generalizations of above inequality.

Applications of the Schwarz lemma to inequalities for rational functions with prescribed poles

By using a boundary refinement of the classical Schwarz lemma some results for the rational functions with prescribed poles and restricted zeros have been proved. The estimates obtained strengthen some known results recently proved by Li, Mohapatra and Rodriguez, Aziz, Shah and others. This in turn improves some polynomial inequalities proved earlier by Lax, Turan etc.

Bernstein type inequalities for rational functions

In this paper, we consider a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree m and prove some sharp results concerning to Bernstein type inequalities for rational functions.

An L 2 inequality for rational functions

We prove an L 2 inequality for rational functions having all their poles in the open unit disc and as a corollary obtain an inequality for functions belonging to the Hardy class H 2 of power series ∑a j z j with ∑|a j |2 < ∞.

A Turán type inequality for rational functions with prescribed poles

A Turán type inequality for rational functions with prescribed poles

A Turán type inequality for rational functions with prescribed poles

By employing a novel idea and simple techniques, we substantially generalize the Turan type inequality for rational functions with real zeros and prescribed poles established by Min [5] to include Lp spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.