The sum of squared logarithms inequality in arbitrary ...

Proceedings in Applied Mathematics and Mechanics, 3 July 2016

The sum of squared logarithms inequality in arbitrary dimensions Lev Borisov1 , Patrizio Neff2 , Suvrit Sra3 , and Christian Thiel2,∗ 1 2

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Department of Mathematics, Rutgers University, 240 Hill Center, Newark, NJ 07102, United States Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, United States

We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ P Rn whose elementary symmetric polynomials satisfy ek (x) ≤ ek (y) (for 1 ≤ k < n) and en (x) = en (y), the inequality i (log xi )2 ≤ P 2 by a suitable extension to the complex plane. In particular, we i (log yi ) holds. Our proof of this inequality follows P show that the function f : M ⊆ Cn → R with f (z) = i (log zi )2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. We conclude by providing applications and wider connections of the SSLI. Copyright line will be provided by the publisher

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Introduction

For x ∈ Rn we denote by ek (x) the k-th elementary symmetric polynomial: ek (x) := Theorem 1.1 (Sum of squared logarithms Pn inequality) LetPnn ∈ N and 2x, y ∈ {1, . . . , n − 1} and en (x) = en (y), then i=1 (log xi )2 ≤ i=1 (log yi ) .

Rn+

P

1≤i1 0 for all k ∈ {1, . . . , n − 1}, provided that the z1 , . . . , zn corresponding to (e1 , . . . , en ) ∈ Rn+ are pairwise different. We obtain the derivatives by means of the chain rule. The exterior as well as the interior derivatives are complex, the composition is then again real, the derivation of the zi with respect to the ek occurs implicitly. In the second step we consider the path esk = (es1 , . . . , esn ) := (1 − s) ek (x) + s ek (y) ,

0 ≤ s ≤ 1.

We can express f in relation to s, this dependence is continuous, because the esk are continuous in s, the zi are continuous Q 2 in ek and f is continuous in zi . For the polynomial h the discriminant is D(h) := 1≤i x2 > . . . > xn we get D(1) 6= 0, the polynomial is not the zero polynomial and D can have not more than a finite number of roots, there can be only a finite number of values s for which we have not shown the positivity of the partial derivatives. The continuity leads to strict monotonicity in s unter the restriction x1 > x2 > . . . > xn . In the third step we get rid of the restriction x1 > x2 > . . . > xn : Because the set of the (e1 , . . . , en ) ∈ Rn+ with pairwise different zi is dense in Rn+ , we can pull apart equal pairs xi = xj a little bit by x0i := xi /(1 + ε) and x0j := xj (1 + ε), such that ek (x0 ) < ek (y) and en (x0 ) = en (y). Due to (log x0i )2 + (log x0j )2 ≥ (log xi )2 + (log xj )2 it follows f (x) ≤ f (x0 ) < f (y). In the fourth step we loosen the requirement on strictness in the ek and transfer the strict monotonicity to simple monotone increase of f in the ek : If ek (x) = ek (y) holds 1 for some k < n, then we consider the sequence xm := z(em ) with em k := ek (x) − m for all k ∈ {1, . . . , n − 1}. We have m m x → x and 0 < ek (x ) < ek (y) for all k < n and with the strict SSLI it follows f (xm ) < f (y) and by continuation we get f (x) ≤ f (y).

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Applications

The SSLI can be applied directly to the quadratic Hencky energy WH (F ) = µ k devn log U k2 +

κ λ [tr(log U )]2 = µ k log U k2 + [log(det U )]2 , 2 2

which was introduced into the theory of nonlinear elasticity in 1929 by H. Hencky [6]. Here,√F ∈ GL+ (n) is the deformation gradient, GL+ (n) is the set of invertible n × n-matrices with positive determinant, U = F T F is the right stretch tensor and devn log U = log U − n1 tr(log U ) · 1 is the deviatoric part of the Hencky strain tensor log U . The material parameters µ, λ with µ > 0 and 3 λ + 2 µ ≥ 0 are called the Lamé constants, while κ ≥ 0 is the bulk modulus. The Hencky energy has recently been characterized by a unique geometric property [7–9]: it measures the squared geodesic distance of F to the special orthogonal group SO(n) with respect to any left-GL(n)-invariant, right-SO(n)-invariant Riemannian metric on GL(n). In terms of the quadratic Hencky energy, Theorem 1.2 can be stated as follows: p √ e = FeT Fe. If det U = det U e and Ik (U ) ≤ Ik (U e ) for all Corollary 4.1 Let F, Fe ∈ GL+ (n) with U = F T F and U e k ∈ {1, . . . , n − 1}, then WH (F ) ≤ WH (F ). Remark 4.2 Corollary 4.1 means that WH satisfies a version of Truesdell’s empirical inequalities [10, pages 158, 171].

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