Inequalities on Generalized Tensor Functions with Diagonalizable and Symmetric Positive Definite Tensors

The main purpose of this paper is to investigate inequalities on symmetric sums of diagonalizable and positive definite tensors. In particular, we generalize the well-known Hlawka and Popoviciu inequalities to the case of diagonalizable and positive definite tensors. As corollaries, we extend Hlawka and Popoviciu inequalities for the combinatorial determinant, permanent and immanant of tensors, and generalized tensor functions.


Introduction
Let G be a subgroup of the symmetric group S I on the set {1, 2, . . ., I} and χ be an irreducible character of G.The G-immanant (also known as the generalized matrix function [20,22]) of A ∈ R I×I is defined as When G = S I , χ(σ) = sgn(σ) yields the determinant and χ(σ) ≡ 1 yields the permanent.Macrcus and Minc [21] revealed a relationship between the generalized matrix function and a function involving the eigenvalues of normal matrices and also considered the relationship between the generalized matrix function and an appropriate function of the singular values of an arbitrary square matrix.Berndt and Sra [3] obtained generalized Hlawka and Popoviciu inequalities for generalized matrix functions with positive definite operators.Huang et al. [14] derived inequalities on no-integer power of products of generalized matrix functions on the sum of positive semi-definite matrices.Chang et al. [5] presented an inequality for Kronecker product (sometimes called tensor product) of positive operators on Hilbert spaces and then applied the inequality to generalized matrix functions.Paksoy et al. [24] obtained some inequalities for generalized matrix functions of positive semi-definite matrices by an embedding and through kronecker products.

INEQUALITIES ON GENERALIZED TENSOR FUNCTIONS
Higher-order equivalents of vectors (first order) and matrices (second order) are called higher-order tensors, multi-dimensional matrices, or multi-way arrays.A tensor is an N -dimensional array of real numbers denoted by script notation A ∈ R I1×I2×•••×IN with entries given by a i1i2...iN ∈ R for i n = 1, 2, . . ., I n and n = 1, 2, . . ., N.
Che et al. [6] defined the generalized tensor function, which is generalized from generalized matrix functions, and indicated that the combinatorial determinant and the permanent of tensors are two special cases of generalized tensor functions.The interested readers can refer to [4,18,23,27] for the combinatorial determinant of tensors and [1,2,10,28,29] for the permanent of tensors.In this paper, we consider the inequalities on generalized tensor functions with diagonalizable and symmetric positive definite tensors, which can be viewed as the generalization from the inequalities on generalized matrix functions with positive definite matrices.
The following notations will be used throughout this paper.We assume that I, J, and N will be reserved to denote the index of upper bounds, unless stated otherwise.We use small letters x, u, v, . . .for scalars, small bold letters x, u, v, . . .for vectors, bold capital letters A, B, C, . . .for matrices, and calligraphic letters A, B, C, . . .for higher-order tensors.These notations are consistently used for lower-order parts of a given structure.For example, the entry with row index i and column index j in a matrix A, i.e., (A) ij , is symbolized by a ij (also (x) i = x i and (A) i1i2...iN = a i1i2...iN ).For some indices, we use MATLAB notation, e.g., the form i = 1 : 2 : 2I − 1 meaning that i increases in steps of 2, taking on only the values 1, 3, . . ., 2I − 1.
The rest of our paper is organized as follows.In Section 2, we introduce some basic definitions and operators about tensors, such as diagonal tensors, symmetric tensors, diagonalizable tensors, Kronecker product of tensors, and so on.In Section 3, we introduce the definition of generalized tensor functions and indicate that the permanent and the combinatorial determinant of tensors are two special cases.In this section, we also consider the basic properties on the Kronecker product of tensors.Three different kinds of inequalities on generalized tensor functions associated with the diagonalizable and symmetric positive definite tensors are considered in Sections 4, 5 and 6, respectively.

Preliminaries
If we set I n := I, then the set of order N dimension I tensors will be denoted by T N,I .The mode-n product [15] of a complex tensor A ∈ T N,I by a matrix B ∈ R I×I , denoted by A × n B, is a tensor C ∈ T N,I , and its entries will be given by where n = 1, 2, . . ., N .
In particular, the mode-n multiplication of a real tensor A ∈ T N,I by a vector z ∈ R I is denoted by For any given tensor A ∈ T N,I and the matrices F, G ∈ R I×I , one has [15] where '•' means the multiplication of two matrices with different integers m and n.
Stat., Optim.Inf.Comput.Vol. 6, December 2018 We introduce the following two notations from [25].For any A ∈ T N,I and x ∈ R I , Ax N −1 is an Idimensional real vector whose ith component is and Ax N is a scalar given by (2) ...i π(N ) where i n = 1, 2, . . ., I and n = 1, 2, . . ., N .The set of all symmetric tensors in T N,I is denoted by ST N,I .A tensor D ∈ T N,I is diagonal [15] if d i1i2...i N = 0, where i 1 , . . ., i N are not identical for i n = 1, 2, . . ., I and n = 1, 2, . . ., N .In particular, for a diagonal tensor D ∈ T N,I , if d ii...i = 1 for i = 1, 2, . . ., I, then D is called the identity tensor, and it is denoted by I. We now introduce the definition of diagonalizable symmetric tensors in T N,I . where More general, we have the following definition.

Definition 2.2
Suppose that A ∈ T N,I .A is called diagonalizable if A can be represented as where B n ∈ R I×I with n = 1, 2, . . ., N , and D ∈ T N,I is diagonal.
In this paper, we suppose that N is even.For a given with all diagonal entries nonzero, and B is nonsingular, then A is positive definite and is called a completely decomposable tensor [17]; c) if D is positive semi-definite with all diagonal entries nonnegative, and B is nonnegative, then A is called a completely positive tensor.For the properties and checkability of completely positive tensors, the interested readers are referred to [11,19,26,30,31] and the references therein.
If Ax N can be decomposed to the sum of squares of polynomial of degree N/2, then Ax N is call a sumof-squares (SOS) polynomial, and the corresponding symmetric tensor A is called an SOS tensor [13].Some important properties of SOS tensors can be referred to [8,9,17] and the references therein.Note that the set of all SOS tensors equal to the set of all completely decomposable tensors with an even N .For two given A, B ∈ ST N,I , the operator inequality A ≥ B denotes the Löwner partial order [3], meaning that A − B is positive definite.

Kronecker product of tensors and Generalized tensor functions
The Kronecker product [16] of A, B ∈ T N,I yields a tensor C = A ⊗ B :≡ AB ∈ T N,I 2 with entries where i n , j n = 1, 2, . . ., I with n = 1, 2, . . ., N .Wherever multiplication is used, we mean Kronecker products (though unusual, we use this notation for esthetic reasons to keep the "visual burden" of our proofs low); thus for arbitrary tensors A and B: where P ≥ 1 and Q ≥ 1 are two given positive integers.Note that this multiplication is nocommutative, so AB ̸ = BA.
Since the entire paper relies extensively on elementary properties of Kronecker products, let us briefly recall these below.

Proposition 3.1
Let A, B, E, F ∈ D N,I be positive definite with an even N .Then we have the following results.
(iv) (A + B) P ≥ A P + B P where P ≥ 1 is any positive integer.

Proof
According to the assumption, we can rewrite A, B, E, F as where D i ∈ D N,I are positive definite and B i ∈ R I×I are nonsingular for i = 1, 2, 3, 4.
[Proof of (i)] According to the Kronecker product of tensors, we have For any nonzero x ∈ R I 2 , we have Hence AB is positive definite.
[Proof of (ii)] Since A ≥ B and E ≥ F, then both A − B and E − F are positive definite.Since then by term (i), AE − BF is positive definite, that is, AE ≥ BF .
[Proof of (iii)] It is trivial.
[Proof of (iv)] Case P = 1 is trivial.Suppose that (A + B) P ≥ A P + B P holds.Now we consider the case of P + 1.Since The second equality holds for term (iii) and the inequality holds for A + B ≥ A, A + B ≥ B and term (i).Let G be a subgroup of the symmetric group S I on the set {1, 2, . . ., I} and χ n (n = 2, 3, . . ., N ) be an irreducible character of G.The G-immanant [6] (also known as the generalized tensor function) of a tensor A ∈ T N,I is defined by According to the definition of generalized tensor functions, it is known that there is a vector x ∈ R I I such that There exist three special cases of generalized tensor functions.One is the combinatorial determinant of the tensor A ∈ T N,I , denoted by det c (A), which is defined as where sgn(π) is the sign of π ∈ S I and a positive integer P satisfies P = (N + 1)/2, for an odd N ; or P = N/2, for an even N.
Another one is the permanent of the tensor A ∈ T N,I , denoted by perm(A), defined as Finally, let λ = (λ 1 , λ 2 , . . . ) be a partition of I and χ λ be the corresponding irreducible representation theoretic character of the symmetric group S I , the immanant of A ∈ T N,I associated with the character χ λ is defined as

Hlawka type inequalities and its generalization
Let f be a convex function on a real interval This inequality is called the functional Hlawka inequality [12].In this section, we generalize Hlawka inequality to the case of generalized tensor functions.
With Proposition 3.1, we are ready to prove our first positive definite tensor Hlawka type inequality.In the matrix case, the following theorem is proved in [3, Theorem 2.1].

Theorem 4.1
Let A, B, C ∈ D N,I be positive definite with an even N .Then for each integer P ≥ 1, we have (4.1)

Proof
The case of P = 1 is trivial and holds with equality.Unsurprisingly, for P = 2, we again have equality, since both sides expend to 2(A 2 + B 2 + C 2 ) + AB + BA + AC + CA + BC + CB.

INEQUALITIES ON GENERALIZED TENSOR FUNCTIONS
We prove the general claim by induction.Assume therefore that (4.1) holds for some integers P ≥ 2.Then, where the inequality follows from the induction hypothesis.The term T is defined as It remains to show that T is positive semi-definite.But this follows immediately upon applying the superadditivity inequality to the first three terms of T and cancelling.Thus, inequality (4.1) is proved.

Remark 4.1
The inequality (4.1) is called strong superadditivity of tensor products; readers familiar with combinatorics may recognize it as supermodularity.
In the special case when N = 2, the following result has been established in [

Proof
Congruence preserves Löwner partial order, so we use (3.2) and (4.1) to derive this theorem.
Suppose that A, B, C ∈ D N,I are positive definite with an even N .Based on Theorem 4.1, for any positive integers L and P , we have In general, we have the following conjecture, which shows inequality involving the non-integer powers of generalized tensor functions.

Conjecture 4.1
Let A, B, C ∈ D N,I be positive definite with an even N .Suppose that d G χ2,...,χN is the generalized tensor function defined on the set T N,I .Then for any More general, we have the following theorem, which can be viewed as a generalization of Theorem 4.1.
Ii be positive definite with an even N and i = 1, 2, . . ., M .Then Consequently, we have

Proof
The proof is similar to one of Theorem 4.1.

Theorem 4.4
Let A i , B i , C i ∈ D N,Ii be positive definite with an even N and i = 1, 2, . . ., M .Suppose that d G,i χ2,...,χN be the generalized tensor function defined on the set T N,Ii .Then we have

Proof
Congruence preserves Löwner partial order, so we use (3.1) and (4.2) to derive this theorem.
The following conjecture is related to Theorem 4.4, as a generalization from Conjecture 4.1.Huang et al. [14] proved that the following conjecture is true for the case of generalized matrix functions on the set of positive (semi-) definite matrices.

Conjecture 4.2
Let the hypothesis be the same as in Theorem 4.4.Then for any r ∈ {1} ∪ [2, ∞), we have

Hlawka type inequalities: generalization
It turns out that the above results can be obtained as corollaries of a more general inequality involving M positive definite tensors.For positive integers M , K and P with K ≤ M , let M = {1, 2, . . ., M } and define the following symmetric sums: The main result is the following theorem.
Theorem 5.1 Let M ≥ 3 and A m ∈ D N,I (m = 1, 2, . . ., M ) be positive definite with an even N .Then for P ∈ N, the inequality holds with an even M , or the inequality holds with an odd M .

Proof
We prove the claim by induction on M .The case that M = 3 is considered in Theorem 4.1.Fix M > 4 and suppose (5.2) or (5.3) holds for all P .We first assume that M is even.We now perform an induction on P .For P = 1, the claim clearly holds as both sides of (5.2) or (5.3) are equal.Assume that the claim holds up to some integer P .Thus, Multiplying (i.e., taking tensor products) both sides by (A 1 + • • • + A M ) on the right and using Proposition 3.1 (ii), we obtain where L and R denote the respective mixed terms.Inequality (5.4) will hold if we show that R ≥ L. In the following, we prove R ≥ L.
For L and R, some tedious multiplications yield that (5.5) Note that the main sums in L and R are only over even and odd sized subsets, respectively.
Stat., Optim.Inf.Comput.Vol. 6, December 2018 The key to the proof is the following regrouping of (5.5), which reveals the underlying inductive structure: and Moreover, we have According to the hypothesis, we conclude that R ≥ L.
If M is odd, the only difference is in the indices of the summations, which now run over j = 1 : 2 : M − 2 for L and j = 2 : 2 : M − 1 for R. We can also conclude that R ≥ L.
The proof is omitted; it can be obtained by following the inductive technique developed above.It should be mentioned that the corresponding inequality for convex functions only holds for certain choices of K, L and T .

Theorem 6.6
Let A m ∈ D N,I (m = 1, 2, . . ., M ) be positive definite with an even N and M ≥ 3 be positive integer.Suppose that d G χ2,...,χN is the generalized tensor function defined on the set T N,I .Then for integers 1 ≤ K < L < T ≤ M , we have Stat., Optim.Inf.Comput.Vol. 6, December 2018

Corollary 4. 1
Let A, B, C ∈ D N,I be positive definite with an even N .Then for each integer P ≥ 1, (A + B + C) P + A P ≥ (A + B) P + (A + C) P .Proof It follows by Proposition 3.1 and inequality (4.1).