Some applications of Dubinin's lemma to rational functions with prescribed poles

Wali, Samad; Shah, W. M.

doi:10.1016/j.jmaa.2017.01.069

Cited by 17 publications

(5 citation statements)

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“…Malik [9]. In addition to these things, we demonstrate how some recently proved results due to S. L. Wali and W. M. Shah, [12] [13], could have been obtained without appealing to the results of Osserman [10] and Dubinin [5]. To Sum up, we asseverate that our work besides improving certain existing estimates also furnishes relatively elementary proofs of the results obtained by S. L. Wali and W. M. Shah in [12] and [13].…”

Section: Resultssupporting

confidence: 78%

“…Although Remark 3 shows that Theorem 1 improves Theorem E, but the bound in (12) demands that all the zeros of R(z) be known beforehand. Since computation of zeros is not always an easy piece of work, therefore in those situations where the zeros of R(z) are unspecified, one may desire to have an improvement which instead of the zeros of R(z) depends upon some coefficients of P (z) in R(z) = P (z) W (z) .…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Inequalities for rational functions with prescribed poles

Rather¹,

Iqbal²,

Dar³

2021

Preprint

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For rational functions R(z) = P (z)/W (z), where P is a polynomial of degree at the most n and W (z) = n J =1 (z − a j ), with |a j | > 1, j ∈ {1, 2, . . . , n}, we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdös-Lax and Turán to rational functions R. In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdös-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we shall demonstrate how some recently obtained results due to S. L. Wali and W. M. Shah, [Some applications of Dubinin's lemma to rational functions with prescribed poles, J. Math.Anal.Appl.450(2017)769-779], could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000Math. 191(12) ( ) 1797Math. 191(12) ( -1807.

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Section: Resultssupporting

confidence: 78%

Section: Resultsmentioning

confidence: 99%

Inequalities for rational functions with prescribed poles

Rather¹,

Iqbal²,

Dar³

2021

Preprint

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“…As a refinement of Theorem 1.1, Wali and Shah [11] in particular proved the following: In this paper we prove some refinements and generalizations of (1.4) and (1.6) besides that we also prove some inequalities for rational functions, which improve the results due to Wali and Shah [11].…”

Section: We Havesupporting

confidence: 62%

Inequalities for Meromorphic Functions Not Vanishing Outside the Disk

2022

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For a polynomial having all zeros inside a unit disk, Dubinin [Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros, Journal of Mathematical Sciences, 143 (2007), 3069-3076] proved:Various refinements and generalizations of this result were proved, we prove some results which include the results of Wali and Shah [Some applications of Dubinin's lemma to rational functions with prescribed poles, J. Math. Anal. Appl., 450, (2017), 769-779] besides the above inequality due to Dubinin as special cases.

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“…The proof of inequalities ( 13) and ( 14) can be found in [15,Lemma 4], [3, Lemma 3 and Theorem 4], and in [3] the geometric function theory was first applied to such a range of problems. Using various versions of the boundary Schwarz lemma, Wali and Shah [24], [26] strengthened (13), (14) in different directions. Kalmykov [13] recently proved two-and three-point distortion theorems for rational functions that generalize some known results on Bernstein-type inequalities for polynomials and rational functions.…”

Section: On Rational Functionsmentioning

confidence: 99%