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Abstract

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.

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REFERENCES

  1. A. Aziz and Q. M. Dawood, ‘‘Inequalities for a polynomial and its derivative,’’ J. Approximation Theory 54, 306–313 (1988).

    Article  MathSciNet  Google Scholar 

  2. A. Aziz and B. A. Zargar, ‘‘Some properties of rational functions with prescribed poles,’’ Can. Math. Bull. 42, 417–426 (1999).

    Article  MathSciNet  Google Scholar 

  3. S. Bernstein, ‘‘Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné,’’ Mem. Cl. Sci., Acad. R. Belg. 4, 1–103 (1912).

    Google Scholar 

  4. V. N. Dubinin, ‘‘On an application of conformal maps to inequalities for rational functions,’’ Izv.: Math. 66, 285–297 (2002).

    Article  MathSciNet  Google Scholar 

  5. V. N. Dubinin, ‘‘Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros,’’ J. Math. Sci. 143, 3069–3076 (2007).

    Article  MathSciNet  Google Scholar 

  6. V. K. Jain, ‘‘Generalization of certain well known inequalities for polynomials,’’ Glas. Mat. 32, 45–51 (1997).

    MathSciNet  MATH  Google Scholar 

  7. P. D. Lax, ‘‘Proof of a conjecture of P. Erdös on the derivative of a polynomial,’’ Bull. Am. Math. Soc. 50, 509–513 (1944).

    Article  Google Scholar 

  8. R. Osserman, ‘‘A sharp Schwarz inequality on the boundary,’’ Proc. Am. Math. Soc. 128, 3513–3517 (2000).

    Article  MathSciNet  Google Scholar 

  9. W. M. Shah, ‘‘A generalization of a theorem of Paul Turán,’’ J. Ramanujan Math. Soc. 1, 67–72 (1996).

    MATH  Google Scholar 

  10. P. Turán, ‘‘Über die Ableitung von Polynomen,’’ Compos. Math. 7, 89–95 (1939).

    MATH  Google Scholar 

  11. Xin Li, ‘‘A comparison inequality for rational functions,’’ Proc. Am. Math. Soc. 139, 1659–1665 (2011).

    Article  MathSciNet  Google Scholar 

  12. Xin Li, R. N. Mohapatra and R. S. Rodriguez, ‘‘Bernstein-type inequalities for rational functions with prescribed poles,’’ J. London Math. Soc. 51, 523–531 (1995).

    Article  MathSciNet  Google Scholar 

  13. S. L. Wali and W. M. Shah, ‘‘Some applications of Dubinin’s lemma to rational functions with prescribed poles,’’ J. Math. Anal. Appl. 450, 769–779 (2017).

    Article  MathSciNet  Google Scholar 

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Correspondence to Abdullah Mir.

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Mir, A. Some Inequalities for Rational Functions with Fixed Poles. J. Contemp. Mathemat. Anal. 55, 105–114 (2020). https://doi.org/10.3103/S1068362320020077

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  • DOI: https://doi.org/10.3103/S1068362320020077

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