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.
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- 1.
Csordas and Dimitrov pointed out in [10] that the name Euler-Laguerre-Pólya-Schur-Turán inequalities would reflect the major contributors.
References
Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd, Edinburgh and London (1965)
Batra, P.: Value-restricted functions for robust and simultaneous stability. Proc. Appl. Math. Mech. 5, 151–152 (2005)
Batra, P.: Necessary stability conditions for differential-difference equations. Proc. Appl. Math. Mech. 6, 617–618 (2006)
Batra, P.: Some Applications of Complex Analysis to Questions of Stability and Stabilizability. Habilitationsschrift, Hamburg University of Technology (2006)
Batra, P.: A family of necessary stability inequalities via quadratic forms. Math. Inequal. Appl. 14(2), 313–321 (2011)
Boas, R.P.: Entire Functions. Academic Press, New York (1954)
Borobia, A., Dormido, S.: Three coefficients of a polynomial can determine its instability. Linear Algebra Appl. 338, 67–76 (2001)
Craven, T., Csordas, G.: Karlin’s conjecture and a question of Pólya. Rocky Mt. J. Math. 35(1), 61–82 (2005)
Craven, T., Csordas, G.: Iterated Laguerre and Turán inequalities. J. Inequal. Pure Appl. Math. 3(3), Article 39 (2002)
Csordas, G., Dimitrov, D.: Conjectures and theorems in the theory of entire functions. Numer. Algorithms 25, 109–122 (2000)
Csordas, G., Norfolk, T., Varga, R.S.: The Riemann hypothesis and the Turan inequalities. Trans. Am. Math. Soc. 296, 521–541 (1986)
Csordas, G., Varga, R.S.: Moment inequalities and the Riemann hypothesis. Constr. Approx. 4, 175–198 (1988)
Dimitrov, D.K.: Counterexamples to a problem of Pólya and to a problem of Karlin. East J. Approx. 4, 479–489 (1998)
Dimitrov, D.K.: Higher Order Turán Inequalities. Proc. Am. Math. Soc. 126(7), 2033–2037 (1998)
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
Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003)
Henrici, P.: Applied and Computational Complex Analysis, vol. 2. John Wiley & Sons, New York (1977)
Hurwitz, A.: Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann. 46, 273–284 (1895)
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)
Krasikov, Ilia: Turán inequalities and zeros of orthogonal polynomials. Methods Appl. Anal. 12, 75–88 (2005)
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
Kritikos, N.: Über ganze transzendente Funktionen mit reellen Nullstellen. Math. Ann. 81, 97–118 (1920)
Levin, B.Ja.: Distribution of Zeros of Entire Functions. Translation of Mathematical Monographs, vol. 5, 2nd revised edn. Am. Math. Soc., Providence (1980)
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
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)
Pólya, G.: Über die Nullstellen gewisser ganzer Funktionen. Math. Z. 2, 352–383 (1918)
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)
Szegö, G.: On an inequality of P. Turán concerning Legendre polynomials. Bull. Am. Math. Soc. 54, 401–405 (1948)
Szegö, G.: Orthogonal Polynomials. Am. Math. Soc., Providence (2003)
Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford University Press, Oxford (1986). Reprint 1988
Tschebotareff, N.: Über die Realität von Nullstellen ganzer transzendenter Funktionen. Math. Ann. 99, 660–686 (1928)
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
Varga, R.S.: Scientific Computation on Mathematical Problems and Conjectures. SIAM, Philadelphia (1990)
Vidyasagar, M.: Control System Synthesis: A Factorization Approach. MIT Press, Cambridge (1985)
Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd reprinted edn. The Syndics of the Cambridge University Press, Cambridge, England (1958)
Yang, X.: Necessary conditions of Hurwitz polynomials. Linear Algebra Appl. 359, 21–27 (2003)
Yang, X.: Some necessary conditions for Hurwitz stability. Automatica 40, 527–529 (2004)
Youla, D.C., Saito, M.: Interpolation with positive-real functions. J. Franklin Inst. 284(2), 77–108 (1967)
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Batra, P. (2012). Deriving Inequalities in the Laguerre-Pólya Class from Properties of Half-Plane Mappings. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_5
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