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Exponential polynomial inequalities and monomial sum inequalities in \(\text{p}\)-Newton sequences. (English) Zbl 1413.11053

Let a sequence \(c:c_0,\dots,c_n\) of positive real numbers be a p-Newton sequence, i.e., \(c_{i-1}c_{i+1}\leq c_i^2\) for \(i=1,\dots,n-1\). A monomial in \(c\) is \(c^a=c_0^{a_0}\cdots c_n^{a_n}\), where \(a_0,\dots,a_n\) are nonnegative real numbers. When does \[ c^a\leq c^b \] hold for all \(c\)? The three first-named authors with O. Walch [in: Notions of positivity and the geometry of polynomials. Basel: Birkhäuser. Trends in Mathematics, 275–282 (2011; Zbl 1250.15025)] answered this and noted that the result can be interpreted as determinantal inequalities in certain classes of matrices.
If \(a_0,\dots,a_n\) are nonnegative integers, then \(c^a\) is simple. Consider a sum of \(m\) simple monomials in \(c\). Writing its \(h\)’th summand as \(c^{a(h)}=c_{i_{h1}}\cdots c_{i_{hk_h}}\), this sum is \[ c^{a,m}=c^{a(1)}+\dots+c^{a(m)}. \] When does \[ c^{a,m}\leq c^{b,m} \] hold for all \(c\)? The three first-named authors [Linear Algebra Appl. 439, No. 7, 2038–2056 (2013; Zbl 1305.15022)] settled the case \(m=k_1=k_2=2\). The present authors settle the case \(m=3\), \(k_1=k_2=k_3=2\), \(n\leq 3\).
An exponential polynomial is a function \[ f(r)=r^{p_1}+\dots+r^{p_n}, \] where \(r>0\) and \(p_1,\dots,p_n\) are real numbers. The authors find inequalities involving these polynomials, and apply them in solving the above problem.

MSC:

11B83 Special sequences and polynomials
11C20 Matrices, determinants in number theory
15A15 Determinants, permanents, traces, other special matrix functions
15A45 Miscellaneous inequalities involving matrices
26D20 Other analytical inequalities
15A18 Eigenvalues, singular values, and eigenvectors
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References:

[1] C. R. Johnson, C. Marijuán, M. Pisonero: Inequalities for linear combinations of monomials in p-Newton sequences. Linear Algebra Appl. 439 (2013), 2038–2056. · Zbl 1305.15022
[2] C. R. Johnson, C. Marijuán, M. Pisonero: Matrices and spectra satisfying the Newton inequalities. Linear Algebra Appl. 430 (2009), 3030–3046. · Zbl 1189.15009
[3] C. R. Johnson, C. Marijuán, M. Pisonero, O. Walch: Monomials inequalities for Newton coefficients and determinantal inequalities for p-Newton matrices. Trends in Mathematics, Notions of Positivity and the Geometry of Polynomials (Brändén, Petter et al., eds.). Springer, Basel, 2011, pp. 275–282. · Zbl 1250.15025
[4] I. Newton: Arithmetica Universalis: Sive de Compositione et Resolutione Arithmetica Liber (William Whiston, ed.). London, 1707.
[5] X. Wang: A simple proof of Descartes’s rule of signs. Am. Math. Mon. 111 (2004), 525–526. · Zbl 1080.26507
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