# The Fekete–Szegö problem for spirallike mappings and non-linear resolvents
in Banach spaces

Mark Elin  Department of Mathematics, Ort Braude College, Karmiel 21982,
Israel<EMAIL_ADDRESS>and  Fiana Jacobzon  Department of
Mathematics, Ort Braude College, Karmiel 21982, Israel<EMAIL_ADDRESS>
###### Abstract.

We study the Fekete–Szegö problem on the open unit ball of a complex Banach
space. Namely, the Fekete–Szegö inequalities are proved for the class of
spirallike mappings relative to an arbitrary strongly accretive operator, and
some of its subclasses. Next, we consider families of non-linear resolvents
for holomorphically accretive mappings vanishing at the origin. We solve the
Fekete–Szegö problem over these families.

###### Key words and phrases:

Fekete–Szegö inequality, holomorphically accretive mapping, spirallike
mapping, non-linear resolvent

###### 2020 Mathematics Subject Classification:

Primary 32H02; Secondary 30C45

Dedicated to the memory of Professor Gabriela Kohr.

##  1\. Introduction

Let $X$ be a complex Banach space equipped with the norm $\|\cdot\|$ and let
$X^{*}$ be the dual space of $X$. We denote by ${\mathbb{B}}$ the open unit
ball in $X$. For each $x\in X\setminus\\{0\\}$, denote

$T(x)=\left\\{\ell_{x}\in X^{*}:\|\ell_{x}\|=1\text{ and
}\ell_{x}(x)=\|x\|\right\\}.$ (1.1)

According to the Hahn–Banach theorem (see, for example, [25, Theorem 3.2]),
$T(x)$ is nonempty and may consists of a singleton (for instance, in the case
of Hilbert space), or, otherwise, of infinitely many elements. Its elements
$\ell_{x}\in T(x)$ are called support functionals at the point $x$.

Let $Y$ be a Banach space (possibly, different from $X$). The set of all
holomorphic mappings from ${\mathbb{B}}$ into $Y$ will be denoted by
$\mathop{\rm Hol}\nolimits({\mathbb{B}},Y)$. It is well known (see, for
example, [20, 9, 15, 24]) that if $f\in\mathop{\rm
Hol}\nolimits({\mathbb{B}},Y)$, then for every $x_{0}\in{\mathbb{B}}$ and all
$x$ in some neighborhood of $x_{0}\in{\mathbb{B}}$, the mapping $f$ admits the
Taylor series representation:

$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}D^{n}f(x_{0})\left[(x-x_{0})^{n}\right],$
(1.2)

where $D^{n}f(x_{0}):\prod\limits_{k=1}^{n}X\to Y$ is a bounded symmetric
$n$-linear operator that is called the $n$-th Fréchet derivative of $f$ at
$x_{0}$. Also we write $D^{n}f(x_{0})\left[(x-x_{0})^{n}\right]$ for
$D^{n}f(x_{0})[x-x_{0},\ldots,x-x_{0}].$ One says that $f$ is normalized if
$f(0)=0$ and $Df(0)=\mathop{\rm Id}\nolimits,$ the identity operator on $X$.

Recall that a holomorphic mapping $f:{\mathbb{B}}\to X$ is called
biholomorphic if the inverse $f^{-1}$ exists and is holomorphic on the image
$f({\mathbb{B}})$. A mapping $f\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$
is said to be locally biholomorphic if for each $x\in{\mathbb{B}}$ there
exists a bounded inverse for the Fréchet derivative $Df(x)$, see [9, 15].

In the one-dimensional case, where $X={\mathbb{C}}$ and
${\mathbb{B}}=\mathbb{D}$ is the open unit disk in ${\mathbb{C}}$, one usually
writes $a_{n}(x-x_{0})^{n}$ instead of
$\frac{1}{n!}D^{n}f(x)\left[(x-x_{0})^{n}\right]$ in (1.2). The classical
Fekete–Szegö problem [12] for a given subclass $\mathcal{F}\subset\mathop{\rm
Hol}\nolimits(\mathbb{D},{\mathbb{C}})$ is to find

$\sup_{f\in\mathcal{F}}\left|a_{3}-\nu a_{2}^{2}\right|,\quad\mbox{where }\
f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\ldots.$

In multi-dimensional settings various analogs of the classical Fekete–Szegö
problem for different classes of holomorphic mappings have been studied by
many mathematicians. Nice survey of the current state of the art and
references can be found in [19] and [22].

H. Hamada, G. Kohr and M. Kohr in [19] introduced a new quadratic functional
that generalizes the Fekete–Szegö functional to infinite-dimensional settings.
Moreover, they estimated this functional over several classes of holomorphic
mappings, including starlike mappings and non-linear resolvents of normalized
holomorphically accretive mappings.

The aim of this paper is to extend the method used in [19] and solve the
Fekete–Szegö problem over the classes of spirallike mappings and resolvents of
non-normalized holomorphically accretive mappings. Along the way we generalize
some results in [19] and [6].

Spirallike mappings in Banach spaces were first introduced and studied in the
mid 1970’s by K. Gurganus and T. J. Suffridge. This study has evolved into a
coherent theory thanks to the influential contributions of Gabriela Kohr and
her co-authors (I. Graham, H. Hamada, M. Kohr and others) over the past
decades (some details can be found below). As for non-linear resolvents, they
seem to have been among the last issues that caught her attention. Progress on
this topic is reflected in [13, 19].

##  2\. Preliminaries

Recall that for a densely defined linear operator $A$ with the domain
$D_{A}\subset X$, the set $V(A)=\\{\ell_{x}(Ax):x\in
D_{A},\|x\|=1,\,\ell_{x}\in T(x)\\}$ is called the numerical range of $A$.

######  Definition 2.1.

Let $A\in L(X)$ be a bounded linear operator on $X$. Then $A$ is called
accretive if

$\mathop{\rm Re}\nolimits\ell_{x}(Ax)\geq 0$

for all $\ x\in X\setminus\\{0\\}$, or, what is the same, if $m(A)\geq 0$,
where $m(A):=\inf\left\\{\mathop{\rm Re}\nolimits\lambda:\lambda\in
V(A)\right\\}$. If for some $k>0$,

$\mathop{\rm Re}\nolimits\ell_{x}(Ax)\geq k\|x\|$

for all $x\in X\setminus\\{0\\}$, the operator $A$ is called strongly
accretive.

The notion of accretivity was extended by Harris [20] to involve holomorphic
mappings (see also [24, 9]).

######  Definition 2.2.

Let $h\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$. This mapping $h$ is said
to be holomorphically accretive if

$m(h):=\liminf_{s\to 1^{-}}\bigl{(}\inf\left\\{\mathop{\rm
Re}\nolimits\ell_{x}(h(sx)):\ \|x\|=1,\ \ell_{x}\in T(x)\right\\}\bigr{)}\geq
0.$

In the case where the last lower limit $m(h)$ is positive, $h$ is called
strongly holomorphically accretive.

######  Remark 2.1.

According to [9, Proposition 2.3.2] if $h(0)=0$ then $V(A)\subset\overline{\rm
conv}\,V(h)$, where $A=Dh(0)$, in particular, $m(A)\geq m(h)$. Consequently,
if $h$ is holomorphically accretive, its linear part at zero $A$ is accretive
too. Furthermore, for such mappings Proposition 2.5.4 in [9] implies that $h$
is holomorphically accretive if and only if $\mathop{\rm
Re}\nolimits\ell_{x}(h(x))\geq 0$ for all $\
x\in{\mathbb{B}}\setminus\\{0\\}$.

The main feature of the class of holomorphically accretive mappings is that
they generate semigroups of holomorphic self-mappings on ${\mathbb{B}}$, so
they are of most importance in dynamical systems [24, 9]. A very fruitful
characterization of holomorphically accretive mappings is:

######  Proposition 2.1 (Theorem 7.3 in [24], see also [9]).

A mapping $h\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$ is holomorphically
accretive if and only if it satisfies the so-called range condition (RC), that
is, $(\mathop{\rm Id}\nolimits+rh)({\mathbb{B}})\supseteq{\mathbb{B}}$ for
each $r>0$, and the inverse mapping $J_{r}:=(\mathop{\rm
Id}\nolimits+rh)^{-1}$ is a well-defined holomorphic self-mapping of
${\mathbb{B}}$.

The mapping $J_{r}$ that occurs in this proposition is called the non-linear
resolvent of $h$. In other words, the non-linear resolvent is the unique
solution $w=J_{r}(x)\in{\mathbb{B}}$ of the functional equation

$w+rh(w)=x\in{\mathbb{B}},\quad r>0.$

Assuming $h(0)=0$, one sees that $J_{r}(0)=0$ for all $r>0$. If, in addition,
$A=Dh(0),$ then $DJ_{r}(0)=(\mathop{\rm Id}\nolimits+rA)^{-1}$. Furthermore,
the accretivity of $A$ mentioned in Remark 2.1, implies $DJ_{r}(0)$ is
strongly contractive because $\|(\mathop{\rm Id}\nolimits+rA)^{-1}\|<1.$

We use the following classes (see [15] and references therein):

$\displaystyle\mathcal{N}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\left\\{h\in\mathop{\rm
Hol}\nolimits({\mathbb{B}},X)\\!:h(0)=0,\mathop{\rm
Re}\nolimits\ell_{x}(h(x))>0,\
x\in{\mathbb{B}}\\!\setminus\\!\\{0\\},\ell_{x}\in T(x)\right\\}\\!,$
$\displaystyle\mathcal{M}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\left\\{h\in\mathcal{N},Dh(0)=\mathop{\rm
Id}\nolimits\right\\}$

and (see [14])

$\displaystyle\mathcal{N}_{A}$ $\displaystyle:=$
$\displaystyle\left\\{h\in\mathcal{N}:\ Dh(0)=A\right\\}.$ (2.1)

To proceed, we note that the inclusion $h\in\mathcal{N}$ can be expressed as
$\ell_{x}(h(x))\in g_{0}(\mathbb{D}),\
x\in{\mathbb{B}}\\!\setminus\\!\\{0\\}$, where $g_{0}(z)=\frac{1+z}{1-z}\,.$
At the same time, $\overline{V(A)}$ is a compact subset of the open right
half-plane, hence the inclusion $\ell_{x}(h(x))\in g_{0}(\mathbb{D})$ is
imprecise. It can be improved by using other functions $g\prec g_{0},$ bearing
in mind that $g(\mathbb{D})$ should contain $V(A)$ by Remark 2.1.

Throughout this paper we suppose that the following conditions hold

######  Assumption 1.

A linear operator $A$ is bounded and strongly accretive. A function
$g=g_{A}\in\mathop{\rm Hol}\nolimits(\mathbb{D},{\mathbb{C}})$ satisfies
$g\prec g_{0}$ and $\overline{V(A)}\subset g(\mathbb{D})$. Therefore
$\Delta:=g^{-1}(V(A))$ is compactly embedded in $\mathbb{D}$.

######  Definition 2.3 (cf. [2, 27]).

Let $A$ and $g$ satisfy Assumption 1. Denote

$\mathcal{N}_{A}(g)\\!:=\\!\left\\{h\in\mathcal{N}_{A}\\!:\frac{\ell_{x}(h(x))}{\|x\|}\\!\in
g(\mathbb{D}),x\in{\mathbb{B}}\\!\setminus\\!\\{0\\},\ell_{x}\in
T(x)\\!\right\\}\\!.$ (2.2)

We now consider specific choices of $g$ providing some properties of
semigroups generated by $h\in\mathcal{N}_{A}(g)$:

  * (a)

$g_{1}^{\alpha}(z):=\left(\frac{1+z}{1-z}\right)^{\alpha},\alpha\in(0,1)$: It
can be shown that the semigroup generated by every
$h\in\mathcal{N}_{A}(g_{1}^{\alpha})$ can be analytically extended with
respect to parameter $t$ to the sector $|\arg t|<\frac{\pi(1-\alpha)}{2}$; for
the one-dimensional case see [11];

  * (b)

$g_{2}^{\alpha}(z):=\alpha+(1-\alpha)\frac{1+z}{1-z},\alpha\in(0,m(A))$: it
follows from Lemma 3.3.2 in [8] that the semigroup $\\{u(t,x)\\}_{t\geq 0}$
generated by any element of $\mathcal{N}_{A}(g_{2}^{\alpha})$ satisfies the
estimate $\|u(t,x)\|\leq e^{-t\alpha}\|x\|$ uniformly on the whole
${\mathbb{B}}$;

  * (c)

$g_{3}^{\alpha}(z):=\frac{1-z}{1-(2\alpha-1)z},\alpha\in(0,1),$ maps
$\mathbb{D}$ onto a disk $\Delta$ tangent the imaginary axis. In a sense this
choice is dual to the previous one (in the one-dimensional case such duality
was investigated in [1]);

In what follows we will refer to these functions as
$g_{0},g_{1}^{\alpha},g_{2}^{\alpha},g_{3}^{\alpha}.$

Another area where holomorphically accretive mappings are widely used is
geometric function theory. The study of spirallike mappings is a good example
of this fruitful connection.

######  Definition 2.4 (see [26, 15, 8, 24]).

Let $A$ be a strongly accretive operator. A biholomorphic mapping
$f\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$ is said to be spirallike
relative to $A$ if its image is invariant under the action of the semigroup
$\\{e^{-tA}\\}_{t\geq 0}$, that is, $e^{-tA}f(x)\in f({\mathbb{B}})$ for all
$t\geq 0$ and $x\in{\mathbb{B}}$. The set of all spirallike mappings relative
to $A$ is denoted by $\mathop{\widehat{S}}\nolimits_{A}({\mathbb{B}})$.

If $f$ is spirallike relative to $A=e^{-i\beta}\mathop{\rm Id}\nolimits$ for
some $|\beta|<\frac{\pi}{2}$, then $f$ is said to be spirallike of type
$\beta$. In the particular case where $\beta=0$, spirallike mappings relative
$A=\mathop{\rm Id}\nolimits$ are called starlike.

The following result is well known (see, for example, Proposition 2.5.3 in [8]
and references therein).

######  Proposition 2.2.

Let $A\in L(X)$ be strongly accretive, and let $f\in\mathop{\rm
Hol}\nolimits({\mathbb{B}},X)$ be a normalized and locally biholomorphic
mapping. Then $f\in\widehat{S}_{A}({\mathbb{B}})$ if and only if the mapping
$h:=(Df)^{-1}Af$ belongs to $\mathcal{N}_{A}$.

This proposition inter alia implies that a spirallike mapping $f$ relative to
$A$ linearizes the semigroup $u(t,x)$ generated by $h=(Df)^{-1}Af$ in the
sense that $f\circ u(t,f^{-1}(x))=e^{-tA}x$ on $f({\mathbb{B}})$. In the one-
dimensional case, any linear operator is scalar, hence can be chosen to be
$A=e^{i\beta}\mathop{\rm Id}\nolimits$. In this case the inclusion
$h=(Df)^{-1}Af\in\mathcal{N}_{A}$ is equivalent to $\mathop{\rm
Re}\nolimits\left(e^{-i\beta}\frac{zf^{\prime}(z)}{f(z)}\right)>0$. This is
the standard definition of spirallike functions of type $\beta$ on
$\mathbb{D}$ (see, for example, [5, 15]).

Moreover, according to Proposition 2.2, it is relevant to consider
biholomorphic functions $g\in\mathop{\rm
Hol}\nolimits(\mathbb{D},{\mathbb{C}})$ satisfying Assumption 1 and to
distinguish subclasses of $\widehat{S}_{A}({\mathbb{B}})$ letting

$\mathop{\widehat{S}}\nolimits_{g}({\mathbb{B}}):=\left\\{f\in\widehat{S}_{A}({\mathbb{B}}):(Df)^{-1}Af\in\mathcal{N}_{A}(g)\right\\}.$
(2.3)

In particular,
$\widehat{S}_{g_{0}}({\mathbb{B}})=\widehat{S}_{A}({\mathbb{B}})$. Further,
$\widehat{S}_{g_{1}^{\alpha}}({\mathbb{B}})$ consists of mappings that are
spirallike relative to operator $e^{i\beta}A$ with any $|\beta|<1-\alpha$. The
classes $\widehat{S}_{g_{2}^{\alpha}}({\mathbb{B}})$ and
$\widehat{S}_{g_{3}^{\alpha}}({\mathbb{B}})$ are also of specific interest.
For instance, if $A=e^{i\beta}\mathop{\rm Id}\nolimits$ and
$\alpha=\lambda\cos\beta$, the class
$\widehat{S}_{g_{3}^{\alpha}}({\mathbb{B}})$ of spirallike mappings of type
$\beta$ of order $\lambda$ is a widely studied object. The intersection
$\widehat{S}_{g_{2}^{\alpha}}({\mathbb{B}})\bigcap\widehat{S}_{g_{3}^{\alpha}}({\mathbb{B}})$
consists of strongly spirallike mappings (for an equivalent definition and
properties of these mappings see [17, 18, 3]).

##  3\. Auxiliary lemmata

Our first auxiliary result essentially coincides with Theorem 2.12 in [19]. We
present it in a somewhat more general form.

######  Lemma 3.1.

Let $p(z)=a+p_{1}z+p_{2}z^{2}+o(z^{2})$ and
$\phi(z)=a+b_{1}z+b_{2}z^{2}+o(z^{2})$ be holomorphic functions on
$\mathbb{D}$ such that $\phi\prec p$. Then for every $\mu\in{\mathbb{C}}$ the
following sharp inequality holds:

$|b_{2}-\mu b_{1}^{2}|\leq\max\left(|p_{1}|,|p_{2}-\mu p_{1}^{2}|\right).$

###### Proof.

Since $\phi\prec p$, there is a function $\omega\in\Omega$ such that
$\phi=p\circ\omega$. Let $\omega(z)=c_{1}z+c_{2}z^{2}+o(z^{2})$. Then

$b_{1}=p_{1}c_{1}\quad\mbox{and}\quad b_{2}=p_{2}c_{1}^{2}+p_{1}c_{2}.$

Therefore

$b_{2}-\mu b_{1}^{2}=(p_{2}-\mu p_{1}^{2})c_{1}^{2}+p_{1}c_{2}.$

Because the inequality $|c_{2}|\leq 1-|c_{1}|^{2}$ holds and is sharp (see,
for example, [5]), one concludes that $|b_{2}-\mu b_{1}^{2}|$ is bounded by a
convex hull of $|p_{1}|$ and $|p_{2}-\mu p_{1}^{2}|$. The result follows. ∎

######  Lemma 3.2.

Let $h\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$ with $h(0)=0$ and $B\in
L(X)$ with $\rho:=\|B\|\leq 1$. For any $x\in\partial{\mathbb{B}}$ and
$\ell\in X^{*}$ denote

$\varphi(t):=\frac{\ell\left(h(tBx)\right)}{t},\qquad
t\in\mathbb{D}\setminus\\{0\\}.$

  * (i)

The function $\varphi$ can be analytically extended to the disk
$\frac{1}{\rho}\mathbb{D}$ with the Taylor expansion
$\varphi(t)=b_{0}+b_{1}t+b_{2}t^{2}+o(t^{2})$, where $b_{0}=\ell(Dh(0)Bx)$,

$b_{1}=\frac{1}{2!}\,\ell\left(D^{2}h(0)[(Bx)^{2}]\right)\quad\mbox{and}\quad
b_{2}=\frac{1}{3!}\,\ell\left(D^{3}h(0)[(Bx)^{3}]\right).$ (3.1)

  * (ii)

If, in addition, $\ell\in T(Bx)$ and $h\in\mathcal{N}_{A}(g)$, then
$\varphi(\mathbb{D})\subset\rho\widehat{g}(\rho\mathbb{D})$, where
$\widehat{g}(t)=g\left(\frac{\tau-t}{1-t\overline{\tau}}\right)$ and
$\tau=g^{-1}\left(\frac{\ell(Dh(0)Bx)}{\|Bx\|}\right)$.

###### Proof.

The function $\varphi$ is holomorphic whenever $\|tBx\|<1$, that is, for
$|t|<\frac{1}{\rho}\leq\frac{1}{\|Bx\|}$. Represent $h$ by the Taylor series
(1.2). A straightforward calculation proves (i).

Recall that $h\in\mathcal{N}_{A}(g)$, hence Definition 2.3 implies
$\frac{\varphi(t)}{\|Bx\|}\in g(\mathbb{D})=\widehat{g}(\mathbb{D})$ as
$|t|<\frac{1}{\rho}$. Therefore the function
$\widehat{g}^{-1}(\frac{\varphi(\cdot)}{\|Bx\|})$ maps the disk of radius
$\frac{1}{\rho}$ into $\mathbb{D}$ and preserves zero. By the Schwarz Lemma
$\widehat{g}^{-1}(\frac{\varphi(t)}{\|Bx\|})\leq\rho|t|$. Thus
$\varphi\prec\|Bx\|\widehat{g}(\rho\,\cdot)$. The proof is complete. ∎

A mapping $f\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$ is said to be of
one-dimensional type if it takes the form $f(x)=s(x)x$ for some
$s\in\mathop{\rm Hol}\nolimits({\mathbb{B}},{\mathbb{C}})$. Such mappings were
studied by many authors (see, for example, [23, 10, 4] and references
therein).

######  Lemma 3.3.

Let $f\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$ be a mapping of one-
dimensional type. Then for every $n\in{\mathbb{N}}$ the entire mapping
$x\mapsto D^{n}f(0)[x^{n}]$ is also of one-dimensional type. Therefore for any
$x\in\partial{\mathbb{B}},\ \ell_{x}\in T(x)$ and constants
$\mu_{j}\in{\mathbb{C}},\ j=1,2,\ldots,$ we have

$\displaystyle\left|\ell_{x}\left(\sum_{j=1}^{n}\mu_{j}D^{j}f(0)[x^{j}]\right)\right|=\left\|\sum_{j=1}^{n}\mu_{j}D^{n}f(0)[x^{j}]\right\|.$

###### Proof.

The first assertion is evident (for detailed calculation see [7]). To prove
the second one we note that there is a function $F\in\mathop{\rm
Hol}\nolimits(X,{\mathbb{C}})$ such that
$\sum_{j=1}^{n}\mu_{j}D^{j}f0)[x^{j}]=F(x)x$. Thus for any
$x\in\partial{\mathbb{B}}$ we have

$\displaystyle\left\|\sum_{j=1}^{n}\mu_{j}D^{j}f(0)[x^{j}]\right\|=|F(x)|\|x\|\qquad\mbox{and}$
$\displaystyle\ell_{x}\left(\sum_{j=1}^{n}\mu_{j}D^{j}f(0)[x^{j}]\right)=F(x)\ell_{x}(x)=F(x),$

which completes the proof. ∎

##  4\. Fekete–Szegö inequalities for spirallike mappings

In what follows $A$ and $g$ satisfy Assumption 1, and the class
$\mathop{\widehat{S}}\nolimits_{g}({\mathbb{B}})$ is defined by formula (2.3).

######  Theorem 4.1.

Let $x\in\partial{\mathbb{B}}$, $\ell_{x}\in T(x)$ and
$\tau=g^{-1}(\ell_{x}(Ax))$. Assume that
$g\left(\frac{\tau-z}{1-z\overline{\tau}}\right)=q_{0}+q_{1}z+q_{2}z^{2}+o(z^{2})$.
Given $f\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$ denote

$\displaystyle\widetilde{a}_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ell_{x}\left(D^{2}f(0)\left[x,D^{2}f(0)[x,Ax]\right]-\frac{1}{2}D^{2}f(0)\left[x,AD^{2}f(0)[x^{2}]\right]\right),$
$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2!}\ell_{x}\left(2D^{2}f(0)[x,Ax]-AD^{2}f(0)[x^{2}]\right),$
(4.1) $\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle\frac{1}{2\cdot
3!}\ell_{x}\left(3D^{3}f(0)[x^{2},Ax]-AD^{3}f(0)[x^{3}]\right).$

If $f\in\mathop{\widehat{S}}\nolimits_{g}({\mathbb{B}})$, then for any
$\nu\in{\mathbb{C}}$ we have

$\left|a_{3}-(\nu-1)a_{2}^{2}-\widetilde{a}_{2}^{2}\right|\leq\frac{|q_{1}|}{2}\max\left\\{1,\left|\frac{q_{2}}{q_{1}}+2(\nu-1)q_{1}\right|\right\\}.$
(4.2)

######  Remark 4.1.

It can be directly calculated that $q_{1}=-g^{\prime}(\tau)(1-|\tau|^{2})$ and
$\frac{q_{2}}{q_{1}}=\overline{\tau}-\frac{g^{\prime\prime}(\tau)}{2g^{\prime}(\tau)}(1-|\tau|^{2})$.
Thus the right-hand side in (4.2) can be expressed by the hyperbolic and pre-
Schwarzian derivatives of $g$.

###### Proof.

Let $h(x)=\left[Df(x)\right]^{-1}Af(x)$. Recall that $f$ is a normalized
biholomorphic mapping. Let the Taylor expansion of $f$ be

$f(x)=x+\frac{1}{2!}D^{2}f(0)[x^{2}]+\frac{1}{3!}D^{3}f(0)[x^{3}]+o(\|x\|^{3}),$
(4.3)

so that

$Df(x)[w]=w+D^{2}f(0)[x,w]+\frac{1}{2}D^{3}f(0)[x^{2},w]+o(\|x\|^{2}).$ (4.4)

Take the Taylor expansion
${h(z)=Ax+\frac{1}{2}D^{2}h(0)[x^{2}]+\frac{1}{6}D^{3}h(0)[x^{3}]+o(\|x\|^{3})}$
and substitute it together with (4.3)–(4.4) into the equality

$Df(x)[h(x)]=Af(x).$

This gives us

$\displaystyle
Ax+\frac{1}{2}D^{2}h(0)[x^{2}]+\frac{1}{6}D^{3}h(0)[x^{3}]+D^{2}f(0)[x,Ax]$
$\displaystyle+$
$\displaystyle\frac{1}{2}D^{2}f(0)[x,D^{2}h(0)x^{2}]+\frac{1}{2}D^{3}f(0)[x^{2},Ax]+o(\|x\|^{3})$
$\displaystyle=$ $\displaystyle
Ax+\frac{1}{2}AD^{2}f(0)[x^{2}]+\frac{1}{6}AD^{3}f(0)[x^{3}]+o(\|x\|^{3}).$

Equating terms of the same order leads to

$\frac{1}{2}D^{2}h(0)[x^{2}]+D^{2}f(0)[x,Ax]=\frac{1}{2}AD^{2}f(0)[x^{2}]$

and

$\frac{1}{6}D^{3}h(0)[x^{3}]+\frac{1}{2}D^{2}f(0)[x,D^{2}h(0)x^{2}]+\frac{1}{2}D^{3}f(0)[x^{2},Ax]=\frac{1}{6}AD^{3}f(0)[x^{3}].$

In turn, these equalities imply

$D^{2}h(0)[x^{2}]=AD^{2}f(0)[x^{2}]-2D^{2}f(0)[x,Ax]$

and

$\displaystyle
D^{3}h(0)[x^{3}]=AD^{3}f(0)[x^{3}]-3D^{2}f(0)[x,D^{2}h(0)x^{2}]-3D^{3}f(0)[x^{2},Ax]$
$\displaystyle=AD^{3}f(0)[x^{3}]-3D^{3}f(0)[x^{2},Ax]$
$\displaystyle-3D^{2}f(0)\left[x,AD^{2}f(0)[x^{2}]\right]+6D^{2}f(0)\left[x,D^{2}f(0)[x,Ax]\right]\\!.$

Recall that $\ell_{x}(Ax)\in V(A)\subset g(\mathbb{D})$, so $\tau\in\Delta$ is
well-defined. Similarly to the proof of the Theorem 3.1 in [19], denote

$\varphi(t)=\left\\{\begin{array}[]{ll}\frac{\ell_{x}\left(h(tx)\right)}{t},&t\in\mathbb{D}\setminus\\{0\\},\vspace{2mm}\\\
\ell_{x}(Ax),&t=0.\end{array}\right.$

Then $\varphi\in\mathop{\rm Hol}\nolimits(\mathbb{D},{\mathbb{C}})$ by
assertion (i) of Lemma 3.2 with $B=\mathop{\rm Id}\nolimits$,

$b_{1}=\frac{1}{2!}\ell_{x}\left(D^{2}h(0)[x^{2}]\right)\quad\mbox{and}\quad
b_{2}=\frac{1}{3!}\ell_{x}\left(D^{3}h(0)[x^{3}]\right).$

Using $a_{2},\,\widetilde{a}_{2}^{2}$ and $a_{3}$ defined in (4.1) we get

$b_{1}=-a_{2}\quad\mbox{and}\quad b_{2}=2\widetilde{a}_{2}^{2}-2a_{3}.$

Therefore,

$\displaystyle\left|a_{3}-\widetilde{a}_{2}^{2}-(\nu-1)a_{2}^{2}\right|=\frac{1}{2}\left|b_{2}-2(1-\nu)b_{1}^{2}\right|.$

Also, by assertion (ii) of the same Lemma 3.2, $\varphi\prec\widehat{g},\
\widehat{g}(t)=g(\frac{\tau-t}{1-\overline{\tau}t})$.

To this end we apply Lemma 3.1 with $p=\widehat{g}$ and $\mu=2(1-\nu)$ and
obtain estimate (4.2). ∎

There are two ways to make the above result more explicit: to fix some
concrete forms of the function $g$, or to put additional restrictions on the
mapping $f$. We start with some concrete choices of $g$.

Recall that for every strongly accretive operator $A$ and every spirallike
mapping $f$ relative to $A$, the mapping $h:=\left(Df\right)^{-1}Af$ is
holomorphically accretive. Hence one can always choose $g=g_{0}$, where
$g_{0}(z)=\frac{1+z}{1-z}$ is defined above. Denoting $\ell:=\ell_{x}(Ax)$ and
using Remark 4.1, we conclude that every spirallike mapping relative to $A$
satisfies

$\displaystyle\left|a_{3}-(\nu-1)a_{2}^{2}-\widetilde{a}_{2}^{2}\right|\leq\mathop{\rm
Re}\nolimits\ell\cdot\max\left(1,\left|1+4(\nu-1)\mathop{\rm
Re}\nolimits\ell\right|\right).$ (4.5)

In the one-dimensional case, this inequality coincides with the result of
Theorem 1 in [21] for $\lambda=0$. By choosing other $g\prec g_{0}$ functions
and denoting $\ell:=\ell_{x}(Ax)$ as above, more precise estimates can be
obtained.

Assume, for example, that $\ell_{x}(h(x))$ belongs to some sector of the form
$\left\\{w:|\arg w|<\frac{\pi\alpha}{2}\right\\}$, $\alpha\in(0,1),$ for all
$x\in{\mathbb{B}}$, where $h=\left(Df\right)^{-1}Af$. Then one can set
$g=g_{1}^{\alpha}$ and to get

######  Corollary 4.1.

Every $f\in\mathop{\widehat{S}}\nolimits_{g_{1}^{\alpha}}({\mathbb{B}})$
satisfies

$\left|a_{3}-(\nu-1)a_{2}^{2}-\widetilde{a}_{2}^{2}\right|\leq\alpha|\ell|\cos\arg\ell^{\frac{1}{\alpha}}\cdot\max\left\\{1,Q_{1,\alpha}\right\\},$

where $Q_{1,\alpha}=\mathop{\rm
Re}\nolimits\ell^{\frac{1}{\alpha}}\left|4\alpha(\nu-1)\ell^{\frac{\alpha-1}{\alpha}}+\frac{1}{\ell^{\frac{1}{\alpha}}}\left(\alpha+i\tan\arg\ell^{\frac{1}{\alpha}}\right)\right|$.

Also assuming that $\frac{\ell_{x}(h(x))}{\|x\|}$ is bounded away from the
imaginary axis, namely, $\mathop{\rm
Re}\nolimits\frac{\ell_{x}(h(x))}{\|x\|}>\alpha,\ \alpha\in(0,1)$, we choose
$g=g_{2}^{\alpha}$. In this situation, we have

######  Corollary 4.2.

Every $f\in\mathop{\widehat{S}}\nolimits_{g_{2}^{\alpha}}({\mathbb{B}})$
satisfies

$\displaystyle\left|a_{3}-(\nu-1)a_{2}^{2}-\widetilde{a}_{2}^{2}\right|\leq\mathop{\rm
Re}\nolimits\ell\cdot\max\left\\{1,Q_{2,\alpha}\right\\},$

where $Q_{2,\alpha}=\left|1+4(\nu-1)(1-\alpha)\mathop{\rm
Re}\nolimits\ell\right|.$

In particular, taking $\alpha=0$, we return to inequality (4.5) for all
spirallike mappings relative to the linear operator $A$.

Another interesting (and, as we mentioned, dual) case occurs when
$\frac{\ell_{x}(h(x))}{\|x\|}$ lies in some circle tangent to the imaginary
axis. We can then set $g=g_{3}^{\alpha}$.

######  Corollary 4.3.

Every $f\in\mathop{\widehat{S}}\nolimits_{g_{3}^{\alpha}}({\mathbb{B}})$
satisfies

$\displaystyle\left|a_{3}-(\nu-1)a_{2}^{2}-\widetilde{a}_{2}^{2}\right|\leq(\mathop{\rm
Re}\nolimits\ell-|\ell|^{2}\alpha)\cdot\max\left\\{1,Q_{3,\alpha}\right\\},$

where $Q_{3,\alpha}=\left|1-2\overline{\ell}\alpha+4(\nu-1)(\mathop{\rm
Re}\nolimits\ell-|\ell|^{2}\alpha)\right|.$

Recall that for $A=e^{i\beta}\mathop{\rm Id}\nolimits$, the class
$\mathop{\widehat{S}}\nolimits_{g_{3}^{\alpha}}({\mathbb{B}})$ consists of so-
called spirallike mappings of type $\beta$ of order $\alpha$.

######  Remark 4.2.

It is worth mentioning that even for the the case in which $A$ is a scalar
operator, the estimates above (starting from (4.5)) are new. Since the class
of spirallike mappings contains the class of starlike mappings, these
estimates generalize Corollary 3.4 (i)–(iv) in [19] for starlike mappings.

In the rest of this section we deal with mappings $f$ that satisfy:

######  Assumption 2.

There exists a function $\kappa:\partial{\mathbb{B}}\to{\mathbb{C}}$ such that

$D^{2}f(0)[x^{2}]=\kappa(x)x,\quad x\in\partial{\mathbb{B}}.$ (4.6)

The Fréchet derivatives of $f$ of second and third order $D^{2}f(0)$ and
$D^{3}f(0)$ commute with the linear operator $A$ in the sense that

$D^{k}f(0)[x^{k-1},Ax]=AD^{k}f(0)[x^{k}],\quad k=2,3.$ (4.7)

Condition (4.6) holds automatically for one-dimensional type mappings
(spirallike mappings of one-dimensional type were studied, for instance, in
[10, 22, 7]), while condition (4.7) holds automatically whenever $A$ is a
scalar operator.

In turn, relations (4.7) in Assumption 2 imply that formulae (4.1) become

$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2!}\ell_{x}\left(AD^{2}f(0)[x^{2}]\right),$
$\displaystyle\widetilde{a}_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ell_{x}\left(AD^{2}f(0)[x,D^{2}f(0)[x^{2}]]\right),$
(4.8) $\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{3!}\ell_{x}\left(AD^{3}f(0)[x^{3}]\right).$

######  Corollary 4.4.

If $f\in\mathop{\widehat{S}}\nolimits_{A}({\mathbb{B}})$ satisfies Assumption
2, then for any $\nu\in{\mathbb{C}},$

$\left|a_{3}-\left(\nu-1+\frac{1}{\ell_{x}\left(Ax\right)}\right)a_{2}^{2}\right|\leq\frac{|q_{1}|}{2}\max\left\\{1,\left|\frac{q_{2}}{q_{1}}+2(\nu-1)q_{1}\right|\right\\}.$
(4.9)

###### Proof.

Indeed, denote $\alpha=\ell_{x}\left(Ax\right)$.Then
$a_{2}=\frac{1}{2}\kappa(x)\alpha$ and

$\displaystyle\widetilde{a}_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\ell_{x}\left(AD^{2}f(0)[x,\kappa(x)x]\right)=\frac{1}{4}\cdot\kappa(x)\ell_{x}\left(AD^{2}f(0)[x^{2}]\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\cdot\kappa(x)\ell_{x}\left(A\kappa(x)x\right)=\frac{\alpha}{4}\cdot(\kappa(x))^{2}.$

Thus $\widetilde{a}_{2}^{2}=\frac{1}{\alpha}a_{2}^{2}$ and hence

$|a_{3}-(\nu-1)a_{2}^{2}-\widetilde{a}_{2}^{2}|=\left|a_{3}-\left(\nu-1+\frac{1}{\alpha}\right)a_{2}^{2}\right|.$

So, estimate (4.9) follows from Theorem 4.1. ∎

Let $A$ be a scalar operator. Without loss of generality, we assume
$A=e^{i\beta}\mathop{\rm Id}\nolimits,\ |\beta|<\frac{\pi}{2}$. Then it
follows from Assumption 2 that formulae (4.1) (or (4)) become

$\begin{array}[]{ll}\displaystyle
a_{2}=\frac{1}{2!}\kappa(x)e^{i\beta}\quad\displaystyle\widetilde{a}_{2}^{2}=\left(\frac{1}{2!}\kappa(x)\right)^{2}e^{i\beta},\quad\displaystyle
a_{3}=\frac{1}{3!}\ell_{x}\left(D^{3}f(0)[x^{3}]\right)e^{i\beta}.\end{array}$

These relations and Lemma 3.3 imply immediately

######  Corollary 4.5.

If $f\in\mathop{\rm Hol}\nolimits({\mathbb{B}},X)$ is a spirallike mapping of
type $\beta$, that satisfies Assumption 2. Then for any $\mu\in{\mathbb{C}}$
we have

$\left|a_{3}-\mu
a_{2}^{2}\right|\leq\frac{|q_{1}|}{2}\max\left\\{1,\left|\frac{q_{2}}{q_{1}}+2(\mu-e^{-i\beta})q_{1}\right|\right\\}.$

If, in addition, $f$ is of one-dimensional type, then for any
$x\in\partial{\mathbb{B}}$ we have

$\displaystyle\left\|\frac{1}{3!}D^{3}f(0)[x^{3}]-\mu\cdot\frac{1}{2!}D^{2}f(0)\left[x,\frac{1}{2!}D^{2}f(0)[x^{2}]\right]\right\|$
$\displaystyle\leq\frac{|q_{1}|}{2}\max\left\\{1,\left|\frac{q_{2}}{q_{1}}+2(\mu-e^{-i\beta})q_{1}\right|\right\\}.$

The last estimate coincides with Theorem 2 in [7].

##  5\. Fekete–Szegö inequalities for normalized non-linear resolvents

As above, we suppose that $A\in L(X)$ and $g\in\mathop{\rm
Hol}\nolimits(\mathbb{D},{\mathbb{C}})$ satisfy Assumption 1 and
$h\in\mathcal{N}_{A}(g)$. In this section we concentrate on the non-linear
resolvent $J_{r}:=(\mathop{\rm Id}\nolimits+rh)^{-1},\ r>0,$ that is well-
defined self-mappings of the open unit ball ${\mathbb{B}}$ that solves the
functional equation

$J_{r}(x)+rh(J_{r}(x))=x\in{\mathbb{B}},\quad r>0.$ (5.1)

######  Lemma 5.1.

  * (a)

For any $r>0$, the operator $B_{r}:=DJ_{r}(0)=(\mathop{\rm
Id}\nolimits+rA)^{-1}$ is strongly contractive, that is,
$\rho_{r}:=\|B_{r}\|<1$.

  * (b)

If $h$ is of one-dimensional type, then $A$ is a scalar operator and $J_{r}$,
$r>0$, is of one-dimensional type too.

###### Proof.

Assertion (a) follows from the strong accretivity of $A$.

Since $h$ is of one-dimensional type, it has the form $h(x)=s(x)x$, where
$s\in\mathop{\rm Hol}\nolimits({\mathbb{B}},{\mathbb{C}})$. Therefore
$A=Dh(0)=s(0)\mathop{\rm Id}\nolimits$. In addition, (5.1) implies

$x=J_{r}(x)+rs(J_{r}(x))J_{r}(x)=(1+rs(J_{r}(x)))J_{r}(x),$

that is, $J_{r}(x)$ is collinear to $x$. ∎

Further, it is natural to consider the family of normalized resolvents
$(\mathop{\rm Id}\nolimits+rA)J_{r}$ and to study the Fekete–Szegö problem for
these mappings.

We now present the main result of this section.

######  Theorem 5.1.

Let $h\in\mathcal{N}_{A}(g)$ and $J_{r}$ be the nonlinear resolvent of $h$ for
some $r>0$. For $x\in\partial{{\mathbb{B}}}$ and $\ell_{r}:=\ell_{B_{r}x}\in
T(B_{r}x)$, let

$\displaystyle\widetilde{a}_{2}^{2}$ $\displaystyle:=$
$\displaystyle\ell_{r}\left((\mathop{\rm
Id}\nolimits+rA)\frac{1}{2!}D^{2}J_{r}(0)\left[x,(\mathop{\rm
Id}\nolimits+rA)\frac{1}{2!}D^{2}J_{r}(0)[x^{2}]\right]\right),$
$\displaystyle a_{2}$ $\displaystyle:=$
$\displaystyle\ell_{r}\left((\mathop{\rm
Id}\nolimits+rA)\frac{1}{2!}D^{2}J_{r}(0)[x^{2}]\right),$ (5.2) $\displaystyle
a_{3}$ $\displaystyle:=$ $\displaystyle\ell_{r}\left((\mathop{\rm
Id}\nolimits+rA)\frac{1}{3!}D^{3}J_{r}(0)[x^{3}]\right).$

Then for any $\nu\in{\mathbb{C}}$ we have

$\left|a_{3}-2\widetilde{a}_{2}^{2}-(\nu-2)a_{2}^{2}\right|\leq
r|q_{1}|\|B_{r}x\|\rho_{r}^{2}\max\left(1,Q_{r}(x)\right),$ (5.3)

where

$Q_{r}(x):=\left|\frac{q_{2}}{q_{1}}-(2-\nu)rq_{1}\|B_{r}x\|\right|$ (5.4)

and $q_{1},q_{2}$ are the Taylor coefficients of
$\widehat{g}(t)=g\left(\frac{\tau-t}{1-t\overline{\tau}}\right)$ with
$\tau=g^{-1}\left(\frac{\ell_{r}(AB_{r}x)}{\|B_{r}x\|}\right)$.

###### Proof.

Denote $x_{r}:=B_{r}x$. Using the functional equation (5.1), one finds

$\left(I+rA\right)D^{2}J_{r}(0)[x,y]=-rD^{2}h(0)\left[x_{r},B_{r}y\right]$

and

$\displaystyle(\mathop{\rm
Id}\nolimits+rA)\frac{1}{2!}D^{2}J_{r}(0)[x^{2}]=-r\frac{1}{2!}D^{2}h(0)\left[(x_{r})^{2}\right]\\!,$
$\displaystyle(\mathop{\rm
Id}\nolimits+rA)\frac{1}{3!}D^{3}J_{r}(0)[x^{3}]=-r\frac{1}{3!}B_{r}D^{3}h(0)\left[(x_{r})^{3}\right]$
$\displaystyle\hskip
31.29802pt+2r^{2}\cdot\frac{1}{2!}B_{r}D^{2}h(0)\left[x_{r},B_{r}\frac{1}{2!}D^{2}h(0)\left[(x_{r})^{2}\right]\right]\\!\\!.$

Thus the quantities $a_{2},\widetilde{a}_{2}^{2}$ and $a_{3}$ defined by (5.1)
can be expressed by the Fréchet derivatives of $h$:

$\displaystyle\widetilde{a}_{2}^{2}$ $\displaystyle=$ $\displaystyle
r^{2}\frac{1}{2!}\ell_{r}\left(D^{2}h(0)\left[x_{r},\frac{1}{2!}B_{r}D^{2}h(0)\left[(x_{r})^{2}\right]\right]\right)$
$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle-r\frac{1}{2!}\ell_{r}\left(D^{2}h(0)\left[(x_{r})^{2}\right]\right)$
(5.5) $\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle-r\displaystyle\frac{1}{3!}\ell_{r}\left(D^{3}h(0)\left[(x_{r})^{3}\right]\right)$
$\displaystyle+$ $\displaystyle
2r^{2}\ell_{r}\left(\frac{1}{2!}D^{2}h(0)\left[x_{r},\frac{1}{2!}B_{r}D^{2}h(0)\left[(x_{r})^{2}\right]\right]\right).$

Denote

$\varphi(t)=\left\\{\begin{array}[]{ll}\frac{\ell_{r}\left(h(tx_{r})\right)}{t},&t\in\mathbb{D}\setminus\\{0\\},\vspace{2mm}\\\
\ell_{r}(Ax_{r}),&t=0.\end{array}\right.$

By assertion (i) of Lemma 3.2 with $B=B_{r}$, the function $\varphi$ is
analytic in the disk of radius $\frac{1}{\rho_{r}}$ and

$b_{1}=\frac{1}{2!}\cdot\ell_{r}\left(D^{2}h(0)[(x_{r})^{2}]\right)\quad\mbox{and}\quad
b_{2}=\frac{1}{3!}\cdot\ell_{r}\left(D^{3}h(0)[(x_{r})^{3}]\right).$ (5.6)

Comparing formulae (5.6) and (5) we see that

$b_{1}=-\frac{1}{r}a_{2}\qquad\text{and}\qquad
b_{2}=-\frac{1}{r}(a_{3}-2\widetilde{a}_{2}^{2}).$

Therefore,

$\left|a_{3}-\widetilde{a}_{2}^{2}-(\nu-2)a_{2}^{2}\right|=r\left|b_{2}-r(2-\nu)b_{1}^{2}\right|.$

Also, by assertion (ii) of Lemma 3.2,
$\varphi\prec\|x_{r}\|\widehat{g}(\rho_{r}\cdot)$.

To complete the proof we apply Lemma 3.1 with
$p=\|x_{r}\|\widehat{g}(\rho_{r}\cdot)$ and $\mu=r(2-\nu)$. ∎

From now on, for any $x\in\partial{\mathbb{B}}$ we will adopt the notations
$x_{r}=B_{r}x$ and $\ell_{r}:=\ell_{x_{r}}\in T(x_{r})$. To compare our
results with the previous ones we consider some special cases.

If, for example, $A=\lambda\mathop{\rm Id}\nolimits$, $\mathop{\rm
Re}\nolimits\lambda>0$, is a scalar operator, then $B_{r}=\frac{1}{1+\lambda
r}\mathop{\rm Id}\nolimits$, $x_{r}=\frac{1}{1+\lambda r}x$ and
$\rho_{r}=\|x_{r}\|=\frac{1}{|1+\lambda r|}$. Thus

$\tau=g^{-1}\left(\frac{\ell_{r}(\lambda
x_{r})}{\|x_{r}\|}\right)=g^{-1}(\lambda).$ (5.7)

Thus inequality (5.3) takes the form

$\left|a_{3}-2\widetilde{a}_{2}^{2}-(\nu-2)a_{2}^{2}\right|\leq\frac{|q_{1}|r}{|1+\lambda
r|^{3}}\max\left(1,\left|\frac{q_{2}}{q_{1}}-\frac{q_{1}r}{|1+\lambda
r|}(2-\nu)\right|\right)\\!,$ (5.8)

where $q_{1},q_{2}$ are the Taylor coefficients of
$\widehat{g}(t)=g\left(\frac{\tau-t}{1-t\overline{\tau}}\right)$ with
$\tau=g^{-1}\left(\lambda\right)$.

######  Corollary 5.1.

Assume that $A=\lambda\mathop{\rm Id}\nolimits$, $\mathop{\rm
Re}\nolimits\lambda>0$ and $g=g_{0}$. Then for any $\nu\in{\mathbb{C}}$ we
have

$\left|a_{3}-2\widetilde{a}_{2}^{2}-(\nu-2)a_{2}^{2}\right|\leq\frac{|1+\lambda^{2}|r}{|1+\lambda
r|^{3}}\max\left(1,\left|\lambda-(2-\nu)r\frac{1+\lambda^{2}}{|1+\lambda
r|}\right|\right)\\!.$ (5.9)

###### Proof.

Since $g=g_{0}$, formula (5.7) is
$\tau=g^{-1}(\lambda)=\frac{\lambda-1}{\lambda+1}.$ Thus
$q_{1}=-(1+\lambda^{2})$ and $q_{2}=\lambda(1+\lambda^{2})$. Then (5.9)
follows from (5.8). ∎

For $A=\mathop{\rm Id}\nolimits$, Corollary 5.1 coincides with [19, Theorem
5.6].

Another interesting case occurs when $h$ satisfies Assumption 2.

######  Corollary 5.2.

If $h\in\mathcal{N}_{A}(g)$ satisfies Assumption 2, then

$\left|a_{3}-(\nu-2+2\delta)a_{2}^{2}\right|\leq
r|q_{1}|\|x_{r}\|\rho_{r}^{2}\max\left(1,Q_{r}(x)\right),$ (5.10)

where $Q_{r}(x)$ is defined by (5.4) and
$\delta=\frac{\ell_{r}(B_{r}x_{r})}{\|x_{r}\|^{2}}$.

###### Proof.

Since $h$ satisfies condition (4.6), there exists a function
$\kappa:\partial{\mathbb{B}}\to{\mathbb{C}}$ such that
$D^{2}h(0)[x^{2}]=\kappa(x)x,$ $x\in\partial{\mathbb{B}}$. Thus,

$\displaystyle a_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{4}\left(\ell_{r}\left(D^{2}h(0)\left[(x_{r})^{2}\right]\right)\right)^{2}=\frac{r^{2}}{4}\left(\ell_{r}\left(\kappa(x_{r})x_{r}\right)\right)^{2}$
$\displaystyle=$
$\displaystyle\left(\frac{r}{2}\kappa(x_{r})\right)^{2}\|x_{r}\|^{2}.$

At the same time,

$\displaystyle\widetilde{a}_{2}^{2}$ $\displaystyle=$ $\displaystyle
r^{2}\frac{1}{2!}\ell_{r}\left(D^{2}h(0)\left[x_{r},\frac{1}{2!}B_{r}D^{2}h(0)\left[(x_{r})^{2}\right]\right]\right)$
$\displaystyle=$ $\displaystyle
r^{2}\frac{1}{4}\ell_{r}\left(D^{2}h(0)\left[x_{r},B_{r}\kappa(x_{r})x_{r}\right]\right)$
$\displaystyle=$ $\displaystyle
r^{2}\frac{1}{4}\kappa(x_{r})\ell_{r}\left(D^{2}h(0)\left[x_{r},B_{r}x_{r}\right]\right).$

The mapping $h$ also satisfies condition (4.7), then

$\displaystyle\widetilde{a}_{2}^{2}$ $\displaystyle=$ $\displaystyle
r^{2}\frac{1}{4}\kappa(x_{r})\ell_{r}\left(B_{r}D^{2}h(0)\left[(x_{r})^{2}\right]\right)$
$\displaystyle=$ $\displaystyle
r^{2}\frac{1}{4}\kappa(x_{r})\ell_{r}\left(B_{r}\kappa(x_{r})x_{r}\right)=\left(\frac{r}{2}\kappa(x_{r})\right)^{2}\ell_{r}\left(B_{r}x_{r}\right).$

Now estimate (5.10) follows from the relation $\widetilde{a}_{2}^{2}=\delta
a_{2}^{2}$ with $\delta=\frac{\ell_{r}(B_{r}x_{r})}{\|x_{r}\|^{2}}$. ∎

If $h$ is of a one-dimensional type, then $A=\lambda\mathop{\rm Id}\nolimits$
for some $\lambda\in{\mathbb{C}}$ by Lemma 5.1. In this case formula (5.10)
gets a simpler form.

######  Corollary 5.3.

If $h\in\mathcal{N}_{A}(g)$ is one-dimensional type with $A=\lambda\mathop{\rm
Id}\nolimits$, then for any $\nu\in{\mathbb{C}}$ we have

$\displaystyle\left\|(\mathop{\rm
Id}\nolimits+rA)\frac{1}{3!}D^{3}J_{r}(0)[x^{3}]-\mu(\mathop{\rm
Id}\nolimits+rA)\frac{1}{2!}D^{2}J_{r}(0)\left[x,(\mathop{\rm
Id}\nolimits+rA)\frac{1}{2!}D^{2}J_{r}(0)[x^{2}]\right]\right\|\vspace{1mm}$
$\displaystyle=\left|a_{3}-\mu
a_{2}^{2}\right|\,\leq\,\frac{r|q_{1}|}{|1+\lambda
r|^{3}}\cdot\max\left(1,\left|\frac{q_{2}}{q_{1}}-(2\delta-\mu)\frac{rq_{1}}{|1+\lambda
r|}\right|\right)$

with $\delta=\frac{|1+\lambda r|}{1+\lambda r}.$

In particular, if $A=\mathop{\rm Id}\nolimits$ and $g=g_{0}$, this coincides
with [19, Corollary 5.7].

###### Proof.

By Lemma 3.3, there is a function $\kappa$ such that $\frac{1+\lambda
r}{2!}D^{2}J_{r}(0)[x^{2}]=\kappa(x)x$. Then the left-hand term equals to

$\displaystyle\left\|\frac{1+r\lambda}{3!}D^{3}J_{r}(0)[x^{3}]-\mu\frac{1+r\lambda}{2!}\kappa(x)D^{2}J_{r}(0)[x^{2}]\right\|.$

Lemma 3.3 states that this is equal to

$\displaystyle\left|\ell_{x}\left(\frac{1+r\lambda}{3!}D^{3}J_{r}(0)[x^{3}]-\mu\frac{1+r\lambda}{2!}\kappa(x)D^{2}J_{r}(0)[x^{2}]\right)\right|$
$\displaystyle=$ $\displaystyle\left|a_{3}-\mu
a_{2}\kappa(x)\right|=\left|a_{3}-\mu a_{2}^{2}\right|.$

Set $\mu=\nu-2+2\delta$. Then we proceed by Corollary 5.2:

$\displaystyle\leq$ $\displaystyle\frac{r|q_{1}|}{|1+\lambda
r|^{3}}\cdot\max\left(1,\left|\frac{q_{2}}{q_{1}}-(2-\nu)\frac{rq_{1}}{|1+\lambda
r|}\right|\right)$ $\displaystyle=$ $\displaystyle\frac{r|q_{1}|}{|1+\lambda
r|^{3}}\cdot\max\left(1,\left|\frac{q_{2}}{q_{1}}-(2\delta-\mu)\frac{rq_{1}}{|1+\lambda
r|}\right|\right).$

∎

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# Human Identity-Preserved Motion Retargeting in Video Synthesis by Feature
Disentanglement

Jingzhe Ma<EMAIL_ADDRESS>Department of Computer Science and
EngineeringSouthern University of Science and TechnologyShenzhenGuangdongChina
and  Shiqi Yu<EMAIL_ADDRESS>Department of Computer Science and
EngineeringSouthern University of Science and TechnologyShenzhenGuangdongChina

(2022)

###### Abstract.

Most motion retargeting methods in human action video synthesis decompose the
input video to motion (dynamic information) and shape (static information).
However, we observe if the dynamic information is directly transferred to
another subject, it will result in unnatural synthesised motion. This
phenomenon is mainly caused by neglecting subject-dependent information in
motion. To solve the problem, we propose a novel motion retargeting method
which can combine both subject-independent (common motion content) information
from a source video and subject-dependent (individualized identity motion)
information from a target video. So it can synthesize videos with a much
natural appearance along with identity-preserved motion (Figure 1). In the
proposed method two encoders are employed to extract identity and motion
content representations respectively. We employ the adaptive instance
normalization (AdaIN) layer in the generator and the instance normalization
(IN) layer in the motion content encoder to synthesize the new motion.
Besides, we also collected a dataset, named $Chuang101$, with 101 subjects in
total. Each subject performs identical dancing movement, and so it is
convenient for feature disentanglement among motion and identity of each
subject. Furthermore, an efficient quantitative metric for identify
information is designed by gait recognition. The experiments show the proposed
method can synthesize videos more naturally when the subject’s identity is
preserved.

Video motion retargeting, Feature disentanglement, Motion analysis, Subject-
dependent dataset

††copyright: acmcopyright††journalyear: 2022††doi:
XXXXXXX.XXXXXXX††conference: ; ††booktitle: ;††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06

##  1\. Introduction

Figure 1. Human identity-preserved motion retargeting. The source videos’
subject-independent information, i.e. common motion content (MC), is
transferred to a target appearance and preserves the subject-dependent
information, i.e. individualized identity motion (ID), in target video.
Figure 2.  The pipeline of inference framework. Different from the training
stage (Figure 3), the framework only has two input videos, i.e., the source
video and the target video. Then the 2D keypoints are estimated by a off-the-
shelf method, OpenPose (Cao et al., 2019). The two motions can be disentangled
as the ID and MC features by disentanglement block, and synthesize a motion by
motion synthesis block. Finally, the pre-trained VQ-GAN (Esser et al., 2021)
is employed to render the synthesis video.

Motion retargeting refers to transferring motion from a source video of one
person to another person in a target video. This technology has great
potential in augmented reality (AR) and virtual reality (VR). With its help,
users can easily generate their own videos with some specific motions even
they do not act those motions. Movie clips can also be generated when the
actors do not perform. It is a key technology in video editing and content
generation.

There are many methods to achieve motion retargeting, which can be divided
into three categories by data modality: image-based (Li et al., 2019; Zhou et
al., 2019; Siarohin et al., 2018; Zhu et al., 2019), video-based (Chan et al.,
2019; Wang et al., 2018, 2019; Mallya et al., 2020; Huang et al., 2021), and
skeleton-based (Chan et al., 2019; Aberman et al., 2019b, 2020a; Yang et al.,
2020; Liu et al., 2019). The image-based methods get encouraging results on
conditional generation. However, those methods ignored the context in the
temporal domain, and the generated motion may not be the concordant with the
appearance. The video-based methods partially solve the temporal neglecting
issue but it may introduce a complex model hard to be trained. There are many
skeleton-based methods to solve both aforementioned problems. The skeleton-
based methods can further be directly 3D keypoints-based (Aberman et al.,
2020a; Liu et al., 2019) and 2D keypoints-based (Chan et al., 2019; Aberman et
al., 2019b; Yang et al., 2020; Aberman et al., 2019a). It is difficult to
obtain accurately 3D keypoints from video through keypoints estimation
algorithms (Yoshiyasu et al., 2018). Therefore, we study motion retargeting
based on the 2D keypoints modality. However, we observe that previous works on
2D skeleton-based methods do not consider the subject-dependent of subjects in
motion features. For example, if the motion of a male subject is directly
transferred to a female, the generated motion will not be concordant with the
female.

Recently, there have been some works (Xie et al., 2021; Su et al., 2021) try
to solve the aforementioned problem. However, it still remains the following
issues: (1) No efficient decoupling of subject-dependent and subject-
independent information, (2) lack of enough labeled data which contains
similar motion videos from different subjects, otherwise it will be difficult
to evaluate the disentanglement, (3) lack of a suitable metric to evaluate
whether synthetic motion contains subject-dependent information.

In this study, we propose a novel identity-preserved motion retargeting
framework as shown in Figure 2, and it is training pipeline as shown in Figure
3. The inference framework is divided into three stages: skeleton estimation,
motion transfer in 2D keypoints space by feature disentanglement (our method),
and skeleton-to-video. In the first stage, we employ an off-the-shelf
algorithm, OpenPose (Cao et al., 2019), to estimate the 2D skeleton sequences
of the dancer in the video. In the second stage, to cope with the
aforementioned first challenge, we propose a novel disentanglement block to
decompose the representations of subject-dependent, i.e. individualized
identity motion (ID), and subject-independent, i.e. common motion content (MC)
in motions. In addition, we propose a motion synthesis block capable of
synthesizing motion and preserving subject identities. Furthermore, adaptive
instance normalization (AdaIN) and instance normalization (IN) can extract
statistically (Huang and Belongie, 2017). We employ AdaIN and IN in our block
to retarget motions. Meanwhile, triplet loss (Schroff et al., 2015) is
utilized to learn contrastively ID and MC information of subjects. To address
the second challenge, a dataset with 101 subjects is collected on the
Internet, which is named $Chuang101$. Compared with the previous dataset,
$Chuang101$ has the following advantages: more subjects (dancers), complex
dance actions, and each subject performing the same group of actions. For the
third challenge, we propose a new metric, named Identity Score ($IDScore$), to
evaluate the synthesized motion whether preserved ID information. Thus, we
propose an efficient quantitative metric by gait recognition algorithm. In the
last stage, we convert the transferred 2D skeleton sequence frame-by-frame to
a pixel-level representation and feed these images into a pre-trained VQ-GAN
(Esser et al., 2021) network to render the video.

We summarize our contributions as follows:

  * 1)

We propose a novel framework in 2D skeleton space, which can disentangle the
ID and MC representations and synthesize a motion that can preserve the ID
information of the target subject.

  * 2)

101 dance videos have been collect online from the public domain, and the
different subjects in this dataset perform the same group of actions.

  * 3)

We propose a new metric, Identity Score (IDScore), to evaluate the identity
representations whether retargeting to the synthetic motion correctly.
Experiments demonstrate that our method can disentangle the identity
representations and apply in the synthesized videos effectively.

Figure 3. The training stage of our proposed method. Before training, we
sample some paired motion videos, and employ the OpenPose method to get 2D
skeleton sequences. During training phase, our task is training
disentanglement block (Section 3.1) and motion synthesis block (Section 3.2),
where $M_{real}^{i}$, $i\in\\{1,2,3\\}$ denotes the real motion from the
$i$-th branch (differnet colors), $\bar{f}_{id}^{i}$ and $f_{mc}^{i}$ denote
the ID and MC representations, and $M_{sys}^{j|k}$, $j,k\in\\{1,2,3\\}$ denote
the synthesized motions, ${j|k}$ denotes the MC feature from $j$-th branch
(left color) and ID feature from $k$-th branch (right color). To ensure that
we correctly disentangle the ID and MC information, we use some loss
functions(Section 3.4).

##  2\. Related work

Over the last several years, some works have been dedicated to skeleton-based
motion retargeting from 2D inputs. Several works (Ma et al., 2017; Dong et
al., 2018; Zhao et al., 2020; Cui et al., 2021) have explored to motion
retargeting in image space. Ma et al. (Ma et al., 2017) proposed the pioneer
top-down method $PG^{2}$, which mainly used conditional GAN (Mirza and
Osindero, 2014). $PG^{2}$ mainly contains two stages: pose integration and
image refinement. The top-down method usually shows poor results for large
differences between source and target poses. To address the above deficiency,
based-deformation methods (Dong et al., 2018; Cui et al., 2021) are proposed
by transferring the feature maps of the source person image into the target
pose. To explore the latent feature representation in motion retargeting, Zhao
et al. (Zhao et al., 2020) extract pose features from an interpolated pose
sequence, from source pose to target pose. Although the works mentioned above
perform impressive results, they only retarget the action in image space and
ignore the temporal information.

Several approaches focus on video motion retargeting from 2D inputs. Aberman
et al. (Aberman et al., 2019a) proposed a two-branch framework with part
confidence map as 2D pose representations to clone the performance in a video.
Abermanet et al. (Aberman et al., 2019b) trained a deep neural network to
decompose temporal sequences of 2D poses into three components: motion,
skeleton, and camera view-angle. Recently, Yang et al. (Yang et al., 2020)
proposed a method can be trained in an unsupervised manner to decompose
aforementioned three components: view, shape and motion. Their work is named
TransMoMo. However, the aforementioned methods directly transfer the motion
from source subject to target subject, which will result in the motion of the
target subject having abnormal motion after synthesis.

More recently, to address the above problem, Xie et al. (Xie et al., 2021)
proposed a novel network to decompose motion to unique and common features.
For disentanglement of unique and common features, they employed the gradient
reversal layer (GRL) (Ganin et al., 2016) to restrict the gradient of common
features. However, their method was not effective for situations where the
identity label information is unseen, i.e. in-the-wild videos. Therefore, we
propose a novel disentanglement block containing the MC encoder and ID
encoder. Meanwhile, the triplet loss (Hoffer and Ailon, 2015) is employed to
decompose the ID and MC representations. In addition, we employ IN layer in
the MC encoder to control the identity information in motions and perform the
global average-pooling in ID motion encoder to suppress the motion content
information. Finally, we employ the AdaIN layer to fuse the MC and ID
representations statistically.

Unlike other related works (Chan et al., 2019; Aberman et al., 2019b, 2020a;
Yang et al., 2020; Liu et al., 2019), our model’s evaluation metrics need to
evaluate whether the identity information is transferred correctly. Xie et al.
(Xie et al., 2021) employed a classification to evaluate the identity
information. However, the metric is not applied in unseen identity situations.
If the identity is unknown, their metric will not be applied. Thus, we propose
an novel quantitative metric by gait recognition algorithm.

##  3\. Methods

Our problem can be defined as: given a target motion
$M_{t}\in\mathbb{R}^{2J\times T}$ and a source motion
$M_{s}\in\mathbb{R}^{2J\times T}$ in 2D space, we disentangle the identity
(ID) and motion content (MC) features by learning the disentanglement block
$E(\cdot)$ and synthesis of a motion by motion synthesis block $S(\cdot)$. The
synthetic motion has the ID information from the target motion $M_{t}$ and the
MC information from the source motion $M_{s}$. The formula can be represented
as:

(1) $M_{sys}=S(E(M_{s}),E(M_{t})),$

where $M_{sys}\in\mathbb{R}^{2J\times T}$, $J$ is the number of skeleton
points, $T$ is the number of frames.

Our framework for the training stage is illustrated in Figure 3. Before
training, we sample three videos. Then, we estimate the 2D keypoints by an
off-the-shelf network, OpenPose (Cao et al., 2019). The estimated results can
be denoted as $M_{real}^{1}$, $M_{real}^{2}$, and $M_{real}^{3}$, where $real$
means the motions from real data, and the shape of each motion is
$\mathbb{R}^{2J\times T}$, where $J$ is the number of keypoints, and $T$ is
the number of motion frames. In particular, $M_{real}^{1}$ and $M_{real}^{2}$
have the same MC label and have different ID labels, in contrast,
$M_{real}^{2}$ and $M_{real}^{3}$ have different MC labels with the same ID
label. Our training stage contains three branches. We feed $M_{real}^{i}$,
$i\in\\{1,2,3\\}$ into disentanglement block of correspond branch
respectively, and each disentanglement block is share parameters. Then the ID
and MC features are obtained, which can be denoted as: $\bar{f}_{id}^{i}$ and
$f_{mc}^{i}$ respectively. Afterwards, we cross MC and ID features to take
full advantage of these samples, thus we perform the Cartesian Product to get
a tuple, i.e.
$\\{f_{mc}^{1},f_{mc}^{2},f_{mc}^{3}\\}\times\\{\bar{f}_{id}^{1},\bar{f}_{id}^{2},\bar{f}_{id}^{3}\\}$.
Nine permutations are obtained, and feed these permutations to motion
synthesis block to get the synthesized motions, which can be denoted as
$M_{sys}^{j|k}$, where $j|k$ denotes the MC information from $j$-th branch,
$j\in\\{1,2,3\\}$, the ID information from $k$-th branch, $k\in\\{1,2,3\\}$.
The discriminator $D(\cdot)$ is employed to ensure that the distribution of
synthetic motion is similar to the real motion. The discriminator is a
temporal convolutional network. Moreover, we use the disentanglement block
again after the motion synthesis block to ensure that the synthesized motion’s
ID and MC information are similar to the input motion. The inference phase
pipeline is shown in the second stage in Figure 2.

###  3.1. Disentanglement block

To get the ID and MC representations in the latent space, we propose a
disentanglement block, as shown in Figure 4. The block mainly contains two
encoder $E_{mc}(\cdot)$ and $E_{id}(\cdot)$, which are capture the MC and ID
representations.

Figure 4. The pipeline of our disentanglement block. Where $IN$ means the
instance normalization, $E_{mc}(\cdot)$ and $E_{id}(\cdot)$ denote the MC and
ID encoders, $g$ means the identity projection head, $Avgpooling$ means the
global average pooling on temporal dimension.

The function $E_{mc}(\cdot)$, i.e. motion content encoder (MC encoder), uses
several one dimensional convolution layers. The MC feature can be represented
as:

(2) $f_{mc}^{i}=E_{mc}(M_{real}^{i})\in\mathbb{R}^{C_{mc}\times N_{mc}},$

where $N_{mc}$ is the sequence number after MC encoder, $C_{mc}$ is the number
of channels, and $i\in\\{1,2,3\\}$. In addition, inspired by (Aberman et al.,
2020b), we use instance normalization(IN) layers in $E_{mc}(\cdot)$, which can
control mainly local statistics, in other words, identity information, and
preserve the motion content.

The function $E_{id}(\cdot)$, i.e. identity motion encoder (ID encoder), has a
similar network structure with $E_{mc}(\cdot)$ apart from the IN layers. The
identity sequence feature can be denoted as:

(3) $f_{id}^{i}=E_{id}(M_{real}^{i})\in\mathbb{R}^{C_{id}\times N_{id}},$

where $N_{id}$ is the sequence number after ID encoder, $C_{id}$ is the number
of channels. It is worth mentioning that $E_{id}(\cdot)$ takes the average
temporal pooling after the last downsampling layer, and the ID feature can be
denoted as:

(4) $\bar{f}_{id}^{i}=AvgPooling(f_{id}^{i})\in\mathbb{R}^{C_{id}\times 1},$

where $AvgPooling$ denotes average pooling on the temporal dimension.
Furthermore, (Chen et al., 2020) proposed a non-linear transformation
structure, projection head, which can improve the performance in their
downstream tasks. Therefore, we employ identity projection $g(\cdot)$ head for
identity sequence feature $f_{id}^{i}$, which contains two fully connected
layers to project the $f_{id}^{i}$ to another latent space. The projected
features can be denoted as:

(5)
$h_{id}^{i}=g(f_{id}^{i})=W^{(2)}\sigma(W^{(1)}f_{id}^{i})\in\mathbb{R}^{C_{p}\times
1},$

where $\sigma$ is the Leaky ReLU activation function, $W^{(\cdot)}$ is the
weight for $g(\cdot)$, and $C_{p}$ is the number of channels after identity
projection head. It is worth mentioning that $h_{id}^{i}$ mainly used to
compute triplet loss (Equation 8), and $\bar{f}_{id}^{i}$ mainly used to the
downstream tasks, i.e. motion synthesis.

###  3.2. Motion synthesis block

Figure 5. The pipeline of our Motion synthesis block. The Conv block contains
a 1D convolution layer and a LeakyReLU activate function. The Upsample layer
use the linear interpolation. The ToImageSpace and FromImageSpace are 1D
convolution layers with one kernel size to map the learning space (channel-
wise) to real motion space and Inverse mapping respectively.

Our motion synthesis block is shown in Fig. 5. Recall that we decompose ID and
MC features by our disentanglement block. Firstly, we extract the statistical
representations (mean $\mu(\bar{f}_{id}^{i})$ and variance
$\sigma(\bar{f}_{id}^{i})^{2}$) of ID feature by a multilayer perceptron
(MLP). Then, we apply the statistical representations to the AdaIN layer, the
formula can be denoted as:

(6)
$AdaIN(f_{mc}^{i},f_{id}^{i})\coloneqq\sigma(\bar{f}_{id}^{i})\frac{f_{mc}^{i}-\mu(f_{mc}^{i})}{\sigma(f_{mc}^{i})}+\mu(\bar{f}_{id}^{i}).$

The AdaIN layer mainly applies an affine transformation to the feature of per-
channel by constituting a normalization layer. Note that the affine
transformation is temporally invariant and hence only affects non-temporal
features of the motion i.e. ID information. For removing the local statistics
of MC feature by the IN layer, the AdaIN layer can fuse the MC information
with the other ID information by scaling and shifting the feature channels.
Afterwards, we employ a skip connection to learn the high-level
representations of fused features. Inspired by inspired by (Karras et al.,
2017) we employ some PG blocks in decoder to improve the quality of
synthesized motion. The PG block mainly contain Conv block, Upsample layer,
ToImageSpace layer and FromImageSpace layer. In particular, the ToImageSpace
and FromImageSpace are 1D convolution layer of kernel size 1, $\alpha$ is a
hyperparameter to control the importance of upsample feature to add features
in image space of channel. In our study, we set $\alpha=0.9$. The conv block
contain a 1D convolution layer and a LeakyReLU activate function.

Figure 6. Schematic diagram for $IDScore$. The $Raw$, $Cross$, and
$Reconstruction$ denote the raw data, cross stage, reconstruction stage
evaluation pipeline respectively.

###  3.3. Identity score

In this section, we introduce our proposed metric, Identity Score (IDScore).
The goal of gait recognition is to capture the unique biometric
characteristics from the common human walking process, which is pretty
suitable for our purpose, i.e. identifying individuals from the generally
similar dance motion. Thus, we employ a gait recognition algorithm, GaitGraph
(Teepe et al., 2021), to capture the ID information and evaluate whether it
transfers correctly in synthesis motion. In gait recognition, the test set is
usually split into gallery and probe set, where the identities of sequences in
former set are considered known while the latter are unknown.

In order to evaluate the identity recognition performance, we conduct three
experiments: test raw probe data, reconstruct probe data and cross a new
subject to probe data. The details are as shown in Fig . 6. In the test raw
probe data stage, the evaluated dataset is divided into gallery and probe
sets, which have non-overlapping motion sequences. In the reconstruction
stage, all probe data is reconstructed and fed the reconstruction results to
the off-the-shelf GaitGraph network to get the recognition performances,
$Rank_{1}$ and $Rank_{5}$. We denoted the recognition performances in
reconstruction stage as $Rank_{1}^{rec}$ and $Rank_{5}^{rec}$ respectively. In
the crossing stage, we collect a motion of a new subject (the identity
information apart from the gallery set). Then the ID information of the new
subject is transferred to all probe data. Similarly, the crossed probe data is
fed into GaitGraph. The recognition performances in crossing stage are named
as $Rank_{1}^{cross}$ and $Rank_{5}^{cross}$ respectively. The cross
performances reflect the recognition effect after transferring the new subject
identity information. Ideally, the performances should be random guessing
because the new subject information is not registered in the gallery set, in
other words, the gallery set does not contain the new subject.

For $Rank_{A}^{rec}$, $A\in\\{1,5\\}$, the larger value means better
reconstruction effect. In constant, for $Rank_{A}^{cross}$, the small value
means a better transfer performance because the transferred identity is not in
the gallery data. It is worth mentioning that another reason for the crossing
performance decrease is the poor reconstruction results. The difference
between the reconstructing and crossing recognition performance is employed to
eliminate this factor, named $IDScore$. The higher results indicate that more
identity information is preserved. The $IDScore$ can be formulated as:

(7) $\begin{split}IDScore_{1}=Rank_{1}^{rec}-Rank_{1}^{cross},\\\
IDScore_{5}=Rank_{5}^{rec}-Rank_{5}^{cross},\end{split}$

(a) MC feature using t-SNE

(b) ID feature using t-SNE

Figure 7. The disentanglement results of ID and MC representations in test
dataset.

###  3.4. Loss functions

In our study, we employ some loss functions in the training phase: identity
motion loss, motion content loss, motion reconstruction loss, and adversarial
loss.

Identity motion loss. Recall that we use three input motions in the training
phase, as shown in Figure 2. To ensure the disentanglement of the ID feature,
we employ a triplet loss (Hoffer and Ailon, 2015) to improve the clustering of
the different ID representations in the latent space, named ID triplet loss,
which is defined as:

(8)
$\mathcal{L}_{id}^{tri}=max\\{\frac{1}{C_{p}}\left\|h_{id}^{2}-h_{id}^{3}\right\|_{2}-\frac{1}{C_{p}}\left\|h_{id}^{2}-h_{id}^{1}\right\|_{2}+\delta,0\\},$

where $\delta$ is the margin hyperparameter and we set $\delta=0.2$,
$\left\|\cdot\right\|^{2}$ is L2 distance.

Furthermore, recall that we employ the disentanglement block again for the
synthesized motion. The ID feature of synthesized motion is formulated as
$\bar{f}_{id}^{i^{\prime}}$. To ensure the ID representation of the
synthesized motion $\bar{f}_{id}^{i^{\prime}}$ similar to the raw ID feature
$f_{id}^{i}$, the mean absolute error (MAE) is employed, named ID
reconstruction loss, which is defined as:

(9)
$\begin{split}\mathcal{L}_{id}^{rec}=\frac{1}{3}[&\frac{1}{3C_{id}}\sum_{i=1}^{3}|\bar{f}_{id}^{i}-\bar{f}_{id}^{i^{\prime}}|\\\
&+\frac{1}{3C_{id}N_{id}}\sum_{i=1}^{3}|f_{id}^{i}-f_{id}^{i^{\prime}}|\\\
&+\frac{1}{3C_{p}}\sum_{i=1}^{3}|h_{id}^{i}-h_{id}^{i^{\prime}}|],\end{split}$

where, the $\frac{1}{3}$ outside the square brackets acts on three terms, the
$\frac{1}{3}$ inside the square brackets acts on the three branches, and
$|\cdot|$ is L1 distance.

Motion content loss. Similarly, triplet loss is designed to extract MC
information, named MC triplet loss, which is defined as:

(10)
$\mathcal{L}_{mc}^{tri}=max\\{\frac{1}{C_{mc}N_{mc}}\left\|f_{mc}^{2}-f_{id}^{1}\right\|_{2}-\frac{1}{C_{mc}N_{mc}}\left\|f_{mc}^{2}-f_{mc}^{3}\right\|_{2}+\delta,0\\},$

where, $\delta=0.2$. Note the difference in negative and positive samples
between Equation 10 and Equation 8.

Similarly, we define the motion content reconstruction loss (MC reconstruction
loss) as:

(11)
$\mathcal{L}_{mc}^{rec}=\frac{1}{3C_{mc}N_{mc}}\sum_{i=1}^{3}|f_{mc}^{i}-f_{mc}^{i^{\prime}}|,$

Motion reconstruction loss. Recall that we cross the ID and MC features before
the motion synthesis block, the 9 permutations are obtained and fed these
permutations to motion synthesis block, we get 9 synthesized motions.
Therefore, the motion reconstruction loss is defined as:

(12)
$\mathcal{L}_{rec}=\frac{1}{9}\sum_{j=1}^{3}\sum_{k=1}^{3}\mathbb{E}_{M_{real}^{j|k}\sim\mathcal{M}}|M_{real}^{j|k}-M_{sys}^{j|k}|,$

where $\mathcal{M}$ is the motion training set, and $M_{real}^{j|k}$ denotes
the ground truth motion of $M_{sys}^{j|k}$.

Adversarial loss. The discriminator is used to measure the domain discrepancy
between the real motions and synthesized motions. Therefore, the adversarial
loss is defined as:

(13)
$\mathcal{L}_{adv}=\frac{1}{9}\sum_{j=1}^{3}\sum_{k=1}^{3}\mathbb{E}_{M_{real}^{j|k}\sim\mathcal{M}}\left[\frac{1}{2}logD(M_{real}^{j|k})+\frac{1}{2}log(1-D(M_{sys}^{j|k}))\right].$

Total loss. We combine the above terms. Finally we get our final loss, that is
defined as:

(14)
$\begin{split}\mathcal{L}=&\lambda_{rec}\mathcal{L}_{rec}+\lambda_{adv}\mathcal{L}_{adv}\\\
&+\lambda_{mc}(\mathcal{L}_{mc}^{rec}+\mathcal{L}_{mc}^{tri})\\\
&+\lambda_{id}(\mathcal{L}_{id}^{rec}+\mathcal{L}_{id}^{tri}).\end{split}$

In our study, we set $\lambda_{rec}$, $\lambda_{adv}$, $\lambda_{mc}$, and
$\lambda_{id}$ are $10$, $2$, $2$, and $6$ respectively.

Table 1. Results on the $IDScore$ metric. Methods | $Rank_{1}^{rec}\uparrow$ | $Rank_{1}^{cross}\downarrow$ | $IDScore_{1}\uparrow$ | $Rank_{5}^{rec}\uparrow$ | $Rank_{5}^{cross}\downarrow$ | $IDScore_{5}\uparrow$
---|---|---|---|---|---|---
TransMoMo (Yang et al., 2020) | 26.33% | 12.24% | 14.09% | 66.53% | 30.82% | 35.71%
LCM (Aberman et al., 2019b) | 18.37% | 14.49% | 3.88% | 44.90% | 29.59% | 15.31%
Ours |  32.86% |  10.41% | 22.45% |  72.86% |  29.59% | 43.27%
| $Rank_{1}$ |  |  | $Rank_{5}$ |  |
Raw data | 34.69% |  |  | 77.35% |  |

##  4\. Experiments

###  4.1. Setup

Dataset. To correctly disentangle ID and MC features, we collect 101 subjects’
solo dance videos online from the public domain, named $Chuang101$, with the
same choreography and background music. We manually align the videos in the
temporal dimension, and the length of each video is 3 minutes and 40 seconds
and has 30fps. Then, each video is resized to $512\times 512$ at the same
scale. To obtain the 2D keypoints in our dataset, OpenPose (Cao et al., 2019)
is employed. Meanwhile, some invalid frames in each video, more than a third
of the 2D keypoints are missing, are removed. Afterward, we pad the keypoints
of missing less than a third according to the five frames before and after. In
addition, we trim each 2D skeleton sequence into a clip of 64 frames. We get
81 dance clips for each video. Finally, the 2D skeleton sequences are split
into training and test sets without overlapping. The training set has 91
subjects with 7371 dance clips, and the test set has 10 subjects with 810
dance clips. Furthermore, 2 videos from YouTube are collected to test the
generated performance of our method.

Implementation details. All 2D skeleton keypoints use the BODY_25 keypoints
format of OpenPose. Our method contains about 1.9M parameters. The detail of
training hypterparameters shown in supplementary material. In addition, we set
the number of input frame number $T=64$ and the number joints $J=25$. In
disentanglement block, we set the MC encoder last layer channel $C_{mc}=128$
and ID encoder last layer channel $C_{id}=144$. Especially, the channel of the
last layer of the identity projection head $C_{p}=144$. In discriminator, we
employ some 1D temporal convolution layers, and the last channel of
discriminator is $256$. Inspired by Least squares generative adversarial
networks (LSGAN) (Mao et al., 2017), we employ their adversarial losses in our
model to constrain the distribution of synthesis motion similarly to real
motion. The specific version of triplet loss adopted in ID and MC triplet
loss, is Batch All (BA+) triplet loss (Hermans et al., 2017).

Moreover, we use the VQ-GAN network (Esser et al., 2021) to render the
synthesized motions to videos. The synthesized motions are changed to pix
level frame-by-frame as an input map. The VQ-GAN contains about 87M
parameters. And all resolution are $512\times 512$. The parameters of the
discriminator network are frozen at the first $10000$ steps.

Figure 8. 2D skeleton reconstruction results, where the red skeletons are
ground truth, green skeletons are our method, blue skeletons are TransMoMo
method, and yellow skeletons are LCM method. Each column means the different
frames in the sequence.

###  4.2. Representation Disentanglement

The ID and MC representations in latent space are visualized in Figure 7 to
demonstrate the effectiveness of the proposed disentanglement block, i.e. the
disentanglement ability of ID and MC representations. Expecially, we randomly
sample some different subjects’ motions to get the ID and MC representations
by our disentanglement block. Afterward, we project the representations into
2D space and 3D space by using t-SNE (Van der Maaten and Hinton, 2008).

Motion content feature. The 2D projection of our MC feature is shown in Figure
7(a), where each MC label name is marked with different colors. Considering 81
MC labels could not be shown in the figure, we only sample 10 labels in test
dataset to show in the figure. Each label contains ten samples/subjects. Our
network learned to cluster the MC information, demonstrating that the MC
information will similarly manipulate the synthesized motion.

Table 2. Ablation study results. Methods | w/o id projector | w/o PG | w/o AdaIN | w/o avg | Ours(full) | Raw data
---|---|---|---|---|---|---
$Rank_{1}^{rec}$ | 28.16% | 30.2% | 28.16% | 28.16% | 32.86% | $Rank_{1}$ | 34.69%
$Rank_{1}^{cross}$ | 10.00% | 11.84% | 10.41% | 10.82% | 10.41%
$IDScore_{1}$ | 18.16% | 18.36% | 17.75% | 17.34% | 22.45%
$Rank_{5}^{rec}$ | 69.59% | 68.78% | 68.98% | 69.18% | 72.86% | $Rank_{5}$ | 77.35%
$Rank_{5}^{cross}$ | 27.96% | 29.18% | 27.96% | 30.00% | 29.59%
$IDScore_{5}$ | 41.63% | 39.60% | 41.02% | 39.18% | 43.27%

Identity motion feature. Figure 7(b) shows the t-SNE result of our ID feature,
colored by their ID label names. We can see that our model has a certain
disentanglement capability in terms of ID representation. There is still some
ID information that is not decoupled. We think it may be disentangled in
higher dimensions. Because we project the ID representation to 3D space by
t-SNE and find that some identities also have a certain degree of
discrimination in the z dimension, such as the 00074 (blue) and 00077 (green),
00079 (light brown), and 00071 (pink) in 3D space are separated. The
visualization of 3D space is shown in the supplementary material (id_3d.html).

###  4.3. Comparison

In this subsection, we quantitatively and qualitatively compare our method
with other state-of-the-art methods using motion transfer results. The
compared methods mainly include: TransMoMo (Yang et al., 2020), and Learning
Character-Agnostic Motion (LCM) (Aberman et al., 2019b).

Quantitative Comparisons. Since the output of TransMoMo and LCM methods is 15
keypoints in OpenPose BODY_25 format, for the fairness of the comparison, we
transform our output(25 keypoints in OpenPose BODY_25 format) to 15 keypoints.
It is worth mentioning that the input to the GaitGraph method is 17 keypoints
in COCO format. Therefore, we train an MLP module to map 15 keypoints in
OpenPose BODY_25 format to 17 keypoints in COCO format. Then, we feed the
synthesis motions to the MLP module to get the 17 keypoints in COCO format.
Furthermore, we directly employ GaitGraph pre-train weight on CASIA-B (Yu et
al., 2006), an open dataset for gait recognition, and do not fine-turn on our
test dataset.

Table 1 shows the $IDScore$ comparing our method with others. The recognition
result of raw data is the upper bound of this experiment. We can see that our
method outperforms other methods in $IDScore$, which means that our method can
effectively transfer ID information. The LCM and TransMoMo methods do not
consider the identity information in the motion, therefore, they have the
lower $IDScore$. Specially, the $Rank_{1}^{cross}$ metric of method LCM is
also better, which is caused by the poor reconstruction performance. In
addition, our method has the 32.86% reconstruction performance and is closer
to the upper bound 34.69%, which means the reconstruction results of our
method contain more identity information instead of the motion clone.

Figure 9. The rendered results of quantitative comparisons. All synthesized
motions are the results of cross the target identity information. The first
two rows are the same source motion and different target identity motion.

Qualitative Comparisons. We sample some motions in our test dataset and
reconstruct these motions to evaluate the reconstruction performance of our
method. Figure 8 visually compares our method with others in 2D skeleton
reconstruction. Note that the red skeletons are ground truth, green skeletons
are our method, blue skeletons are the TransMoMo method, and yellow skeletons
are the LCM method. We can see that our method is most similar to the ground
truth skeleton in complex motion, such as the last column. These results
demonstrate that our method can reconstruct high quality skeleton and preserve
their identity information.

In addition, to explore the cross-domain of our methods, the two in-the-wild
videos are collected on YouTube. In this experiment, we combine our test
dataset’s ID information of in-the-wild videos and MC information. Then, we
render the synthesized motion of state-of-the-art methods and ours to RGB
videos. The state-of-the-art methods mainly contain TransMoMo (Yang et al.,
2020) and Learning Character-Agnostic Motion (LCM) (Aberman et al., 2019b).
The results are as shown in Figure 9. Where the first two columns are the
source and target video clips, and the first column of each method is a
synthesized skeleton. We can see that our method can learn the ID information
of the target video, such as the first row, previous methods perform the same
action (raise arms) as the source video. In addition, the target video has the
right view in the last row, and the source video is not. The view of our
rendered video is the same as the target video. It aligns with the conjecture
we mentioned above, i.e. our method does not have the same action but acts
with the target video ID information.

###  4.4. Ablation study

We train some ablated models to study the impact of the different blocks in
our method. The results are shown in Table 2. We set up $5$ ablated models.
The w/o id projector means we removing the id projector block in identity
encoder. The w/o PG means we removing the ToImageSpace and FromImageSpace in
PG block. The w/o AdaIN means we removing the IN in MC encoder and AdaIN in
decoder.The w/o avg means we removing the global average pooling in ID
encoder. We can see that each block is beneficial and that our full method
(full) achieves the best results in reconstruction performance. In particular,
w/o id projector can achieve the lowest recognition performance in crossing
stage, that is they have the poor reconstruction performance. The ablation
study of the skeleton visualization is shown in the supplementary material.

##  5\. Conclusion and Future Work

To make the synthesized videos are more natural according to properties of the
person, the human identity is involved in our proposed method. Human
identities can be disentangled from motions, and can also be used to
synthesize motions by the disentanglement block and the motion synthesis block
respectively. With the help of our dataset $Chuang101$, the model can be
trained and identifies can be easily disentangled. The evaluation with gait
recognition, a human identification technology by walking style, also proves
the human identity plays an important role in video synthesis.

In our proposed work, it is shown that the human identity can help human video
synthesis. From the results we can easily expand the idea to more attributes
of a subject. That means if we can consider the gender, body shape, age and
other attributes of a person along with the identity, the synthesized video
should be improved and more realistic.

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