We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.
Citation: Arnon Ploymukda, Pattrawut Chansangiam. Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products[J]. AIMS Mathematics, 2023, 8(11): 26153-26167. doi: 10.3934/math.20231333
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We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.
To understand in a clear way the notions used in our main results, we need to add here some basic literature of Geometric function theory. For this we start first with the notation A which denotes the class of holomorphic or analytic functions in the region D={z∈C:|z|<1} and if a function g∈A, then the relations g(0)=g′(0)−1=0 must hold. Also, all univalent functions will be in a subfamily S of A. Next we consider to define the idea of subordinations between analytic functions g1 and g2, indicated by g1(z)≺g2(z), as; the functions g1,g2∈A are connected by the relation of subordination, if there exists an analytic function w with the restrictions w(0)=0 and |w(z)|<1 such that g1(z)=g2(w(z)). Moreover, if the function g2∈S in D, then we obtain:
g1(z)≺g2(z)⇔[g1(0)=g2(0) & g1(D)⊂g1(D)]. |
In 1992, Ma and Minda [16] considered a holomorphic function φ normalized by the conditions φ(0)=1 and φ′(0)>0 with Reφ>0 in D. The function φ maps the disc D onto region which is star-shaped about 1 and symmetric along the real axis. In particular, the function φ(z)=(1+Az)/(1+Bz), (−1≤B<A≤1) maps D onto the disc on the right-half plane with centre on the real axis and diameter end points 1−A1−B and 1+A1+B. This interesting familiar function is named as Janowski function [10]. The image of the function φ(z)=√1+z shows that the image domain is bounded by the right-half of the Bernoulli lemniscate given by |w2−1|<1, [25]. The function φ(z)=1+43z+23z2 maps D into the image set bounded by the cardioid given by (9x2+9y2−18x+5)2−16(9x2+9y2−6x+1)=0, [21] and further studied in [23]. The function φ(z)=1+sinz was examined by Cho and his coauthors in [3] while φ(z)=ez is recently studied in [17] and [24]. Further, by choosing particular φ, several subclasses of starlike functions have been studied. See the details in [2,4,5,11,12,14,19].
Recently, Ali et al. [1] have obtained sufficient conditions on α such that
1+zg′(z)/gn(z)≺√1+z⇒g(z)≺√1+z,for n=0,1,2. |
Similar implications have been studied by various authors, for example see the works of Halim and Omar [6], Haq et al [7], Kumar et al [13,15], Paprocki and Sokól [18], Raza et al [20] and Sharma et al [22].
In 1994, Hayman [8] studied multivalent (p-valent) functions which is a generalization of univalent functions and is defined as: an analytic function g in an arbitrary domain D⊂C is said to be p-valent, if for every complex number ω, the equation g(z)=ω has maximum p roots in D and for a complex number ω0 the equation g(z)=ω0 has exactly p roots in D. Let Ap (p∈N={1,2,…}) denote the class of functions, say g∈Ap, that are multivalent holomorphic in the unit disc D and which have the following series expansion:
g(z)=zp+∞∑k=p+1akzk, (z∈D). | (1.1) |
Using the idea of multivalent functions, we now introduce the class SL∗p of multivalent starlike functions associated with lemniscate of Bernoulli and as given below:
SL∗p={g(z)∈Ap:zg′(z)pg(z)≺√1+z, (z∈D)}. |
In this article, we determine conditions on α such that for each
1+αz2+p(j−1)g′(z)pgj(z), for each j=0,1,2,3, |
are subordinated to Janowski functions implies g(z)zp≺√1+z, (z∈D). These results are then utilized to show that g are in the class SL∗p.
Let w be analytic non-constant function in D with w(0)=0. If
|w(z0)|=max{|w(z)|, |z|≤|z0|}, z∈D, |
then there exists a real number m (m≥1) such that z0w′(z0)=mw(z0).
This Lemma is known as Jack's Lemma and it has been proved in [9].
Let g∈Ap and satisfying
1+αz1−pg′(z)p≺1+Az1+Bz, |
with the restriction on α is
|α|≥232p(A−B)1−|B|−4p(1+|B|). | (2.1) |
Then
g(z)zp≺√1+z. |
Proof
Let us define a function
p(z)=1+αz1−pg′(z)p, | (2.2) |
where the function p is analytic in D with p(0)=1. Also consider
g(z)zp=√1+w(z). | (2.3) |
Now to prove our result we will only require to prove that |w(z)|<1. Logarithmically differentiating (2.3) and then using (2.2), we get
p(z)=1+αzw′(z)2p√1+w(z)+α√1+w(z), |
and so
|p(z)−1A−Bp(z)|=|αzw′(z)2p√1+w(z)+α√1+w(z)A−B(1+αzw′(z)2p√1+w(z)+α√1+w(z))|=|αzw′(z)+2pα(1+w(z))2p(A−B)√1+w(z)−B(αzw′(z)+2pα(1+w(z)))|. |
Now, we suppose that a point z0∈D occurs such that
max|z|≤|z0||w(z)|=|w(z0)|=1. |
Also by Lemma 1.1, a number m≥1 exists with z0w′(z0)=mw(z0). In addition, we also suppose that w(z0)=eiθ for θ∈[−π,π]. Then we have
|p(z0)−1A−Bp(z0)|=|αmw(z0)−2pα(1+w(z0))2p(A−B)√1+w(z0)−B(αmw(z0)+2pα(1+w(z0)))|,≥|α|m−2p|α|(|1+eiθ|)2p(A−B)√|1+eiθ|+|B|(|α|m+2pα|1+eiθ|),≥|α|m−4p|α|232p(A−B)+|B||α|(m+4p). |
Now if
ϕ(m)=|α|(m−4p)232p(A−B)+|B||α|(m+4p), |
then
ϕ′(m)=232p(A−B)|α|+8|α|2p|B|(232p(A−B)+|B||α|(m+4p))2>0, |
which illustrates that the function ϕ(m) is increasing and hence ϕ(m)≥ϕ(1) for m≥1, so
|p(z0)−1A−Bp(z0)|≥|α|(1−4p)232p(A−B)+|B||α|(1+4p). |
Now, by using (2.1), we have
|p(z0)−1A−Bp(z0)|≥1 |
which contradicts the fact that p(z)≺1+Az1+Bz. Thus |w(z)|<1 and so we get the desired result.
Taking g(z)=zp+1f′(z)pf(z) in the last result, we obtain the following Corollary:
Let f∈Ap and satisfying
1+αzf′(z)p2f(z)(p+1+zf′′(z)f′(z)−zf′(z)f(z))≺1+Az1+Bz, | (2.4) |
with the condition on α is
|α|≥232p(A−B)1−|B|−4p(1+|B|). |
Then f∈SL∗p.
If g∈Ap such that
1+αpzg′(z)g(z)≺1+Az1+Bz, | (2.5) |
with
|α|≥8p(A−B)1−|B|−4p(1+|B|), | (2.6) |
then
g(z)zp≺√1+z. |
Proof
Let us choose a function p by
p(z)=1+αzg′(z)pg(z), |
in such a way that p is analytic in D with p(0)=1. Also consider
g(z)zp=√1+w(z). |
Using some simple calculations, we obtain
p(z)=1+αzw′(z)2p(1+w(z))+α, |
and so
|p(z)−1A−Bp(z)|=|αzw′(z)2p(1+w(z))+αA−B(1+αzw′(z)2p(1+w(z))+α)|=|αzw′(z)+2pα(1+w(z))2p(A−B)(1+w(z))−B(αzw′(z)+2pα(1+w(z)))|. |
Let a point z0∈D exists in such a way
max|z|≤|z0||w(z)|=|w(z0)|=1. |
Then, by virtue of Lemma 1.1, a number m≥1 occurs such that z0w′(z0)=mw(z0). In addition, we set w(z0)=eiθ, so we have
|p(z0)−1A−Bp(z0)|=|αmw(z0)+2pα(1+w(z0))2p(A−B)(1+w(z0))−B(αmw(z0)+2pα(1+w(z0)))|,≥|α|m−2p|α||1+eiθ|2(A−B)|1+eiθ|+|B||α|m+2p|B||α||1+eiθ|,=|α|m−2p|α|√2+2cosθ2(2(A−B)+|B||α|)p√2+2cosθ+|B||α|m,≥|α|(m−4p)4p(2(A−B)+|B||α|)+|B||α|m. |
Now let
ϕ(m)=|α|(m−4p)4p(2(A−B)+|B||α|)+|B||α|m, |
it implies
ϕ′(m)=|α|8p((A−B)+|α||B|)(4p(2(A−B)+|B||α|)+|B||α|m)2>0, |
which illustrates that the function ϕ(m) is increasing and so ϕ(m)≥ϕ(1) for m≥1, hence
|p(z0)−1A−Bp(z0)|≥|α|(1−4p)4p(2(A−B)+|B||α|)+|B||α|. |
Now, by using (2.6), we have
|p(z0)−1A−Bp(z0)|≥1, |
which contradicts (2.5). Thus |w(z)|<1 and so the desired proof is completed.
Putting g(z)=zp+1f′(z)pf(z) in last Theorem, we get the following Corollary:
If f∈Ap and satisfying
1+αp(p+1+zf′′(z)f′(z)−zf′(z)f(z))≺1+Az1+Bz, |
with
|α|≥8p(A−B)1−|B|−4p(1+|B|), |
then f∈SL∗p.
If g∈Ap and satisfy the subordination relation
1+αz1−pg′(z)p(g(z))2≺1+Az1+Bz, | (2.7) |
with the condition on α
|α|≥252p(A−B)1−|B|−4p(1+|B|) | (2.8) |
is true, then
g(z)zp≺√1+z. |
Proof
Let us define a function
p(z)=1+αz1−pg′(z)p(g(z))2. |
Then p is analytic in D with p(0)=1. Also let us consider
g(z)zp=√1+w(z). |
Using some simplification, we obtain
p(z)=1+αzw′(z)2p(1+w(z))32+α√1+w(z), |
and so
|p(z)−1A−Bp(z)|=|αzw′(z)2p(1+w(z))32+α√1+w(z)A−B(1+αzw′(z)2p(1+w(z))32+α√1+w(z))|=|αzw′(z)+2pα(1+w(z))2p(A−B)(1+w(z))32−Bαzw′(z)−2pαB(1+w(z))|. |
Let us choose a point z0∈D such a way that
max|z|≤|z0||w(z)|=|w(z0)|=1. |
Then, by the consequences of Lemma 1.1, a number m≥1 occurs such that z0w′(z0)=mw(z0) and also put w(z0)=eiθ,for θ∈[−π,π], we have
|p(z0)−1A−Bp(z0)|=|αmw(z0)+2pα(1+w(z0))2p(A−B)(1+w(z0))32−Bαmw(z0)−2pαB(1+w(z0))|,≥|α|m−2p|α||1+eiθ|2p(A−B)|1+eiθ|32+|B||α|m+2p|α||B||1+eiθ|,=|α|m−4p|α|252p(A−B)+|B||α|m+4p|α||B|,≥|α|(m−4p)252p(A−B)+|B||α|m+4p|α||B|=ϕ(m) (say). |
Then
ϕ′(m)=252p(A−B)+8|α|2|B|p(252p(A−B)+B|α|m+4p|α|B)2>0, |
which demonstrates that the function ϕ(m) is increasing and thus ϕ(m)≥ϕ(1) for m≥1, hence
|p(z0)−1A−Bp(z0)|≥|α|(1−4p)252p(A−B)+|B||α|+4p|α||B|. |
Now, using (2.8), we have
|p(z0)−1A−Bp(z0)|≥1, |
which contradicts (2.7). Thus |w(z)|<1 and so we get the required proof.
If we set g(z)=zp+1f′(z)pf(z) in last theorem, we easily have the following Corollary:
Assume that
|α|≥252p(A−B)1−|B|−4p(1+|B|), |
and if f∈Ap satisfy
1+αf(z)z2p+1f′(z)(p+1+zf′′(z)f′(z)−zf′(z)f(z))≺1+Az1+Bz, |
then f∈SL∗p.
If g∈Ap satisfy the subordination
1+αz1−2pg′(z)p(g(z))3≺1+Az1+Bz, |
with restriction on α is
|α|≥8p(A−B)1−|B|−4p(1+|B|). | (2.9) |
then
g(z)zp≺√1+z. |
Proof. Let us define a function
p(z)=1+αz1−2pg′(z)p(g(z))3, |
where p is analytic in D with p(0)−1=0. Also let
g(z)zp=√1+w(z). |
Using some simple calculations, we obtain
p(z)=1+αzw′(z)2p(1+w(z))2+α1+w(z), |
and so
|p(z)−1A−Bp(z)|=|αzw′(z)2p(1+w(z))2+α1+w(z)A−B(1+αzw′(z)2p(1+w(z))2+α1+w(z))|=|αzw′(z)+2pα(1+w(z))2p(A−B)(1+w(z))2−Bαzw′(z)−2pαB(1+w(z))|. |
Let us pick a point z0∈D in such a way that
max|z|≤|z0||w(z)|=|w(z0)|=1. |
Then, by using Lemma 1.1, a number m≥1 exists such that z0w′(z0)=mw(z0) and put w(z0)=eiθ, for θ∈[−π,π], we have
|p(z0)−1A−Bp(z0)|=|αmw(z0)+2pα(1+w(z0))2p(A−B)(1+w(z0))2−Bαmw(z0)−2pαB(1+w(z0))|≥|α|m−2p|α||1+eiθ|2p(A−B)|1+eiθ|2+|B||α|m+2p|α||B||1+eiθ|=|α|m−2p|α|√2+2cosθ2p(A−B)(√2+2cosθ)2+|B||α|m+2p|α||B|√2+2cosθ≥|α|(m−4p)8p(A−B)+|B||α|m+4p|α||B|, |
Now let
ϕ(m)=|α|(m−4p)8p(A−B)+|B||α|m+4p|α||B|, |
then
ϕ′(m)=8p|α|(A−B)+8|α|2|B|p(8p(A−B)+|B||α|m+4p|α||B|)2>0 |
which shows that ϕ(m) is an increasing function and hence it will have its minimum value at m=1, so
|p(z0)−1A−Bp(z0)|≥|α|(1−4p)8p(A−B)+|B||α|+4p|α||B|. |
Using (2.9), we easily obtain
|p(z0)−1A−Bp(z0)|≥1, |
which is a contradiction to the fact that p(z)≺1+Az1+Bz, and so |w(z)|<1. Hence we get the desired result.
If we put g(z)=zp+1f′(z)pf(z) in last Theorem, we achieve the following result:
If f∈Ap and satisfy the condition
|α|≥8p(A−B)1−|B|−4p(1+|B|), |
and
1+αp(f(z))2z3p+2(f′(z))2(p+1+zf′′(z)f′(z)−zf′(z)f(z))≺1+Az1+Bz, |
then f∈SL∗p.
All authors declare no conflict of interest in this paper.
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