# Some Propositions on Generalized Nevanlinna Functions of the ClassNk

Authors: Yan-Ping Song; Hui-Feng Hao; Yong-Jian Hu; Gong-Ning Chen
Journal: Advances in Mathematical Physics
 (2014)

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## Abstract

Some propositions on the generalized Nevanlinna functions are derived. We indicate mainly that (1) the negative inertia index of a Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function in the classNκ does not exceed κ. This leads to an equivalent definition of a generalized Nevanlinna function; (2) if a generalized Nevanlinna function in the class Nκ has a uniform asymptotic expansion at a real point α or at infinity, then the negative inertia index of the Hankel matrix constructed with the partial coefficients of that asymptotic expansion does not exceed κ. Also, an explicit formula for the negative index of a real rational function is given by using relations among Loewner, Bézout, and Hankel matrices. These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the class Nκ based on the so-called Hankel vector approach.


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## Body

## 1. Introduction

LetC, R, and N be the complex plane, the real axis in the complex plane, and the set of nonnegative integers, respectively. Denote by C+ and N+ the open upper half complex plane and the set of positive integers, respectively. For a meromorphic function f(z) of one complex variable, its domain of holomorphy of f(z) is written as D(f). Denote by ν(A) the number of negative eigenvalues of a Hermitian matrix A.In 1977, Krein and Langer [1, Section 1] introduced the function class Nκ of all generalized Nevanlinna functions with negative index κ∈N.Recall that a meromorphic functionf(z) in C∖R is called a generalized Nevanlinna function with negative index κ∈N, if it satisfies the symmetric condition: f(z¯)=f(z)¯ for all z∈D(f), and, for each choice of m∈N+, distinct points z1,…,zm∈C+∩D(f), the Hermitian matrix
(1)Pf(z1,…,zm):=(f(zi)-f(zj)¯zi-z¯j)i,j=1m
has at most κ negative eigenvalues (counted with multiplicities) and for at least one such choice exactly κ negative eigenvalues; that is,
(2)max⁡⁡{ν(Pf(z1,…,zm))∣z1,…,zm∈C+∩D(f),m∈N+ν(Pf(z1,…,zm))}=κ.In particular, forκ=0, the class N0 is actually the set of all Nevanlinna functions which are holomorphic in C∖R such that f(z¯)=f(z)¯ and Im⁡f(z)/Im⁡z≥0 for all z∈C∖R. It is well known that (see, e.g., [2, 3]) each f(z)∈N0 admits an integral representation of the form:
(3)f(z)=αz+β+∫R‍(1u-z-u1+u2)dσ(u),∀z∈C∖R,
in which α≥0, β∈R, and σ(u) is a positive measure on R satisfying
(4)∫R‍(1+u2)-1dσ(u)<+∞.By definition, we check easily that (see, e.g., [1, 4]) f(z)∈Nκ and f(z)≢0 if and only if f(-z-1)∈Nκ, and also, if and only if -f(z)-1∈Nκ. Moreover, if f(z)∈Nκ1 and g(z)∈Nκ2, then f(z)+g(z)∈Nκ, where κ≤κ1+κ2.Forα∈R, the symbol πɛ(α) stands for the sector in C+ with the vertex α:
(5)πɛ(α)={z∣ɛ≤arg(z-α)≤π-ɛ}(0<ɛ<π2).
The notation ∠lim⁡z→αf(z)=A means that the function f has a nontangential limit A at t∈R; that is, f(z) tends to A as z tends to α in each sector πɛ(α). Also, the notation ∠lim⁡z→∞f(z)=A stands for the nontangential limit of f as z→∞ in each sector πɛ(0).Now, we introduce the notions of generalized zeros and generalized poles of nonpositive type of a generalized Nevanlinna functionf(z)∈Nκ (see [5–8] for details).A pointα∈R is called a generalized zero of nonpositive type of multiplicity π of f(z)∈Nκ if
(6)-∞<∠lim⁡z→α(z-α)-2π+1f(z)≤0,0<∠lim⁡z→α(z-α)-2π-1f(z)≤+∞.
Similarly, the point ∞ is called a generalized zero of nonpositive type of multiplicity π of f(z)∈Nκ if
(7)0≤∠lim⁡z→∞z2π-1f(z)<+∞,-∞≤∠lim⁡z→∞z2π+1f(z)<0.
A point β∈R is called a generalized pole of nonpositive type of multiplicity π of f(z)∈Nκ if
(8)0<∠lim⁡z→β(z-β)2π-1f(z)≤+∞,-∞<∠lim⁡z→β(z-β)2π+1f(z)≤0.
Similarly, the point ∞ is called a generalized pole of nonpositive type of multiplicity π of f(z)∈Nκ if
(9)-∞≤∠lim⁡z→∞z-2π+1f(z)<0,0≤∠lim⁡z→∞z-2π-1f(z)<+∞.Remark that the point∞ is a generalized zero of nonpositive type of multiplicity π of f(z)∈Nκ if and only if 0 is a generalized zero of nonpositive type of multiplicity π of f(-z-1)∈Nκ and that the point ∞ is a generalized pole of nonpositive type of multiplicity π of f(z)∈Nκ if and only if the point ∞ is a generalized zero of nonpositive type of multiplicity π of -f(z)-1∈Nκ (see [8] for the proof of sufficiency).In 1981, Krein and Langer [7] showed that the total multiplicity of poles (zeros, respectively) in C+ and generalized poles (generalized zeros, respectively) of nonpositive type in R∪{∞} of f(z)∈Nκ is equal to κ, which implies that each f(z)∈Nκ has finite number of zeros and poles in C+. As a corollary of this result, we infer ([9, Proposition 3.2]) that if f(z)∈Nκ1 and g(z)∈Nκ2 have no common poles in C+ and common generalized poles of nonpositive type in R∪{∞}, then f(z)+g(z)∈Nκ1+κ2.By means of an operator representation off(z)∈Nκ, in 1985, Daho and Langer [9, Proposition 2.1] derived an integral representation of f(z)∈Nκ as follows:
(10)f(z)=1∏j=1κ1‍(z-αj)(z-α¯j)×((z2+y02)κ∫R‍uz+y02u-zdσ(u)+∑i=02κ+1‍bizi),
where σ(u) is a positive measure on R satisfying ∫R‍dσ(u)<+∞, α1,…,ακ1 are either the poles of f(z) in C+ or its generalized poles of nonpositive type on R, counted according to their multiplicities, b0,b1,…,b2κ+1∈R with b2κ+1≥0, and y0 is a given positive number such that iy0∈D(f).In 2000, Dijksma et al. [6] gave an interesting factorization for each function f(z)∈Nκ. More precisely, if α1,…,ακ1 are all the zeros in C+ and the generalized zeros of nonpositive type of f(z)∈Nκ on R and β1,…,βκ2 are all the poles in C+ and the generalized poles of nonpositive type of f(z)∈Nκ on R, all counted according to their multiplicities, then
(11)f(z)=∏i=1κ1‍(z-αi)(z-α¯i)∏i=1κ2‍(z-βi)(z-β¯i)f0(z),
where f0(z)∈N0. The factorization result (11) of f(z)∈Nκ, together with the integral representation (3) of f0(z)∈N0, gives rise to the following result. Each f(z)∈Nκ admits an integral representation
(12)f(z)=∏i=1κ1‍(z-αi)(z-α¯i)∏i=1κ2‍(z-βi)(z-β¯i)×(αz+β+∫R‍(1u-z-u1+u2)dσ(u)),
where αi, βi, κ1, κ2 are the same as in (11), and α≥0, β∈R, and σ(u) is a positive measure on R satisfying ∫R‍(1+u2)-1dσ(u)<+∞.It should be noted that generalized Nevanlinna functions are closely related to the theory of extensions of symmetric operators in the Pontryagin spaces. The reader can refer to [1, 7, 10–13] and the references therein.In this paper, we present some interesting propositions on generalized Nevanlinna functions in the classNκ. Firstly, we show that the number of negative eigenvalues of each Hermitian generalized Loewner matrix generated by a function f(z)∈Nκ is not greater than κ. This leads to an equivalent definition of the generalized Nevanlinna function with negative index κ (see Theorem 7 below), which extends the original one to the multiple point case. Secondly, we indicate that the number of negative eigenvalues of the Hankel matrix, constructed with partial coefficients of the asymptotic expansion of a function f(z)∈Nκ at a real point or at infinity, does not exceed κ (see Theorems 14 and 15). Finally, we give an explicit formula of the negative index of real rational functions by using the relations among Loewner, Bézout, and Hankel matrices (see Theorem 21). These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the class Nκ, based on the so-called Hankel vector approach, in subsequent work.


## 2. An Equivalent Definition of Generalized Nevanlinna Functions of the ClassNκ

In this section, we will show that the Hermitian Loewner matrixPf(z1,…,zm) appearing in the definition of generalized Nevanlinna functions with negative index κ can be substituted by a certain Hermitian generalized Loewner matrix. For a more general form of generalized Loewner matrices, the reader can refer to [14, 15].For convenience, we introduce the notion of the divided difference of a complex function of one complex variable. Letf(z) be such a function of one complex variable z, let (zi)i∈N+ be a sequence of distinct points located in the domain of f(z), and let [zi]f:=f(zi),i=1,2,…, and, in the case k≥1, moreover,
(13)[z1,…,zk+1]f:=[z1,…,zk]f-[z2,…,zk+1]fz1-zk+1.
Such [z1,…,zk+1]f is called the kth divided difference of f(z) at z1,…,zk+1.In the sequel, we give some properties of the higher divided difference of the complex functions in one complex variable. These properties are very similar to those of the higher divided difference of the real functions of one real variable. But the proofs of them need to be checked carefully.Proposition 1.
LetD be an open disk in the complex plane, and let z1,…,zn∈D be distinct points. If f(z) is holomorphic in the closed disk D¯, then
(14)[zπ(1),…,zπ(n)]f=[z1,…,zn]f,
in which π is an arbitrary n-permutation.Proof.
By using Cauchy’s Integral Formula, we infer that(15)f(zj)=12πi∫∂D‍f(ξ)ξ-zjdξ,j=1,…,n.
Then, for each n-permutation π, we have
(16)[zπ(1),zπ(2)]f=12πi∫∂D‍f(ξ)(ξ-zπ(1))(ξ-zπ(2))dξ,[zπ(1),zπ(2),zπ(3)]f=12πi∫∂D‍f(ξ)(ξ-zπ(1))(ξ-zπ(2))(ξ-zπ(3))dξ,⋮[zπ(1),…,zπ(n)]f=[zπ(1),…,zπ(n-1)]f-[zπ(2),…,zπ(n)]fzπ(1)-zπ(n)=1zπ(1)-zπ(n)×(12πi∫∂D‍f(ξ)dξ(ξ-zπ(1))⋯(ξ-zπ(n-1))ssssss-12πi∫∂D‍f(ξ)dξ(ξ-zπ(2))⋯(ξ-zπ(n)))=12πi∫∂D‍f(ξ)(ξ-zπ(1))⋯(ξ-zπ(n))dξ=12πi∫∂D‍f(ξ)(ξ-z1)⋯(ξ-zn)dξ.
The last equation implies that (14) holds for each n-permutation π.With the proof of Proposition1, we can prove the following two statements.Proposition 2.
Iff(z) is holomorphic at the point α∈C and z1,z2,…,zn+1 are distinct and variable points in C, then
(17)lim⁡zi→α,∀i[z1,…,zn+1]f=1n!f(n)(α)=:[α,…,α︸n+1]f
holds for an arbitrary n∈N+.Proof.
Sincef(z) is holomorphic at α, there exists δ>0 such that f(z) is holomorphic in D¯:={z∈C∣|z-α|≤δ}. It follows from the proof of Proposition 1 that
(18)lim⁡zi→α,∀i[z1,…,zn+1]f=lim⁡zi→α,∀i12πi∫∂D‍f(ξ)(ξ-z1)⋯(ξ-zn+1)dξ=12πi∫∂D‍f(ξ)(ξ-α)n+1dξ=1n!f(n)(α),∀n∈N+.
Then (17) follows.Proposition 3.
Iff(z) is holomorphic at two distinct points α,β∈C and z1,…,zm+1,w1,…,wn+1 are distinct and variable points in C, then
(19)lim⁡zj→α,∀jwk→β,∀k⁡[z1,…,zm+1,w1,…,wn+1]f=1m!n!∂m+n∂zm∂wn(f(z)-f(w)z-w)|z=α,w=β=:[α,…,α︸m+1,β,…,β︸n+1]f
holds for any m,n∈N+.Proof.
Sincef(z) is holomorphic at the points α and β, there exist two disks D1 and D2 with centers at α and β, respectively, such that D¯1∩D¯2=Ø and f(z) is holomorphic in D¯1 and D¯2. By using Cauchy’s Integral Formula,
(20)f(zj)=12πi∫∂D1∪∂D2‍f(ξ)ξ-zjdξ,f(wk)=12πi∫∂D1∪∂D2‍f(ξ)ξ-wkdξ,
where zj∈D1,wk∈D2(j=1,…,m+1,k=1,…,n+1). Therefore,
(21)lim⁡zj→α,∀jwk→β,∀k⁡[z1,…,zm+1,w1,…,wn+1]f=lim⁡zj→α,∀jwk→β,∀k⁡12πi×∫∂D1∪∂D2‍f(ξ)(ξ-z1)⋯(ξ-zm+1)(ξ-w1)⋯(ξ-wn+1)dξ=12πi∫∂D1∪∂D2‍f(ξ)(ξ-α)m+1(ξ-β)n+1dξ.
On the other hand,
(22)1m!n!∂m+n∂zm∂wn(f(z)-f(w)z-w)|z=α,w=β=1m!n!∂m+n∂zm∂wn(12πi∫∂D1∪∂D2‍f(ξ)dξ(ξ-z)(ξ-w))|z=α,w=β=12πi∫∂D1∪∂D2‍f(ξ)dξ(ξ-α)m+1(ξ-β)n+1.
Combining (21) and (22), we obtain (19) immediately.The following lemma plays an important role by proving the fact that the number of negative eigenvalues of the Hermitian generalized Loewner matrices generated by a functionf(z)∈Nκ does not exceed κ.Lemma 4.
Letf(z) be a complex function of one complex variable, and let z1,…,zm,w1,…,wn be distinct points in the domain of f(z). Then
(23)M([zi,wj]f)i,j=1m,nNT=([z1,…,zi,w1,…,wj]f)i,j=1m,n,
in which M:=Mm-1⋯M2M1 and N:=Nn-1⋯N2N1 with(24)Mk:=(Ik-11-(zk+1-z1)-1(zk+1-z1)-1⋱⋱-(zm-zm-k)-1(zm-zm-k)-1),Nl:=(Il-11-(wl+1-w1)-1(wl+1-w1)-1⋱⋱-(wn-wn-l)-1(wn-wn-l)-1)being nonsingular matrices for k=1,…,m-1 and l=1,…,n-1.Proof .
LetPk:=(AkBk) and Ql:=(Cl,Dl), in which, for k=0,1,…,m-1,(25)Ak≔([z1,w1]f⋯[z1,wn]f⋮⋮[z1,…,zk+1,w1]f⋯[z1,…,zk+1,wn]f),Bk:=([z2,…,zk+2,w1]f⋯[z2,…,zk+2,wn]f⋮⋮[zm-k,…,zm,w1]f⋯[zm-k,…,zm,wn]f),
and, for l=0,1,…,n-1,
(26)Cl:=([z1,w1]f⋯[z1,w1,…,wl+1]f⋮⋮[z1,…,zm,w1]f⋯[z1,…,zm,w1,…,wl+1]f),Dl:=([z1,w2,…,wl+2]f⋯[z1,wn-l,…,wn]f⋮⋮[z1,…,zm,w2,…,wl+2]f⋯[z1,…,zm,wn-l,…,wn]f).
Obviously, P0=([zi,wj]f)i,j=1m,n, Q0=Pm-1, and Qn-1=([z1,…,zi,w1,…,wj]f)i,j=1m,n. By Proposition 1, we check easily that
(27)Mk(Ak-1Bk-1)=(AkBk),(Cl-1,Dl-1)NlT=(Cl,Dl);
that is, MkPk-1=Pk and Ql-1NlT=Ql(k=1,…,m-1,l=1,…,n-1). Hence we have
(28)MP0NT=Mm-1⋯M2M1P0N1TN2T⋯Nn-1T=Qn-1,
as asserted.In particular, if, in Lemma4, f(z)∈Nκ, m=n, and wi=z¯i(i=1,…,m), then NT=M* and thus (23) in this case can be rewritten as follows:
(29)M([zi,z¯j]f)i,j=1mM*=([z1,…,zi,z¯1,…,z¯j]f)i,j=1m.Suppose now thatf(z)∈Nκ and z1,…,zρ∈C+∩D(f) are mutually different, with multiplicities τ1,…,τρ, respectively. For each zi, there exist mutually different points zij(k)∈C+∩D(f) (j=1,…,τi,k=1,2,…) such that
(30)lim⁡k→+∞zij(k)=zi,i=1,…,ρ,j=1,…,τi.
Therefore, Proposition 3 says
(31)lim⁡k→+∞[zi1(k),…,ziu(k),zj1(k)¯,…,zjν(k)¯]f=[zi,…,zi︸u,z¯j,…,z¯j︸ν]f,
in which i,j=1,…,ρ, u=1,…,τi, and v=1,…,τj. Since f(z)∈Nκ and all the zij(k)∈C+∩D(f) are distinct points, we have
(32)ν(Pf(z11(k),…,z1τ1(k),…,zρ1(k),…,zρτp(k)))≤κ,k=1,2,….
Observe that
(33)Pf(z11(k),…,z1τ1(k),…,zρ1(k),…,zρτρ(k))=(Pij(k))i,j=1ρ,
in which
(34)Pij(k):=([ziu(k),zjν(k)¯]f)u,ν=1τi,τj,i,j=1,…,ρ.
By use of (23), there exist nonsingular matrices Ti(k)(i=1,…,ρ) such that
(35)P~ij(k):=Ti(k)Pij(k)Tj(k)*=([zi1(k),…,ziu(k),zj1(k)¯,…,zjν(k)¯]f)u,ν=1τi,τj,i,j=1,…,ρ,
or equivalently,
(36)P~f(k):=(P~ij(k))i,j=1ρ=diag⁡(T1(k),…,Tρ(k))×Pf(z11(k),…,z1τ1(k),…,zρ1(k),…,zρτρ(k))×diag⁡(T1(k),…,Tρ(k))*.
The last equation together with (32) implies that
(37)ν(P~f(k))=ν(Pf(z11(k),…,z1τ1(k),…,zρ1(k),…,zρτρ(k)))≤κ,k=1,2,….Letf(z)∈Nκ and let z1,…,zρ∈C+∩D(f) be distinct points with multiplicities τ1,…,τρ, respectively. Define the generalized Loewner matrix generated by the function f(z)∈Nκ and such z1,…,zρ by
(38)Pf(z1,…,z1︸τ1,…,zρ,…,zρ︸τρ)≔(Pij)i,j=1ρ,Pij:=([zi,…,zi︸u,z¯j,…,z¯j︸ν]f)u,ν=1τi,τj,i,j=1,…,ρ.
Proposition 3 indicates that
(39)lim⁡k→+∞P~f(k)=Pf(z1,…,z1︸τ1,…,zρ,…,zρ︸τρ)=(Pij)i,j=1ρ,
where P~f(k) is defined by (36).Thus, from (2) and the fact that each eigenvalue of a square matrix is a continuous function of its entries, we immediately obtain the following result.Theorem 5.
Letf(z)∈Nκ, and let z1,…,zρ∈C+∩D(f) be distinct points with multiplicities τ1,…,τρ(eachτi≥1), respectively. Then the Hermitian generalized Loewner matrix
(40)Pf(z1,…,z1︸τ1,…,zρ,…,zρ︸τρ)
defined by (38) has at most κ negative eigenvalues.Proof.
Sincef(z)∈Nκ, by (2), we have ν(P~f(k))≤κ for k=1,2,…. It follows from (39) and the continuity of eigenvalues of a square matrix that
(41)ν(Pf(z1,…,z1︸τ1,…,zρ,…,zρ︸τρ))≤κ.
Thus we complete the proof.Observe that, by Proposition3, each entry of Pij defined by (38) can be rewritten as follows:
(42)[zi,…,zi︸k+1,z¯j,…,z¯j︸l+1]f=1k!l!∂k+l∂zk∂wl(f(z)-f(w)z-w)|z=zi,w=z¯j,∀i,j,k,l.
Then the Loewner matrix appearing in the definition of generalized Nevanlinna functions of the class Nκ is a special case of the generalized Loewner matrix (Pij)i,j=1ρ defined by (38).Theorem 6.
Letf(z) be meromorphic in C∖R satisfying f(z¯)=f(z)¯ for z∈D(f). If
(43)max⁡⁡{ν(Pf(z1,…,z1︸τ1,…,zρ,…,zρ︸τρ))∣ρ,τ1,…,τρ∈N+,z1,…,zρ∈C+∩D(f)different(Pf(z1,…,z1︸τ1,…,zρ,…,zρ︸τρ))}=κ,
thenf(z)∈Nκ.Proof.
It follows from (43) that, for an arbitrary m∈N+ and for any distinct points z1′,…,zm′∈C+∩D(f), ν(Pf(z1′,…,zm′))≤κ, and, for some choice of ρ,τ1,…,τρ and distinct points z1,…,zρ∈C+∩D(f),
(44)ν(Pf(z1,…,z1︸τ1,…,zρ,…,zρ︸τρ))=κ.
Let nowm:=τ1+⋯+τρ∈N+ and let Dzi,ɛ:={z∈C∣|z-zi|<ɛ},∀ɛ>0. Taking ɛ>0 small enough such that Dzi,ɛ∩Dzj,ɛ=∅,∀i≠j, we can choose different points zi1,…,ziτi in Dzi,ɛ∩C+∩D(f) for i=1,…,ρ instead of the original multiple point zi(i=1,…,ρ) such that
(45)ν(Pf(z11,…,z1τ1,…,zρ1,…,zρτρ))=κ.
This is possible, because of (36) and (39) together with the continuity of the eigenvalues of a square matrix for its entries. By definition, we obtain from the afore-stated results that f(z)∈Nκ. Thus the proof is complete.In view of Theorems5 and 6, we present now the announced result, that is, an equivalent definition of generalized Nevanlinna functions of the class Nκ in some sense, which extends the original one to the multiple point case.Theorem 7.
Letf(z) be a meromorphic function in C∖R such that f(z¯)=f(z)¯ for z∈D(f). Then f(z) is a generalized Nevanlinna function of the class Nκ, if and only if (41) holds for any ρ∈N+ and any distinct points z1,…,zρ∈C+∩D(f) with multiplicities τ1,…,τρ(eachτi≥1), respectively, and, for at least one such choice, the equality in (41) holds.


## 3. Some Propositions on Asymptotic Expansions off(z) of the Class Nκ

In this section, we study some propositions on asymptotic expansions of a functionf(z)∈Nκ at t∈R and at infinity. To begin with, we introduce the notion of angular derivative of a meromorphic function f(z) defined on C+ at a real point t.Letf(z) be meromorphic in C+ and let t∈R. If the nontangential limits
(46)f(t):=∠lim⁡z→t[z]f,f′(t):=∠lim⁡z→t[z,t]f=∠lim⁡z→tf(z)-f(t)z-t
exist, then we say that f(z) has an angular derivative (of order 1) at t, denoted by f′(t). Sometimes we write f(t) and f′(t) as [t]f and [t,t]f, respectively.More generally, for an arbitraryk∈N+, we say that f(z) has the kth angular derivative at a real point t, denoted by f(k)(t), if the nontangential limits
(47)[t,…,t︸j+1]f=∠lim⁡z→t[z,t,…,t︸j]f=∠lim⁡z→t[z,t,…,t︷j-1]f-[t,…,t︷j]fz-t(j=0,1,…,k)
exist, and in this case f(k)(t):=k![t,…,t︸k+1]f,∀k≥0.In [16], Chen and Hu proved an interesting property on higher angular derivatives of a function f(z) in the class N0 at a real point.Lemma 8.
Lett∈R and let f(z)∈N0 have an integral representation (3), in which α≥0, β∈R, and σ(u) is a positive measure on R such that (4) holds. Then, for some m∈N+, the angular derivatives f(k)(t)(k=0,1,…,2m-1) exist and are real numbers if and only if ∫R‍(u-t)-2mdσ(u)<+∞. In that case,
(48)f(k)(t)={αt+β+∫R‍(1u-t-u1+u2)dσ(u),k=0;αδ1k+∫R‍k!(u-t)k+1dσ(u),1≤k≤2m-1,
in which δij stands for Kronecker symbol, that is, when i=j, δij=1; when i≠j, δij=0.It follows from Lemma8 that if f(k)(t)(k=0,1,…,2m-1) exist and are real numbers then, for z∈C+,
(49)|f(k)(z)-f(k)(t)|≤k!|∫R‍(1(u-z)k+1-1(u-t)k+1)dσ(u)|≤k!|∫|u-t|>1‍(1(u-z)k+1-1(u-t)k+1)dσ(u)|+k!|∫|u-t|≤1‍(z-t)∑i=0k‍(u-z)k-i(u-t)i(u-z)k+1(u-t)k+1dσ(u)|.
Obviously, the first term on the right hand side of the inequality (49) tends to 0 when z tends nontangentially to t. As for the second term therein, noting |u-z|≥sinɛ|z-t| for all z∈C+, in the case of |u-t|≤1, we have
(50)|(z-t)∑i=0k‍(u-z)k-i(u-t)i(u-z)k+1(u-t)k+1|≤(u-t)-2m∑i=0k‍|(z-t)(u-t)2m+i-k-1(u-z)i+1|=(u-t)-2m∑i=0k‍|z-tu-z|·|u-tu-z|i|u-t|2m-1-k≤(u-t)-2m∑i=0k‍|z-tu-z|·|u-tu-z|i=(u-t)-2m∑i=0k‍|z-tu-z|·|u-z+z-tu-z|i≤(u-t)-2m∑i=0k‍(1+|z-tu-z|)i+1≤(u-t)-2m∑i=0k‍(1+1sinɛ)i+1.
Since ∫|u-t|≤1‍(u-t)-2mdσ(u)<+∞, by Lebesgue’s dominated convergence theorem, the second term on the right hand side of the inequality (49) tends to 0 as well when z tends nontangentially to t. Thus
(51)|f(k)(z)-f(k)(t)|⟶0(z∈πɛ(t),z⟶t),k=0,1,…,2m-1.
Therefore, the following statement is valid.Proposition 9.
Lett∈R and f(z)∈N0. If, for some m∈N+, the angular derivatives f(k)(t)(k=0,1,…,2m-1) exist and are real numbers, then
(52)∠lim⁡z→tf(k)(z)=f(k)(t),k=0,1,…,2m-1.Next, we verify that each functionf(z) in the class Nκ has a similar property to that given in Proposition 9. For this purpose, we need the following lemma.Lemma 10.
Letf(z), g(z) be meromorphic in C+, let t∈R, and let n∈N+. If f(z) and g(z) have the angular derivatives of order from 1 to (n-1) at t, then
(53)[z,t,…,t︸n]fg=[z,t,…,t︸n]f[z]g+∑i=1n‍[t,…,t︸i]f[z,t,…,t︸n+1-i]g,z∈C+∩D(f)∩D(g).Proof.
In the case ofn=1, we have
(54)[z,t]fg=f(z)g(z)-f(t)g(t)z-t=f(z)-f(t)z-tg(z)+f(t)g(z)-g(t)z-t=[z,t]f[z]g+[t]f[z,t]g.
Thus the equality (53) holds for n=1. Now we assume that the equality (53) holds for n=k(k≥1); that is,
(55)[z,t,…,t︸k]fg=[z,t,…,t︸k]f[z]g+∑i=1k‍[t,…,t︸i]f[z,t,…,t︸k+1-i]g.
Then, for the case of n=k+1,(56)[z,t,…,t︸k+1]fg=[z,t,…,t︷k]fg-[t,…,t︷k+1]fgz-t=[z,t,…,t︷k]f[z]g-[t,…,t︷k+1]f[t]g+∑i=1k‍[t,…,t︷i]f([z,t,…,t︷k+1-i]g-[t,…,t︷k+2-i]g)z-t=[z,t,…,t︸k+1]f[z]g+[t,…,t︸k+1]f[z,t]g+∑i=1k‍[t,…,t︸i]f[z,t,…,t︸k+2-i]g=[z,t,…,t︸k+1]f[z]g+∑i=1k+1‍[t,…,t︸i]f[z,t,…,t︸k+2-i]g,which means that, for n=k+1, the equality (53) is also true. Therefore, by the principle of mathematical induction, we conclude that the equality (53) is valid for all n∈N+.Letf(z) and g(z) have the angular derivatives of order from 1 to n at t∈R. Then observe that f(z)+g(z) has also the angular derivatives of order from 1 to n at t; moreover,(57)(f+g)(k)(t)=f(k)(t)+g(k)(t),k=0,1,…,n.
On the other hand, by Lemma 10, f(z)g(z) has then the angular derivatives of order from 1 to n at real t. As z tends nontangentially to real t in (53), we have
(58)[t,…,t︸n+1]fg=[t,…,t︸n+1]f[t]g+∑i=1n[t,…,t︸i]f[t,…,t︸n+2-i]g
at t∈R, which implies in turn that
(59)1n!(fg)(n)(t)=1n!f(n)(t)g(t)+∑i=1n1(i-1)!f(i-1)(t)1(n-i+1)!g(n-i+1)(t),
and consequently
(60)(fg)(n)(t)=f(n)(t)g(t)+∑i=0n-1(ni)f(i)(t)g(n-i)(t)=∑i=0n(ni)f(i)(t)g(n-i)(t),
where f(0)(t):=f(t), g(0)(t):=g(t).In view of the facts (57) and (60), we can prove the following result.Theorem 11.
Lett∈R and let f(z)∈Nκ have an integral representation of the form (10). Then for some m∈N+, the angular derivatives f(k)(t)(k=0,1,…,2m-1) exist and are real numbers if and only if
(61)∫R‍(u-t)-2mdσ(u)<+∞.
In that case,
(62)f(k)(t)=∠lim⁡z→tf(k)(z),k=0,1,…,2m-1.Proof.
Observe that the integrand in (10) can be rewritten as z+(z2+y02)(u-z)-1, so the integral representation (10) of f(z) can be rewritten as
(63)f(z)=1∏j=1κ1(z-αj)(z-α¯j)×((z2+y02)κ+1∫R‍dσ(u)u-z+∑j=02κ+1b~jzj),
where b~0,…,b~2κ+1∈R with b~2κ+1≥0, and αj,κ1,y0, and σ(u) are as in (10). Let now
(64)s(z)=(z2+y02)κ+1(∏j=1κ1(z-αj)(z-α¯j))-1,g(z)=∫R‍dσ(u)u-z,r(z)=(∏j=1κ1(z-αj)(z-α¯j))-1∑j=02κ+1b~jzj.
Then f(z)=s(z)g(z)+r(z), or equivalently, g(z)=s~(z)f(z)+r~(z), where s~(z)=s(z)-1, r~(z)=s(z)-1r(z). Since, for real t,s(k)(t),r(k)(t),s~(k)(t),r~(k)(t)∈R and s(t),s~(t)>0, from (57) and (60), we deduce that the angular derivatives f(k)(t)(k=0,1,…,2m-1) exist and are real numbers if and only if the angular derivatives g(k)(t)(k=0,1,…,2m-1) exist and are real numbers. Thus, by Lemma 8, the angular derivatives f(k)(t)(k=0,1,…,2m-1) exist and are real numbers if and only if the inequality (61) holds.
From the relationf(z)=s(z)g(z)+r(z), together with Proposition 9 and (57)-(60), it follows in turn that
(65)∠lim⁡z→tf(k)(z)=∠lim⁡z→t(∑i=0k‍(ki)s(k-i)(z)g(i)(z)+r(k)(z))=∑i=0k‍(ki)s(k-i)(t)g(i)(t)+r(k)(t)=f(k)(t),k=0,1,…,2m-1.In the sequel, we study the property of the asymptotic expansion of a functionf(z) in the class Nκ at a real point t.Proposition 12.
Letf(z)∈Nκ and t∈R. If, for some m∈N+, f(z) has an asymptotic expansion at t of the form:
(66)f(z)=h-1+∑i=02m‍hi(z-t)i+1+o((z-t)2m+1)(z∈πϵ(t),z⟶t),
where hi∈R(i=-1,0,…,2m), then f(z) has the angular derivatives of order from 1 to 2m+1 at the real point t; moreover,
(67)hi-1=1i!f(i)(t),i=0,1,…,2m+1.Proof.
From the asymptotic expansion (66) of f(z) at z=t, we deduce
(68)[t]f=∠lim⁡z→tf(z)=h-1,[t,t]f=∠lim⁡z→tf(z)-h-1z-t=h0,[t,t,t]f=∠lim⁡z→t[z,t]f-[t,t]fz-t=∠lim⁡z→tf(z)-h-1-h0(z-t)(z-t)2=h1,⋮[t,…,t︸2m+2]f=∠lim⁡z→t[z,t,…,t︷2m]f-[t,…,t︷2m+1]fz-t=∠lim⁡z→tf(z)-h-1-h0(z-t)-⋯-h2m-1(z-t)2m(z-t)2m+1=h2m.
Observe that f(i)(t)=i![t,…,t︸i+1]f, i=0,1,…,2m+1, and thus (67) holds.Theorem11 and Proposition 12 say actually that if, for some m∈N+, f(z)∈Nκ has an asymptotic expansion of the form (66) at t∈R, then
(69)∠lim⁡z→tf(i)(z)=i!hi-1,i=0,1,…,2m+1.Theorem 13.
Letf(z)∈Nκ and t∈R. If, for some m∈N+, f(z) has an asymptotic expansion of the form (66) at t∈R, then
(70)lim⁡y→0[t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]f=hk+l,k,l=0,1,…,m.Proof.
Letp(z)=h-1+h0(z-t)+⋯+h2m(z-t)2m+1 and r(z)=f(z)-p(z). Then we have
(71)∠lim⁡z→tp(k)(z)=k!hk-1=∠lim⁡z→tf(k)(z),k=0,1,…,2m+1.
In view of the relation f(z)=p(z)+r(z), we obtain, for real y,
(72)[t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]f=[t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]p+[t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]r.
Since p(z) is a polynomial, we have
(73)lim⁡y→0[t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]p=[t,t,…,t︸k+l+2]p=1(k+l+1)!p(k+l+1)(t)=hk+l,0≤k,l≤m.
Therefore, in order to prove (70), we need only to prove
(74)lim⁡y→0[t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]r=0,0≤k,l≤m.
By Proposition 3, we have
(75)[t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]r=1k!l!∂k+l∂λk∂μl(r(λ)-r(μ)λ-μ)|λ=t+iy,μ=t-iy=1k!dkdλkr(λ)(λ-t+iy)l+1|λ=t+iy+1l!dldμlr(μ)(μ-t-iy)k+1|μ=t-iy.
A direct computation leads to
(76)1k!dkdλkr(λ)(λ-t+iy)l+1|λ=t+iy=∑s=0k‍(-1)s(l+1)⋯(l+s)s!(k-s)!r(k-s)(t+iy)(2iy)l+s+1=∑s=0k‍(-1)s(l+1)⋯(l+s)s!(k-s)!(2i)l+s+1ssssss×(f(k-s)(t+iy)-p(k-s)(t+iy))yl+s+1.
By L’Hospital’s rule, together with (69), we have
(77)lim⁡y→0f(k-s)(t+iy)-p(k-s)(t+iy)yl+s+1=ilim⁡y→0f(k-s+1)(t+iy)-p(k-s+1)(t+iy)(l+s+1)yl+s=il+s+1lim⁡y→0f(k+l+1)(t+iy)-p(k+l+1)(t+iy)(l+s+1)!=il+s+1(l+s+1)!lim⁡y→0(f(k+l+1)(t+iy)-(k+l+1)!hk+l)=0(0≤k,l≤m,0≤s≤k).
This means that
(78)lim⁡y→01k!dkdλkr(λ)(λ-t+iy)l+1|λ=t+iy=0,0≤k,l≤m.
In a similar way, we can verify that
(79)lim⁡y→01l!dldμlr(μ)(μ-t-iy)k+1|μ=t-iy=0,0≤k,l≤m.
Then (74) holds, as needed.From Theorems7 and 13, we obtain an interesting result about the number of negative eigenvalues of the Hankel matrix constructed with the partial coefficients of the asymptotic expansion of f(z)∈Nκ at t∈R.Theorem 14.
Letf(z)∈Nκ and t∈R. If, for some m∈N+, f(z) has an asymptotic expansion of the form (66) at real t, then ν(H)≤κ, in which H:=(hk+l)k,l=0m.Proof.
By Theorem13, we have
(80)lim⁡y→0Pf(t+iy,…,t+iy︸m+1)=lim⁡y→0([t+iy,…,t+iy︸k+1,t-iy,…,t-iy︸l+1]f)k,l=0m=(hk+l)k,l=0m=H,
and therefore H is a real Hankel matrix of order m+1. On the other hand, it follows from Theorem 7 that ν(Pf(t+iy,…,t+iy︸m+1))≤κ. Since the eigenvalues of a matrix are continuous functions of its entries, we obtain ν(H)≤κ.Finally, we consider two properties of the asymptotic expansion off(z)∈Nκ at infinity.Theorem 15.
If, for somem∈N+, f(z)∈Nκ has an asymptotic expansion at infinity:
(81)f(z)=-h-1-∑i=02m‍hizi+1+o(z-2m-1)(z∈πɛ(0),z⟶∞),
where h-1,h0,…,h2m∈R, then ν(H)≤κ, where H:=(hk+l)k,l=0m.Proof.
Letf~(z)=f(-z-1). Then f~(z)∈Nκ. It follows from (81) that f~(z) has an asymptotic expansion at the point 0 of the form:
(82)f~(z)=h~-1+h~0z+h~1z2+⋯+h~2mz2m+1+o(z2m+1),
where
(83)h~i={hi,ifiiseven;-hi,ifiisodd.
By Theorem 14, we obtain further that ν(H~)≤κ, in which H~:=(h~k+l)k,l=0m. Observe that H~=(h~i+j)i,j=0m=DHD*, where D=diag⁡(1,-1,…,(-1)m). Therefore, ν(H~)=ν(H), so that ν(H)≤κ, as desired.Theorem 16.
Letf(z)∈Nκ have an asymptotic expansion at infinity of the form:
(84)f(z)=-∑i=02n-2‍hizi+1+o(z-2n+1)(z∈πɛ(0),z⟶∞),
where n∈N+, h0,…,h2n-2∈R, and let H:=(hi+j)i,j=0n-1. Then, for an arbitrary m∈N+ and mutually different points z1,…,zm∈C+∩D(f),
(85)ν(Lf(z1,…,zm))≤κ,
where
(86)Lf(z1,…,zm)≔(HC(z1)*⋯C(zm)*C(z1)⋮Pf(z1,…,zm)C(zm)),C(z):=(f(z),z(f(z)+h0z),…,dddddddzn-1(f(z)+h0z+h1z2+⋯+hn-2zn-1)).Proof.
Letf~(z)=f(-z-1). Then f~(z)∈Nκ. For an arbitrary m∈N+ and mutually different points z1,…,zm∈C+∩D(f), -z1-1,…,-zm-1∈C+∩D(f~) are also mutually different points. Then, for each z∈C+∩D(f~),
(87)ν(Pf~(z,…,z︸n,-z1-1,…,-zm-1))≤κ.
Observe that ∠lim⁡z→0Pf~(z,…,z︸n,-z1-1,…,-zm-1)=Pf~(0,…,0︸n,-z1-1,…,-zm-1). Thus we infer
(88)ν(Pf~(0,…,0︸n,-z1-1,…,-zm-1))≤κ.
On the other hand, it follows from (84) that
(89)f~(z)=h0z-h1z2+⋯+h2n-2z2n-1+o(z2n-1)(z∈πϵ(0),z⟶0).
Then we have
(90)Pf~(0,…,0︸n)=((-1)i+jhi+j)i,j=0n-1=diag⁡(1,-1,…,(-1)n-1)Hdiag⁡(1,-1,…,(-1)n-1)*,
where H=(hi+j)i,j=0n-1 is a real Hankel matrix of order n. Choose S:=diag⁡(1,-1,…,(-1)n-1,z1-1,…,zm-1) which is a nonsingular matrix of order n+m. A direct computation leads to
(91)SPf~(0,…,0︸n,-z1-1,…,-zm-1)S*=Lf(z1,…,zm).
Therefore, ν(Lf(z1,…,zm))=ν(Pf~(0,…,0︸n,-z1-1,…,-zm-1))≤κ.


## 4. An Explicit Formula of the Negative Index of Real Rational Functions

In this section, we give an explicit formula of the negative index of real rational functions by decomposing each real rational function into the sum of a real polynomial and a strictly proper real rational function.From definition of a generalized Nevanlinna function, we can prove that each real polynomial is a generalized Nevanlinna function in the classNκ with some negative index κ determined by the following lemma (see, e.g., [1], [4, Section 3]).Lemma 17.
Letp(z)=p0+p1z+⋯+pmzm(pm≠0) be a real polynomial. Then p(z)∈Nκ, where
(92)κ={n,ifm=2n+1,pm>0;n+1,ifm=2n+1,pm<0;n,ifm=2n.Proof.
Obviously, the real polynomialp(z) is holomorphic in C such that p(z¯)=p(z)¯. Then p(z) has no poles in C+ and no generalized poles of nonpositive type in R. In what follows we will determine the multiplicity of the point ∞ as a generalized pole of nonpositive type of p(z).(1)
Ifm=2n+1 and pm>0, then
(93)∠lim⁡z→∞z-2n-1p(z)=pm>0,∠lim⁡z→∞z-2n+1p(z)=∞, so that, by definition, the point ∞ is a generalized pole of nonpositive type of p(z) with multiplicity n.(2)
Ifm=2n+1 and pm<0, then
(94)∠lim⁡z→∞z-2(n+1)-1p(z)=0,∠lim⁡z→∞z-2(n+1)+1p(z)=pm<0, so that, by definition, the point ∞ is a generalized pole of nonpositive type of p(z) with multiplicity n+1.(3)
Ifm=2n, then
(95)∠lim⁡z→∞z-2n-1p(z)=0,∠lim⁡z→∞z-2n+1p(z)=∞, so that, by definition, the point ∞ is a generalized pole of nonpositive type of p(z) with multiplicity n.
Since the total multiplicity of the poles ofp(z) in C+ and the generalized poles of nonpositive type of p(z) in R∪{∞} is equal to κ, the multiplicity of the point ∞ as a generalized pole of nonpositive type of p(z) is equal to κ. Hence the equality (92) holds.Letf1(z)∈Nκ1 and f2(z)∈Nκ2. Daho and Langer [9, Proposition 3.2] proposed a sufficient condition for f1(z)+f2(z)∈Nκ1+κ2 mentioned in introduction part. In view of this result, we can obtain a modified version of that sufficient condition (see also [1, Satz 1.13]).Lemma 18.
Letf1(z)∈Nκ1 and f2(z)∈Nκ2 have no common poles in C+ and no common generalized poles of nonpositive type in R∪{∞} and satisfy the following two conditions:(1)
ifα∈R is a generalized pole of nonpositive type of fi(z)(i=1,2), then ∠lim⁡z→α(z-α)fj(z)=0,    j≠i;(2)
if the point∞ is a generalized pole of nonpositive type of fi(z)(i=1,2), then ∠lim⁡z→∞z-1fj(z)=0,    j≠i. Then f(z):=f1(z)+f2(z)∈Nκ1+κ2. Moreover, the poles of fi(z)(i=1,2) in C+ and the generalized poles of nonpositive type of fi(z)(i=1,2) in R∪{∞} are also the poles of f(z) in C+ and the generalized poles of nonpositive type of f(z) in R∪{∞} with the same multiplicities.Proof.
Without loss of generality, we need only to verify that Lemma18 is valid in the case of i=1 and j=2. If α∈C+ is a pole of f1(z) of multiplicity π, then f2(z) is holomorphic at α, and hence α is a pole of f(z)=f1(z)+f2(z) of multiplicity π.
Ifα∈R is a generalized pole of nonpositive type of f1(z) with multiplicity π, then ∠lim⁡z→α(z-α)2π-1f1(z)>0 or is equal to ∞ and ∠lim⁡z→β(z-α)2π+1f1(z)≤0. It follows from condition (1) that
(96)∠lim⁡z→α(z-α)2π-1f(z)=∠lim⁡z→α(z-α)2π-1(f1(z)+f2(z))>0orisequalto∞,∠lim⁡z→α(z-α)2π+1f(z)=∠lim⁡z→β(z-α)2π+1(f1(z)+f2(z))≤0.
Thus, by definition, α∈R is a generalized pole of nonpositive type of f(z)=f1(z)+f2(z) with multiplicity π.
If the point∞ is a generalized pole of nonpositive type of f1(z) with multiplicity π, then ∠lim⁡z→∞z-2π+1f1(z)<0 or is equal to ∞ and ∠lim⁡z→∞z-2π-1f1(z)≥0. It follows from condition (2) that
(97)∠lim⁡z→∞z-2π+1f(z)=∠lim⁡z→∞z-2π+1(f1(z)+f2(z))<0orisequalto∞,∠lim⁡z→∞z-2π-1f(z)=∠lim⁡z→∞z-2π-1(f1(z)+f2(z))≥0.
Thus the point ∞ is a generalized pole of nonpositive type of f(z)=f1(z)+f2(z) with multiplicity π by definition.
The above analysis shows that the poles offi(z)(i=1,2) in C+ and the generalized poles of nonpositive type of fi(z)(i=1,2) in R∪{∞} are also the poles of f(z) in C+ and the generalized poles of nonpositive type of f(z) in R∪{∞} with the same multiplicities. However, since f1(z) and f2(z) have no common poles in C+ and no common generalized poles in R∪{∞}, the total multiplicity of poles of f(z) in C+ and the generalized poles of nonpositive type of f(z) in R∪{∞} is greater than or equal to κ1+κ2, and hence f(z)∈Nκ with κ≥κ1+κ2. On the other hand, as stated in the introduction part, we have κ≤κ1+κ2. Thus κ=κ1+κ2.To describe the negative index of a strictly proper real rational function, we have to consider the Bézout matrix of a pair of polynomials (see, e.g., [17–19]) and the generating functions of a Hankel matrix.Letf(z) and g(z) be real polynomials with max⁡{deg⁡f(z),deg⁡g(z)}=n. We define the Bézout matrix of polynomials f(z) and g(z) as the n by n matrix B(f,g):=(bij)i,j=0n-1 generated by
(98)f(z)g(ω)-f(ω)g(z)z-ω=∑i,j=0n-1bijziωj.
Clearly, we have that B(f,g)=-B(g,f) and B(f,g)=B(f,g)T.Leth0,h1,…,h2n-2∈R be given. Let f(z) be a real polynomial of degree l, and let g(z) be a real polynomial of degree not more than (l-1) such that gcd(f(z),g(z))=1. If the strictly proper rational function g(z)/f(z) has an asymptotic expansion at the point ∞ as follows:
(99)g(z)f(z)=h0z+h1z2+⋯+h2n-2z2n-1+o(z-2n+1)(z⟶∞),
then g(z)/f(z) is called a generating function, of McMillan degree l, of the Hankel matrix H:=(hi+j)i,j=0n-1, and in this case that Hankel matrix H is also represented as H=Hn(g/f).It is well-known that the Bézout matrix and the Hankel matrix have the following relation; see, for example, [17, 20].Lemma 19.
Letf(z)=∑i=0nfizi(fn≠0) and g(z)=∑i=0ngizi be two real polynomials such that the rational function g(z)/f(z) has an asymptotic expansion at the point ∞ as
(100)g(z)f(z)=h-1+h0z+h1z2+⋯+h2n-2z2n-1+o(z-2n+1).
Then
(101)B(f,g)=S(f)Hn(gf)S(f),
where
(102)S(f):=(f1f2⋯fnf2⋰⋮⋰fn0),Hn(gf)=(hi+j)i,j=0n-1.In the following, we give the result on the negative index of a strictly proper real rational function and its detailed proof.Theorem 20.
Letf(z) be a real polynomial of degree n, and let g(z) be a real polynomial of degree not more than (n-1). If r(z):=g(z)/f(z) has an asymptotic expansion at the point ∞ as follows:
(103)r(z)=g(z)f(z)=-h0z-h1z2-⋯-h2n-2z2n-1+o(z-2n+1)(z⟶∞),
then r(z)∈Nκ, where κ=ν(Hn(-g/f))=ν(H) and H:=(hi+j)i,j=0n-1.Proof.
Observe thatr(z¯)=r(z)¯ and, for an arbitrary m∈N+ and distinct points z1,…,zm∈C+∩D(r),
(104)Pr(z1,…,zm)=(r(zi)-r(z¯j)zi-z¯j)i,j=1m=(g(zi)/f(zi)-g(z¯j)/f(z¯j)zi-z¯j)i,j=1m=diag⁡(f(z1)-1,…,f(zm)-1)V(z1,…,zm)ddddd×B(g,f)V(z1,…,zm)*×diag⁡(f(z1)-1,…,f(zm)-1)*,
where
(105)V(z1,…,zm)=(1z1⋯z1n-1⋮⋮⋮1zm⋯zmn-1).
Thus we obtain that ν(Pr(z1,…,zm))≤ν(B(g,f)), and hence r(z)∈Nκ and κ≤ν(B(g,f)).
On the other hand, sincer(z) has an asymptotic expansion of the form (103), by Theorem 15, we have r(z)∈Nκ, with κ≥ν(H), where H=(hi+j)i,j=0n-1. Moreover, Lemma 19 provides that B(f,g)=S(f)Hn(g/f)S(f)=-S(f)HS(f)*, where S(f) is invertible, so that B(g,f)=S(f)HS(f)*, and hence ν(B(g,f))=ν(H)≤κ. Therefore, we have r(z)∈Nκ with κ=ν(H).For an arbitrary real rational functionR(z):=g(z)/f(z), where g(z):=∑i=0m‍gizi(gm≠0) and f(z):=∑j=0n‍fjzj(fn=1) are real polynomials such that gcd(f(z),g(z))=1 and m,n∈N+. By the Euclidean division of polynomials, g(z) has an expression of the form g(z)=f(z)q(z)+r(z), in which r(z)=0 or deg⁡r(z)<deg⁡f(z); hence, we have R(z)=g(z)/f(z)=q(z)+r(z)/f(z), where q(z)=0(ifm<n) or q(z) is the form q(z)=∑i=0m-n‍qizi(ifm≥n), with qm-n=gm.Ifm<n, then, by Theorem 20, the negative index of R(z) is equal to the negative inertia index of the n by n Hankel matrix generated by the function -R(z).Ifm≥n, we define q(z) as the polynomial part of R(z) and r(z)/f(z) as the strictly proper rational function part of R(z). In this case, the negative index of q(z) is given by use of Lemma 17 and the negative index of r(z)/f(z) is derived by use of Theorem 20. Therefore, by Lemma 18, the negative index of a real rational function R(z) is the sum of the negative indexes of q(z) and r(z)/f(z). Thereby, we obtain an explicit formula of the negative index of real rational functions.Theorem 21.
LetR(z)=g(z)/f(z) be a real rational function, where g(z)=∑i=0m‍gizi(gm≠0) and f(z)=∑j=0n‍fjzj(fn=1) are real polynomials with gcd(f(z),g(z))=1 and m,n∈N+. Then(1)
Ifm<n, then R(z)∈Nκ, where κ=ν(Hn(-R)).(2)
Ifm≥n, then R(z)∈Nκ1+κ2, where
(106)κ1={l,ifm-n=2l+1,gm>0;l+1,ifm-n=2l+1,gm<0;l,ifm-n=2l,κ2=ν(Hn(-rf)), where r(z)/f(z) is the strictly proper rational function part of R(z).


## 5. Conclusion

In this paper, we consider some propositions on generalized Nevanlinna functions with detailed proofs. We prove that the Hermitian matrix([zi,z¯j]f)i,j=1m is congruent to the Hermitian matrix ([z1,…,zi,z¯1,…,z¯j]f)i,j=1m, given in (29). This result is quite important in the proof of an equivalent definition of a generalized Nevanlinna function (see Theorem 7), because it can be used to verify the fact that the negative inertia index of every Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function of the class Nκ does not exceed κ. We also obtain that the negative inertia index of the Hankel matrix constructed with the partial coefficients of the asymptotic expansions of a generalized Nevanlinna function of the class Nκ at a real point or at infinity is less than or equal to κ, and the same result holds also for the Loewner matrix Lf(z1,…,zm) defined by (86). Finally, based on the relations among Loewner, Bézout, and Hankel matrices, we give an explicit formula of the negative index of real rational functions. These results stated in this paper will be applied to solve the indefinite Hamburger moment problems and the multiple Nevanlinna-Pick interpolation problems for the class Nκ in subsequent work.



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