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On the generalised sum of squared logarithms inequality
- Waldemar Pompe^{1}Email author and
- Patrizio Neff^{2}
https://doi.org/10.1186/s13660-015-0623-6
© Pompe and Neff; licensee Springer. 2015
- Received: 7 October 2014
- Accepted: 11 February 2015
- Published: 18 March 2015
Abstract
Assume \(n\geq2\). Consider the elementary symmetric polynomials \(e_{k}(y_{1},y_{2},\ldots, y_{n})\) and denote by \(E_{0},E_{1},\ldots,E_{n-1}\) the elementary symmetric polynomials in reverse order \(E_{k}(y_{1},y_{2},\ldots,y_{n}):=e_{n-k}(y_{1},y_{2},\ldots,y_{n})= \sum_{i_{1}<\cdots<i_{n-k}} y_{i_{1}}y_{i_{2}}\cdots y_{i_{n-k}}\), \(k\in\{0,1,\ldots,n-1 \}\). Let, moreover, S be a nonempty subset of \(\{0,1,\ldots,n-1\}\). We investigate necessary and sufficient conditions on the function \(f:I\to\mathbb{R}\), where \(I\subset\mathbb{R}\) is an interval, such that the inequality \(f(a_{1})+f(a_{2})+\cdots+f(a_{n})\leq f(b_{1})+f(b_{2})+\cdots+f(b_{n})\) (∗) holds for all \(a=(a_{1},a_{2},\ldots,a_{n})\in I^{n}\) and \(b=(b_{1},b_{2},\ldots,b_{n})\in I^{n}\) satisfying \(E_{k}(a)< E_{k}(b)\) for \(k\in S\) and \(E_{k}(a)=E_{k}(b)\) for \(k\in\{0,1,\ldots,n-1\}\setminus S\). As a corollary, we obtain our inequality (∗) if \(2\leq n\leq4\), \(f(x)=\log^{2}x\) and \(S=\{1,\ldots,n-1\}\), which is the sum of squared logarithms inequality previously known for \(2\le n\le3\).
Keywords
- elementary symmetric polynomials
- logarithm
- matrix logarithm
- inequality
- characteristic polynomial
- invariants
- positive definite matrices
- inequalities
MSC
- 26D05
- 26D07
1 Introduction - the sum of squared logarithms inequality
Definition 1.1
Conjecture 1.2
(Sum of squared logarithms inequality)
Remark 1.3
Recently, the case \(n=2\) was used to verify the polyconvexity condition in nonlinear elasticity [4, 5] for a certain class of isotropic energy functions. For more background information on the sum of squared logarithms inequality we refer the reader to [1].
In this paper we extend the investigation as to the validity of Conjecture 1.2 by considering arbitrary functions f instead of \(f(x)=\log^{2} x\). We formulate this more general problem and we are able to extend Conjecture 1.2 to the case \(n=4\). The same methods should also be useful for proving the statement for \(n=5,6\). However, the necessary technicalities prevent us from discussing these cases in this paper.
In addition, we present ideas which might be helpful in attacking the fully general case, namely arbitrary f and arbitrary n.
2 The generalised inequality
Remark 2.1
The formulation of the above problem has a certain monotonicity structure: we assume that ‘\(E(a)< E(b)\)’ and want to prove that ‘\(F(a)< F(b)\)’. Therefore our idea is to consider a curve y connecting the points a and b, such that \(E(y(t))\) ‘increases’. Then the function \(g(t)=F(y(t))\) should also increase and therefore \(g'(t)>0\) must hold. From this we are able to derive necessary and sufficient conditions on the function f.
This approach motivates the following definition.
Definition 2.2
(b dominates a, \(a\preceq b\))
If \(a\preceq b\), then \(E_{k}(a)=A_{k}(0)\leq A_{k}(1)=E_{k}(b)\), so it follows from Definition 2.2 that a, b satisfy assumption (2.4) with S being the set of all k for which \(A_{k}(t)\) is not a constant function on \([0,1]\).
We are ready to formulate the main results of this section.
Theorem 2.3
A partially reverse statement is also true.
Theorem 2.4
In this respect, we can formulate another conjecture.
Conjecture 2.5
Remark 2.6
In concrete applications of Theorem 2.3 and Theorem 2.4 one would like to know whether condition (2.4) already implies \(a\preceq b\). This is Conjecture 2.5. Unfortunately, we are able to prove Conjecture 2.5 only for \(2\leq n\leq4\), \(I=(0,\infty)\) and \(S\subseteq\{1,2,\ldots,n-1\}\) (see the next section).
Example 2.7
Note also that property (2.5) is not true for \(k=0\). Therefore Theorem 2.3 and Theorem 2.4 for \(f(x)=\log^{2}x\) attain the following formulation.
Corollary 2.8
Remark 2.9
Corollary 2.8 is a weaker statement than Conjecture 1.2 since we assume that \(a\preceq b\). If Conjecture 2.5 is true, then Conjecture 1.2 follows.
Example 2.10
Similarly, the function \(f(x)=x^{p}\) for \(p\in(-1,0)\) satisfies property (2.5) for the set \(S=\{ 1,2,\ldots,{n-1}\}\), because \(p<0\) and among the factors \(k+p-1,k+p-2,\ldots,k+p-(n-1)\) there are exactly \(n-k\) negative ones. On the other hand, property (2.5) is not true for \(k=0\).
Thus, like above, we have the following.
Corollary 2.11
Proof of Theorem 2.3
If S is empty, then \(E_{k}(a)=E_{k}(b)\) for all \(k\in\{0,1,\ldots,n-1\} \) and hence \(a=b\), which immediately implies the inequality. We therefore assume that S is nonempty.
Proof of Theorem 2.4
3 Construction of the connecting curve
In this section we prove that condition (2.4) implies \(a\preceq b\), if \(2\leq n\leq4\), \(I=(0,\infty)\) and \(S\subseteq\{1,2,\ldots,n-1\}\). However, we start with a construction of the desired curve for a general interval I, integer \(n\geq2\) and set \(S\subseteq\{0,1,\ldots,n-1\}\).
For \(a,b\in\Delta_{n}\), we say that \(a< b\), if \(a\neq b\) and \(E_{k}(a)\leq E_{k}(b)\) for all \(k=0,1,\ldots,n-1\). We say that \(a\leq b\), if \(a< b\) or \(a=b\).
Definition 3.1
- (a)
the curve \(y(t)\) starts at a (i.e. \(y(0)=a\), if the curve \(y(t)\) is parametrised by the interval \([0,\varepsilon]\));
- (b)
\(y(t)\in\operatorname{int}(\Delta_{n})\) for all but at most countable many values t;
- (c)
the mappings \(E_{k}(y(t))\) are nondecreasing in t and \(E_{k}(y(t))\leq E_{k}(b)\) for all t and each \(k=0,1,\ldots,n-1\).
Proposition 3.2
Let \(n\geq2\) be a positive integer and let S be a nonempty subset of \(\{0,1,\ldots,n-1\}\). Let, moreover, \(a,b\in\Delta_{n}\) be such that (2.4) holds. Furthermore, suppose that for all \(c\in\Delta_{n}\) with \(a\leq c< b\) the set \(\mathcal{C}(c,b)\) is nonempty. Then \(a\preceq b\).
Proof
Each element (curve) of \(\mathcal{C}(a,b)\) is a (closed) subset of \(\Delta_{n}\). We equip the set \(\mathcal{C}(a,b)\) with the inclusion relation ⊆, obtaining a nonempty partially ordered set \((\mathcal{C}(a,b),\subseteq)\). We are going to show that each chain \(\{y_{i}\}_{i\in\mathcal{I}}\) has an upper bound in \(\mathcal{C}(a,b)\).
Now, by the Kuratowski-Zorn lemma, there exists a maximal element y in \((\mathcal{C}(a,b),\subseteq)\). We show that y is a desired curve connecting the points a and b, which will imply that \(a\preceq b\).
To this end, it is enough to show that, if the curve y is parametrised on \([0,1]\), then \(y(1)=b\). Suppose, to the contrary, that \(y(1)=c\neq b\). Then \(a\leq c< b\), and hence the set \(\mathcal{C}(c,b)\) is nonempty. Thus the curve y can be extended beyond the point c, which contradicts the fact that y is a maximal element in \(\mathcal{C}(a,b)\). This completes the proof of Proposition 3.2. □
From now on assume that \(I=(0,\infty)\) and S is a nonempty subset of \(\{1,2,\ldots,n-1\}\).
In order to prove that (2.4) implies \(a\preceq b\), it suffices to show that the sets \(\mathcal{C}(a,b)\) for \(a,b\in\Delta_{n}\) with \(a< b\) are nonempty. This is implied by the following conjecture, which we will prove later for \(n\leq4\).
Conjecture 3.3
Now we show how Conjecture 3.3 implies that the sets \(\mathcal{C}(a,b)\) are nonempty.
Proposition 3.4
Let n and S be such that the conjecture holds. Let, moreover, \(a,b\in\Delta_{n}\) be such that (2.4) holds. Then the set \(\mathcal{C}(a,b)\) is nonempty.
Proof
Lemma 3.5
Proof
That P has exactly one root in \((-a_{1},0)\) follows immediately from the observation that \(P(-a_{1})<0\), \(P(0)>0\) and \(P'(x)>0\) on \((-a_{1},0)\).
Now we show that Q has exactly one root in \((-\infty,-a_{n})\).
Now we prove that Q has at most two roots in the interval \((-a_{2},-a_{1})\). To the contrary, suppose that Q has at least three roots in \((-a_{2},-a_{1})\). Since \(Q(-a_{2})>0\) and \(Q(-a_{1})>0\), it follows that Q has an even number, and hence at least four, roots in the interval \((-a_{2},-a_{1})\).
The same proof yields an analogous result for even values of n.
Lemma 3.6
Proof
The same proof as that for Lemma 3.5 can be used. □
Now we turn to the proof of Conjecture 3.3 for \(2\leq n\leq 4\) and an arbitrary nonempty set \(S\subseteq\{1,2,\ldots,n-1\}\).
We first make some useful general remarks.
For \(n=2\) the only possibility for the set S is \(\{1\}\) and it is enough to notice that the polynomial \((x+a_{1})(x+a_{2})+t x\) has two distinct real roots for any \(t>0\).
Assume now \(n=3\). Then, in view of the above remarks, we have to consider two cases: (1) \(a_{1}< a_{2}=a_{3}\); (2) \(a_{1}=a_{2}=a_{3}\).
(2) According to the above remarks, \(S=\{1,2\}\). Then the polynomial \((x+a_{1})^{3}+t a_{1}x+t x^{2}\) has three distinct real roots for all sufficiently small \(t>0\).
Assume \(n=4\). In this case we have five possibilities: (1) \(a_{1}=a_{2}< a_{3}<a_{4}\); (2) \(a_{1}< a_{2}=a_{3}<a_{4}\); (3) \(a_{1}< a_{2}=a_{3}=a_{4}\); (4) \(a_{1}=a_{2}< a_{3}=a_{4}\); (5) \(a_{1}=a_{2}=a_{3}=a_{4}\).
Suppose that the polynomial \(Q(x)=(x+a_{1})(x+a_{2})^{3}+A_{1}x+A_{2}x^{2}\) has four real roots. Let \(Q_{1}(x)=(x+a_{1})(x+a_{2})^{3}\) and \(Q_{2}(x)=A_{1}x+A_{2}x^{2}\). Let \(-c\neq a_{2}\) be the root of the polynomial \(Q_{1}'(x)\) and let −d be the root of \(Q_{2}'(x)\).
If \(d< c\), then Q is decreasing on \((-\infty,-c]\), so Q has at most one root in this interval. Therefore Q has at least three roots in the interval \((-c,0)\). Thus \(Q''(x)\) has a root in the interval \((-c,0)\), which is impossible, since \(Q''(x)>0\) on \((-c,0)\).
If \(a_{2}\geq d\geq c\), then Q is increasing on the interval \([-c,0)\) and decreasing on the interval \((-\infty,-d]\), so Q must have at least two roots in the interval \((-d,-c)\). But \(Q(x)<0\) on this interval.
(4) Since the polynomial \((x+a_{1})^{2}(x+a_{3})^{2}+A_{2}x^{2}\) has no real roots, \(1\in S\) or \(3\in S\). Then the polynomial \((x+a_{1})^{2}(x+a_{3})^{2}+t x^{k}\) for \(k=1,3\) has, for all sufficiently small \(t>0\), four distinct real roots.
Thus we have proved the following.
Corollary 3.7
Conjecture 3.3 is true if \(2\leq n\leq4\) and S is an arbitrary nonempty subset of \(\{1,2,\ldots,n-1\}\).
This implies that the sum of squared logarithms inequality (Conjecture 1.2) holds also for \(n=4\).
Corollary 3.8
(Sum of squared logarithms inequality for \(n=4\))
Proof
Use Corollary 3.7 and observe that S may be an arbitrary subset of \(\{1,2,3\}\). □
Corollary 3.9
Proof
Assume first (3.9) holds and let \(a,b\in\Delta_{n}\) satisfy (3.8). Consider any \(c\in\Delta_{n}\) with \(a\leq c< b\). Then the pair c, b satisfies condition (2.4) for some nonempty subset S of T. Therefore by Proposition 3.4, the set \(\mathcal{C}(c,b)\) is nonempty and hence by Proposition 3.2, \(a\preceq b\). Now Theorem 2.3 implies that inequality (2.6) holds.
Conversely, if (2.6) holds for all \(a,b\in\Delta_{n}\) satisfying (3.8), then (2.6) also holds for all \(a,b\in\Delta_{n}\) satisfying condition (2.4) with \(S=T\). Thus Theorem 2.4 implies (3.9). This completes the proof. □
4 Outlook
Our result generalises and extends previous results on the sum of squared logarithms inequality. Indeed, compared to the proof in [1] our development here views the problem from a different angle in that it is not the logarithm function that defines the problem, but a certain monotonicity property in the geometry of polynomials, explicitly stated in Conjecture 3.3.
If one tries to adopt the above proof of Conjecture 3.3 for \(n\leq4\) to the case \(n\geq5\), one has to deal with approximately \(2^{n}\) cases considered separately. Therefore it is clear that the extension to natural numbers n beyond \(n=6\), say, is out of reach with such a method. Instead, a general argument should be found to prove or disprove Conjecture 3.3 for general n. Furthermore, it might be worthwhile to develop a better understanding of the differential inequality condition \((-1)^{n+k}(x^{k} f'(x))^{(n-1)}\leq0\).
Declarations
Acknowledgements
We thank Johannes Lankeit (Universität Paderborn) as well as Robert Martin (Universität Duisburg-Essen) for their help in revising this paper.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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