Deriving Inequalities in the Laguerre-Pólya Class from Properties of Half-Plane Mappings

Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 161)

Abstract

Newton, Euler and many after them gave inequalities for real polynomials with only real zeros. We show how to extend classical inequalities ensuring a guaranteed minimal improvement. Our key is the construction of mappings with bounded image domains such that existing coefficient criteria from complex analysis are applicable. Our method carries over to the Laguerre-Pólya class \(\mathcal{L}\)\(\mathcal{P}\) which contains real polynomials with exclusively real zeros and their uniform limits. The class \(\mathcal{L}\)\(\mathcal{P}\) covers quasi-polynomials describing delay-differential inequalities as well as infinite convergent products representing entire functions, while it is at present not known whether the Riemann ξ-function belongs to this class. For the class \(\mathcal{L}\)\(\mathcal{P}\) we obtain a new infinite family of inequalities which contains and generalizes the Laguerre-Turán inequalities.

Keywords

Coefficient inequalities Reality of zeros Moment problem Hankel determinants Logarithmic derivative 

Mathematics Subject Classification

26D05 30D20 33C45 

References

  1. 1.
    Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd, Edinburgh and London (1965) MATHGoogle Scholar
  2. 2.
    Batra, P.: Value-restricted functions for robust and simultaneous stability. Proc. Appl. Math. Mech. 5, 151–152 (2005) CrossRefGoogle Scholar
  3. 3.
    Batra, P.: Necessary stability conditions for differential-difference equations. Proc. Appl. Math. Mech. 6, 617–618 (2006) CrossRefGoogle Scholar
  4. 4.
    Batra, P.: Some Applications of Complex Analysis to Questions of Stability and Stabilizability. Habilitationsschrift, Hamburg University of Technology (2006) Google Scholar
  5. 5.
    Batra, P.: A family of necessary stability inequalities via quadratic forms. Math. Inequal. Appl. 14(2), 313–321 (2011) MathSciNetMATHGoogle Scholar
  6. 6.
    Boas, R.P.: Entire Functions. Academic Press, New York (1954) MATHGoogle Scholar
  7. 7.
    Borobia, A., Dormido, S.: Three coefficients of a polynomial can determine its instability. Linear Algebra Appl. 338, 67–76 (2001) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Craven, T., Csordas, G.: Karlin’s conjecture and a question of Pólya. Rocky Mt. J. Math. 35(1), 61–82 (2005) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Craven, T., Csordas, G.: Iterated Laguerre and Turán inequalities. J. Inequal. Pure Appl. Math. 3(3), Article 39 (2002) MathSciNetGoogle Scholar
  10. 10.
    Csordas, G., Dimitrov, D.: Conjectures and theorems in the theory of entire functions. Numer. Algorithms 25, 109–122 (2000) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Csordas, G., Norfolk, T., Varga, R.S.: The Riemann hypothesis and the Turan inequalities. Trans. Am. Math. Soc. 296, 521–541 (1986) MathSciNetMATHGoogle Scholar
  12. 12.
    Csordas, G., Varga, R.S.: Moment inequalities and the Riemann hypothesis. Constr. Approx. 4, 175–198 (1988) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dimitrov, D.K.: Counterexamples to a problem of Pólya and to a problem of Karlin. East J. Approx. 4, 479–489 (1998) MathSciNetMATHGoogle Scholar
  14. 14.
    Dimitrov, D.K.: Higher Order Turán Inequalities. Proc. Am. Math. Soc. 126(7), 2033–2037 (1998) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Grommer, J.: Ganze transzendente Funktionen mit lauter reellen Nullstellen. J. Reine Angew. Math. 144, 114–166 (1914). Available via: http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=255746 MATHGoogle Scholar
  16. 16.
    Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003) MATHCrossRefGoogle Scholar
  17. 17.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 2. John Wiley & Sons, New York (1977) MATHGoogle Scholar
  18. 18.
    Hurwitz, A.: Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann. 46, 273–284 (1895) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Jones, W.B., Thron, W.J.: Continued Fractions. Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Company, Reading (1980) MATHGoogle Scholar
  20. 20.
    Krasikov, Ilia: Turán inequalities and zeros of orthogonal polynomials. Methods Appl. Anal. 12, 75–88 (2005) MathSciNetMATHGoogle Scholar
  21. 21.
    Krein, M.: Concerning a special class of entire and meromorphic functions. In: Ahiezer, N.I., Krein, M. (eds.): Some Questions in the Theory of Moments. Translations of Mathematical Monographs, vol. 2, Chap. 3. Am. Math. Soc., Providence (1962). Reprint 1974 Google Scholar
  22. 22.
    Kritikos, N.: Über ganze transzendente Funktionen mit reellen Nullstellen. Math. Ann. 81, 97–118 (1920) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Levin, B.Ja.: Distribution of Zeros of Entire Functions. Translation of Mathematical Monographs, vol. 5, 2nd revised edn. Am. Math. Soc., Providence (1980) Google Scholar
  24. 24.
    Mařik, J.: O polynomech, které mají jen reálné kořeny. Čas. Pěst. Mat. 89, 5–9 (1964) Available online: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN31311157X_0089&DMDID=dmdlog12 MATHGoogle Scholar
  25. 25.
    Pólya, G.: Über die algebraisch-funktionentheoretischen Untersuchungen von J.L.W.V. Jensen. In: Boas, R.P. (ed.) Location of Zeros. Collected Papers of George Pólya, vol. 2, pp. 278–313. MIT Press, Cambridge (1974). Originally in: Kgl. Danske Vidensk. Selsk. Math.-Fys. Medd. 7(17), 3–33 (1927) Google Scholar
  26. 26.
    Pólya, G.: Über die Nullstellen gewisser ganzer Funktionen. Math. Z. 2, 352–383 (1918) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Pólya, G., Szegö, G.: Problems and Theorems in Analysis; Volume II. Revised and enlarged translation of ‘Aufgaben und Lehrsätze aus der Analysis II’. Springer-Verlag, New York, (1976) CrossRefGoogle Scholar
  28. 28.
    Szegö, G.: On an inequality of P. Turán concerning Legendre polynomials. Bull. Am. Math. Soc. 54, 401–405 (1948) MATHCrossRefGoogle Scholar
  29. 29.
    Szegö, G.: Orthogonal Polynomials. Am. Math. Soc., Providence (2003) Google Scholar
  30. 30.
    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford University Press, Oxford (1986). Reprint 1988 MATHGoogle Scholar
  31. 31.
    Tschebotareff, N.: Über die Realität von Nullstellen ganzer transzendenter Funktionen. Math. Ann. 99, 660–686 (1928) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Turán, P.: On the zeros of the polynomials of Legendre. Čas. Pěst. Mat. Fys. 75, 113–122 (1950) Available online: http://gdz.sub.uni-goettingen.de/index.php?id=resolveppn&PPN=PPN31311028X_0075&DMDID=dmdlog89 MATHGoogle Scholar
  33. 33.
    Varga, R.S.: Scientific Computation on Mathematical Problems and Conjectures. SIAM, Philadelphia (1990) MATHCrossRefGoogle Scholar
  34. 34.
    Vidyasagar, M.: Control System Synthesis: A Factorization Approach. MIT Press, Cambridge (1985) MATHGoogle Scholar
  35. 35.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd reprinted edn. The Syndics of the Cambridge University Press, Cambridge, England (1958) Google Scholar
  36. 36.
    Yang, X.: Necessary conditions of Hurwitz polynomials. Linear Algebra Appl. 359, 21–27 (2003) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Yang, X.: Some necessary conditions for Hurwitz stability. Automatica 40, 527–529 (2004) MATHCrossRefGoogle Scholar
  38. 38.
    Youla, D.C., Saito, M.: Interpolation with positive-real functions. J. Franklin Inst. 284(2), 77–108 (1967) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institute f. Computer Technology (E-13)Hamburg University of TechnologyHamburgGermany

Personalised recommendations