Coefficient Inequalities for Biholomorphic Mappings on the Unit Ball of a Complex Banach Space

Abstract

In the first part of this paper, we give generalizations of the Fekete–Szegö inequalities for quasiconvex mappings F of type B and the first elements F of g-Loewner chains on the unit ball of a complex Banach space, recently obtained by H. Hamada, G. Kohr and M. Kohr. We obtain the Fekete–Szegö inequalities using the norm under the restrictions on the second and third order terms of the homogeneous polynomial expansions of the mappings F. In the second part of this paper, we give the estimation of the difference of the moduli of successive coefficients for the first elements of g-Loewner chains on the unit disc. We also give the estimation of the difference of the moduli of successive coefficients for the first elements F of g-Loewner chains on the unit ball of a complex Banach space under the restrictions on the second and third order terms of the homogeneous polynomial expansions of the mappings F.

1. Introduction

Let S denote the class of all normalized univalent holomorphic functions

$$f\left(\zeta \right)=\zeta +\sum _{m=2}^{\infty }{a}_m{\zeta }^m$$

on the unit disc $\mathbb{D}$ in $\mathbb{C}$. Fekete and Szegö [1] obtained the following inequality for the coefficients of $f\in S$ by means of Loewner’s method (see also [2] [Theorem 6.4])

$$|a_3-\lambda a_2^2|\le 1+2exp(-2\lambda /(1-\lambda )),\mathrm{for}\lambda \in [0,1).$$

(1)

The above inequality is known as the Fekete–Szegö inequality. After that, there have been many papers interested to consider the corresponding problems for various subclasses of the class S. For example, for normalized convex functions on $\mathbb{D}$, Keogh and Merkes [3] obtained the following result:

$$|a_3-\lambda a_2^2|\le max\left\{\frac{1}{3},|1-\lambda |\right\},\mathrm{for}\lambda \in \mathbb{C}.$$

(2)

They also obtained sharp estimates over the classes of starlike functions and spirallike functions of arbitrary order. Koepf [4] obtained the sharp Fekete–Szegö inequalities for normalized starlike functions and for normalized close-to-convex functions on $\mathbb{D}$. Recently, a unified treatment of the Fekete–Szegö inequality for the subclasses of normalized univalent functions on $\mathbb{D}$ was given by Hamada, Kohr and Kohr [5].

Let $\mathbb{B}$ be the unit ball of a complex Banach space X. In the case of several complex variables, Xu and Liu [6] generalized the above result of Koepf for normalized starlike functions to subclasses of normalized starlike mappings on $\mathbb{B}$. Xu, Yang, Liu and Xu [7] generalized the above result due to Keogh and Merkes to a subclass of normalized quasiconvex mappings of type B on $\mathbb{B}$. A unified approach to the Fekete–Szegö problem for various subclasses of starlike mappings in several complex variables has been given by Xu, Liu and Liu [8] (see also [9]). Hamada, Kohr and Kohr [10] gave the Fekete–Szegö inequality for $(1+r)J_r$, where $J_r$ is the nonlinear resolvent mapping of f in the Carathéodory family $\mathcal{M}\left(\mathbb{B}\right)$. Długosz and Liczberski [11] studied the Fekete–Szegö type problem for Bavrin’s families of holomorphic functions in ${\mathbb{C}}^n$. Lai and Xu [12] and Elin and Jacobzon [13] studied the Fekete–Szegö type problem for spirallike mappings of type $\beta$ on $\mathbb{B}$.

Hamada, Kohr and Kohr [10] generalized the Fekete–Szegö inequality for normalized starlike functions on $\mathbb{D}$ by Koepf to all normalized starlike mappings on the unit ball $\mathbb{B}$ of a complex Banach space or the Fekete–Szegö Inequality (2) for normalized convex functions on $\mathbb{D}$ to all normalized quasiconvex mappings of type B on $\mathbb{B}$ (cf. [14]). This result generalizes the results in [7,8,9]. In these results, the Fekete–Szegö inequalities have the extended form (see Proposition 1). As a corollary, they obtained the Fekete–Szegö inequalities of the form (2) in the case $F\left(z\right)=f\left(z\right)z$, where $f\in H(\mathbb{B},\mathbb{C})$. The latter result for quasi-convex mappings of type B can be written as follows.

Theorem 1. 

Let F be a holomorphic mapping on $\mathbb{B}$ such that $F\left(z\right)=f\left(z\right)z$, where $f\in H(\mathbb{B},\mathbb{C})$. For $\parallel z_0\parallel =1$ and ${\ell }_{z_0}\in T\left(z_0\right)$, let

$$a_3=\frac{1}{3!}{\ell }_{z_0}\left(D^{3}F\left(0\right)\left(z_0^3\right)\right)$$

and

$$a_2=\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0^2\right)\right).$$

Let F be a quasiconvex mapping of type B on $\mathbb{B}$. Then, for any $\lambda \in \mathbb{C}$, we have

$$\left|a_3-\lambda a_2^2\right|\le max\left\{\frac{1}{3},|1-\lambda |\right\}.$$

This estimate is sharp.

Under the restriction on the second order term $P_2(·)$ of the homogeneous polynomial expansion of a spirallike mapping F on the unit ball of a complex Banach space, Elin and Jacobzon [15] obtained the Fekete–Szegö inequality. Under the restriction on the second-order term $P_2(·,t)$ of the homogeneous polynomial expansion of a g-Loewner chain $F(z,t)$, Hamada, Kohr and Kohr [5] [Theorem 4.1] gave a generalization of the classical Fekete–Szegö inequality (1) for $f\in S$ to the first element of g-Loewner chains, which coincides with inequality (1) in the case $g\left(\zeta \right)=(1+\zeta )/(1-\zeta )$ as follows.

Theorem 2. 

Let $g\left(\zeta \right)=(1+\zeta )/(1-\zeta )$ and let $F\in S_g^0\left(\mathbb{B}\right)$ be the first element of a g-Loewner chain $F(z,t)$ defined on $\mathbb{B}\times [0,\infty )$ such that

$$\frac{1}{2!}{D}^{2}F(0,t)\left(z^2\right)=L_F(z,t)z,z\in X,t\ge 0,$$

where $L_F(·,t)\in L(X,\mathbb{C})$ for each $t\ge 0$. For $\parallel z_0\parallel =1$ and ${\ell }_{z_0}\in T\left(z_0\right)$, let

$$a_3=\frac{1}{3!}{\ell }_{z_0}\left(D^{3}F\left(0\right)\left(z_0^3\right)\right)$$

and

$$a_2=\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0^2\right)\right).$$

Then, for $\lambda \in [0,1)$, we have

$$|a_3-\lambda a_2^2|\le 1+2exp(-2\lambda /(1-\lambda )).$$

This estimation is sharp.

Recently, Elin and Jacobzon [13] obtained the Fekete–Szegö inequality for spirallike mappings F of type $\beta$ on the unit ball $\mathbb{B}$ of a complex Banach space X using the norm

$$∥\frac{D^{3}F\left(0\right)\left(z^3\right)}{3!}-\lambda \frac{1}{2}{D}^{2}F\left(0\right)\left(x,\frac{D^{2}F\left(0\right)\left(z^2\right)}{2!}\right)∥,$$

in the case $F\left(z\right)=f\left(z\right)z$, where $f\in H(\mathbb{B},\mathbb{C})$ is such that $f\left(0\right)=1$. Xu, Liu and Lu [16] [Theorem 3.1] gave the Fekete–Szegö inequality for close-to-quasiconvex mappings F of type B on the unit polydisc ${\mathbb{D}}^n$ with respect to a quasiconvex mapping G on ${\mathbb{D}}^n$ using the norm in the case $F\left(z\right)=f\left(z\right)z$ and $G\left(z\right)=g\left(z\right)z$, where $f,g\in H({\mathbb{D}}^n,\mathbb{C})$. The first author [17] extended these results to the mappings F, which satisfy

$$\frac{1}{2!}{D}^{2}F\left(0\right)\left(z^2\right)=L_F\left(z\right)z,z\in X,$$

where $L_F(·)\in L(X,\mathbb{C})$ and

$$\frac{1}{3!}{D}^{3}F\left(0\right)\left(z^3\right)=Q_F\left(z\right)z,z\in X,$$

where $Q_F\left(z\right)$ is a homogeneous polynomial of degree 2 with values in $\mathbb{C}$.

Natural problems arise as follows.

Problem 1. 

(i) Can we weaken the assumptions on F in Theorem 1?

(ii) Can we generalize the Fekete–Szegö inequalities in Theorems 1 and 2 to those using the norm?

In the first part of this paper, we give positive answers to the above problems. For quasiconvex mappings of type B, we will show that if the second order term $P_2\left(z\right)$ of the homogeneous polynomial expansion of $F\in H(\mathbb{B},X)$ has the form $P_2\left(z\right)=L_F\left(z\right)z$, where $L_F$ is a bounded linear operator from X into $\mathbb{C}$, then the Fekete–Szegö inequality of the form (2) hold. Next, we will show that if the second order term $P_2\left(z\right)$ of the homogeneous polynomial expansion of $F\in H(\mathbb{B},X)$ has the form $P_2\left(z\right)=L_F\left(z\right)z$, where $L_F$ is a bounded linear operator from X into $\mathbb{C}$, and the third order term $P_3\left(z\right)$ of the homogeneous polynomial expansion of F has the form $P_3\left(z\right)=Q_F\left(z\right)z$, where $Q_F\left(z\right)$ is a homogeneous polynomial of degree 2 with values in $\mathbb{C}$, then the Fekete–Szegö inequality using the norm hold. We also give a similar result for the first elements of g-Loewner chains. In the second part of this paper, we give the estimate of the difference of the moduli of successive coefficients for the first elements of g-Loewner chains on the unit disc. We also give the estimation of the difference of the moduli of successive coefficients for the first elements F of g-Loewner chains on the unit ball of a complex Banach space under the restrictions on the second and third order terms of the homogeneous polynomial expansions of the mappings F.

Other bounds for coefficients of various subclasses of normalized biholomorphic mappings in higher dimensions have been obtained by Bracci [18], Gong [19], Graham, Hamada and Kohr [20,21], Hamada and Kohr [22], Liu, Wu and Yang [14], Liu and Liu [23,24], Xu and Liu [25,26,27].

Let X be a complex Banach space with respect to a norm $\parallel ·\parallel$. Let $\mathbb{B}=\{z\in X:\parallel z\parallel <1\}$ be the open unit ball of X. When $X=\mathbb{C}$, $\mathbb{B}=\mathbb{D}$ is the unit disc in $\mathbb{C}$. Let $L(X,Y)$ denote the set of continuous linear operators from X into a complex Banach space Y. Let I be the identity in $L(X,X)$. For each $z\in X\backslash \left\{0\right\}$, let

$$T\left(z\right)=\left\{{\ell }_z\in L(X,\mathbb{C}):{\ell }_z\left(z\right)=\parallel z\parallel ,\parallel {\ell }_z\parallel =1\right\}.$$

This set is nonempty by the Hahn–Banach theorem.

For a domain $G\subset X$, let $H\left(G\right)$ be the set of holomorphic mappings from G into X. If $F\in H\left(\mathbb{B}\right)$ and $z\in \mathbb{B}$, then for each $k=1,2,\cdots$, there exists a bounded symmetric k-linear mapping

$$D^{k}F\left(z\right):\prod _{j=1}^{k}X\to X,$$

called the k-th Fréchet derivative of F at z such that

$$F\left(w\right)=\sum _{k=0}^{\infty }\frac{1}{k!}{D}^{k}F\left(z\right)\left({(w-z)}^k\right),$$

for all w in some neighbourhood of z, where $D^{0}F\left(z\right)=F\left(z\right)$. In particular, we have

$$F\left(w\right)=\sum _{k=0}^{\infty }{P}_k\left(w\right),$$

for all w in some neighbourhood of 0, where

$$P_k\left(w\right)=\frac{1}{k!}{D}^{k}F\left(0\right)\left(w^k\right)$$

is a bounded homogeneous polynomial of degree k on X.

When $F\in H\left(\mathbb{B}\right)$, we say that F is normalized if $F\left(0\right)=0$ and $DF\left(0\right)=I$. When $F\in H\left(\mathbb{B}\right)$, we say that F is biholomorphic on $\mathbb{B}$ if $F\left(\mathbb{B}\right)$ is a domain and the inverse exists and is holomorphic on $F\left(\mathbb{B}\right)$. Let $S\left(\mathbb{B}\right)$ denote the family of normalized biholomorphic mappings $F\in H\left(\mathbb{B}\right)$.

2. Fekete–Szegö Inequalities

2.1. Fekete–Szegö Inequalities for Quasi-Convex Mapping of Type B

Definition 1.

(see [28] for $X={\mathbb{C}}^n$) A normalized locally biholomorphic mapping $F\in H\left(\mathbb{B}\right)$ is said to be quasiconvex of type B if

$$\Re {\ell }_z\left({\left[DF\left(z\right)\right]}^{-1}(D^{2}F\left(z\right)\left(z^2\right)+DF\left(z\right)z)\right)>0,\forall z\in \mathbb{B}\backslash \left\{0\right\},{\ell }_z\in T\left(z\right).$$

Hamada, Kohr and Kohr [10] [Theorem 4.1] obtained the following result, which is a generalization of [7] [Theorem 2] and [3] [Corollary 1] (see also [9] [Theorem 3.2], in the case $\mathbb{B}={\mathbb{D}}^n$). In particular, the following result holds for every convex mapping f on $\mathbb{B}$.

Proposition 1. 

Let F be a quasiconvex mapping of type B on $\mathbb{B}$. For $\parallel z_0\parallel =1$ and ${\ell }_{z_0}\in T\left(z_0\right)$, let

$$a_3=\frac{1}{3!}{\ell }_{z_0}\left(D^{3}F\left(0\right)\left(z_0^3\right)\right)$$

$$a_2=\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0^2\right)\right),$$

$${\tilde{a}}_2^2=\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0,\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)\right)\right).$$

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

$$\left|a_3-\left(\lambda -\frac{2}{3}\right)a_2^2-\frac{2}{3}{\widetilde{a_2}}^2\right|\le max\left\{\frac{1}{3},|1-\lambda |\right\}.$$

This estimate is sharp.

In particular, we obtain the following results, which generalize [10] [Corollary 4.2]. Note that Elin and Jacobzon [15] [Corollary 4.7] obtained a similar result to Corollary 1 for spirallike mappings under the assumption (3).

Corollary 1. 

Let F be a quasiconvex mapping of type B on $\mathbb{B}$ such that

$$\frac{1}{2!}{D}^{2}F\left(0\right)\left(z^2\right)=L_F\left(z\right)z,z\in X,$$

(3)

where $L_F(·)\in L(X,\mathbb{C})$. For $\parallel z_0\parallel =1$ and ${\ell }_{z_0}\in T\left(z_0\right)$, let

$$a_3=\frac{1}{3!}{\ell }_{z_0}\left(D^{3}F\left(0\right)\left(z_0^3\right)\right)$$

$$a_2=\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0^2\right)\right).$$

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

$$\left|a_3-\lambda a_2^2\right|\le max\left\{\frac{1}{3},|1-\lambda |\right\}.$$

This estimation is sharp.

Proof. 

It suffices to show that ${\tilde{a}}_2^2=a_2^2$. Indeed, we have

$$\begin{array}{ccc}\hfill {\tilde{a}}_2^2& =& \frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0,\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)\right)\right)\hfill \\ & =& L_F{\left(z_0\right)}^2\hfill \\ & =& {\left(\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0^2\right)\right)\right)}^2\hfill \\ & =& a_2^2.\hfill \end{array}$$

This completes the proof. □

If F satisfies Conditions (4) and (5), then we obtain the following Fekete–Szegö inequality using the norm.

Corollary 2. 

Let F be a quasiconvex mapping of type B on $\mathbb{B}$ such that

$$\frac{1}{2!}{D}^{2}F\left(0\right)\left(z^2\right)=L_F\left(z\right)z,z\in X,$$

(4)

where $L_F(·)\in L(X,\mathbb{C})$ and

$$\frac{1}{3!}{D}^{3}F\left(0\right)\left(z^3\right)=Q_F\left(z\right)z,z\in X,$$

(5)

where $Q_F\left(z\right)$ is a homogeneous polynomial of degree 2 with values in $\mathbb{C}$. For $\parallel z_0\parallel =1$ and for any $\lambda \in \mathbb{C}$, we have

$$∥\frac{1}{3!}{D}^{3}F\left(0\right)\left(z_0^3\right)-\lambda \frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0,\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)\right)∥\le max\left\{\frac{1}{3},|1-\lambda |\right\}.$$

(6)

This estimate is sharp.

Proof. 

If F satisfies the Conditions (4) and (5), then we have

$$a_3-\lambda a_2^2=Q_F\left(z_0\right)-\lambda L_F{\left(z_0\right)}^2.$$

(7)

We have

$$\frac{1}{3!}{D}^{3}F\left(0\right)\left(z_0^3\right)-\lambda \frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0,\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)\right)=(Q_F\left(z_0\right)-\lambda L_F{\left(z_0\right)}^2)z_0.$$

(8)

Therefore, by formulas (7) and (8) and Corollary 1, we obtain inequality (6), as desired. Sharpness follows from Corollary 1. This completes the proof. □

2.2. Fekete–Szegö Inequalities for the First Elements of g-Loewner Chains

Assumption 1. 

Let $g:\mathbb{D}\to \mathbb{C}$ be a univalent holomorphic function such that $g\left(0\right)=1$ and $\Re g\left(\zeta \right)>0$ on $\mathbb{D}$.

Definition 2.

(see, e.g., [20,29]) Let $g:\mathbb{D}\to \mathbb{C}$ satisfy Assumption 1. Let $H:\mathbb{B}\to X$ be a normalized holomorphic mapping. We say that H belongs to the family ${\mathcal{M}}_g\left(\mathbb{B}\right)$ if

$$\frac{1}{\parallel z\parallel }{\ell }_z\left(H\left(z\right)\right)\in g\left(\mathbb{D}\right),z\in \mathbb{B}\backslash \left\{0\right\},{\ell }_z\in T\left(z\right).$$

If $g\left(\zeta \right)=(1+\zeta )/(1-\zeta )$, ${\mathcal{M}}_g\left(\mathbb{B}\right)$ will be denoted by $\mathcal{M}\left(\mathbb{B}\right)$. Also, if $\mathbb{B}=\mathbb{U}$, ${\mathcal{M}}_g\left(\mathbb{D}\right)$ will be denoted by ${\mathcal{M}}_g$ and $\mathcal{M}\left(\mathbb{D}\right)$ will be denoted by $\mathcal{M}$.

Definition 3.

(cf. [29,30]) Let X be a complex Banach space. A mapping $H=H(z,t):\mathbb{B}\times [0,\infty )\to X$ is called a generating vector field (Herglotz vector field) if the following conditions hold:

(i) 

$H(·,t)\in \mathcal{M}\left(\mathbb{B}\right)$, for a.e. $t\ge 0$;

(ii) 

$H(z,·)$ is strongly measurable on $[0,\infty )$, for all $z\in \mathbb{B}$.

Next, we recall the notions of subordination and Loewner chain.

Definition 4.

(see, e.g., [29,31]) Let X be a complex Banach space and let $\Omega \subseteq X$ be a domain which contains the origin.

(i) If $F,G\in H(\Omega )$, we say that F is subordinate to G ($F\prec G$) if there exists a Schwarz mapping V (i.e., $V\in H(\Omega )$, $V\left(0\right)=0$ and $V(\Omega )\subseteq \Omega$) such that $F=G\circ V$.

(ii) A mapping $F:\Omega \times [0,\infty )\to X$ is called a univalent subordination chain if $F(·,t)$ is univalent on Ω, $F(0,t)=0$ for $t\ge 0$, and $F(·,s)\prec F(·,t)$, $0\le s\le t<\infty$. A univalent subordination chain $F:\Omega \times [0,\infty )\to X$ is called a Loewner chain if $F(·,t)$ is biholomorphic on Ω and $DF(0,t)=e^{t}I$, for all $t\ge 0$.

In addition, we have the following result.

Remark 1. 

(see, e.g., [29,31]) If $F:\Omega \times [0,\infty )\to X$ is a Loewner chain, then the subordination condition is equivalent to the existence of a unique biholomorphic Schwarz mapping $V=V(·,s,t)$, called the transition mapping associated with $F(z,t)$, such that $F(z,s)=F\left(V\right(z,s,t),t)$ for $z\in \Omega$ and $t\ge s\ge 0$. $DV(0,s,t)=e^{s-t}I$ for $t\ge s\ge 0$.

We may define the notion of a g-Loewner chain in the case of complex Banach spaces (not necessarily separable or reflexive), where $g:\mathbb{D}\to \mathbb{C}$ satisfies Assumption 1. In the case $X={\mathbb{C}}^n$, see [20].

Definition 5.

Let $g:\mathbb{D}\to \mathbb{C}$ satisfy Assumption 1. We say that a mapping $F=F(z,t):\mathbb{B}\times [0,\infty )\to X$ is a g-Loewner chain if the following conditions hold:

(i) 

$F(z,t)$ is a Loewner chain such that the family ${\left\{e^{-t}F(·,t)\right\}}_{t\ge 0}$ is uniformly bounded on each ball $\rho \mathbb{B}$$(0<\rho <1)$;

(ii) 

$\frac{\partial F}{\partial t}(z,t)$ exists for a.e. $t\ge 0$, for all $z\in \mathbb{B}$ and there exists a generating vector field $H=H(z,t):\mathbb{B}\times [0,\infty )\to X$ with $H(·,t)\in {\mathcal{M}}_g\left(\mathbb{B}\right)$ for a.e. $t\ge 0$, which satisfies the Loewner differential equation

$$\frac{\partial F}{\partial t}(z,t)=DF(z,t)H(z,t),a.e.t\ge 0,\forall z\in \mathbb{B}.$$

By the condition (i) in Definition 5, every g-Loewner chain on $\mathbb{B}\times [0,\infty )$ has a homogeneous polynomial expansion which converges uniformly on $\rho \mathbb{B}$ for each $\rho \in (0,1)$ and $t\ge 0$.

Let $S_g^0\left(\mathbb{B}\right)$ denote the family of the first elements of g-Loewner chains on $\mathbb{B}$. When $\mathbb{B}=\mathbb{D}$, $S_g^0\left(\mathbb{D}\right)$ is denoted by $S_g$.

Under the restriction (9) on the second order term $P_2(·,t)$ of the homogeneous polynomial expansion of a g-Loewner chain $F(z,t)$, Hamada, Kohr and Kohr [5] [Theorem 4.1] gave the following Fekete-Szegö inequality. When $\mathbb{B}$ is the unit disc $\mathbb{D}$, the Condition (9) is satisfied for all $F\in S_g$, which implies that the following result is a generalization of estimate (1) to the unit ball of a complex Banach space.

Proposition 2. 

Let $g\left(\zeta \right)=1+g_1\zeta +g_2{\zeta }^2+\cdots$ satisfy Assumption 1 and let $F\in S_g^0\left(\mathbb{B}\right)$ be the first element of a g-Loewner chain $F(z,t)$ defined on $\mathbb{B}\times [0,\infty )$ such that

$$\frac{1}{2!}{D}^{2}F(0,t)\left(z^2\right)=L_F(z,t)z,z\in X,t\ge 0,$$

(9)

where $L_F(·,t)\in L(X,\mathbb{C})$ for each $t\ge 0$. For $\parallel z_0\parallel =1$ and ${\ell }_{z_0}\in T\left(z_0\right)$, let

$$a_2=\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0^2\right)\right)$$

and

$$a_3=\frac{1}{3!}{\ell }_{z_0}\left(D^{3}F\left(0\right)\left(z_0^3\right)\right).$$

Assume that $g_1,g_2\in \mathbb{R}$, $g_1>0$, $g_2\le g_1$ and $g_1^2\le g_1+g_2$. Then, for $\lambda \in [0,1)$, we have

$$|a_3-\lambda a_2^2|\le \frac{g_1}{2}+\frac{g_1+g_2}{2}exp\left(-\frac{2(g_1+g_2)}{g_1^2(1-\lambda )}+2\right).$$

In particular, we have

$$|a_3|\le \frac{g_1}{2}+\frac{g_1+g_2}{2}exp\left(-\frac{2(g_1+g_2)}{g_1^2}+2\right).$$

The above estimates are sharp.

Using arguments similar to those in the proof of Corollary 2, we obtain the following result.

Corollary 3. 

Let $g\left(\zeta \right)=1+g_1\zeta +g_2{\zeta }^2+\cdots$ satisfy Assumption 1 and let $F\in S_g^0\left(\mathbb{B}\right)$ be the first element of a g-Loewner chain $F(z,t)$ defined on $\mathbb{B}\times [0,\infty )$ such that

$$\frac{1}{2!}{D}^{2}F(0,t)\left(z^2\right)=L_F(z,t)z,z\in X,t\ge 0,$$

where $L_F(·,t)\in L(X,\mathbb{C})$ for each $t\ge 0$ and

$$\frac{1}{3!}{D}^{3}F\left(0\right)\left(z^3\right)=Q_F\left(z\right)z,z\in X,$$

where $Q_F\left(z\right)$ is a homogeneous polynomial of degree 2 with values in $\mathbb{C}$.

Assume that $g_1,g_2\in \mathbb{R}$, $g_1>0$, $g_2\le g_1$ and $g_1^2\le g_1+g_2$. Then, for $\lambda \in [0,1)$ and $\parallel z_0\parallel =1$, we have

$$\begin{array}{c}{\displaystyle ∥\frac{1}{3!}{D}^{3}F\left(0\right)\left(z_0^3\right)-\lambda \frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0,\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)\right)∥}\hfill \\ \hfill \le \frac{g_1}{2}+\frac{g_1+g_2}{2}exp\left(-\frac{2(g_1+g_2)}{g_1^2(1-\lambda )}+2\right).\end{array}$$

In particular, we have

$$∥\frac{1}{3!}{D}^{3}F\left(0\right)\left(z_0^3\right)∥\le \frac{g_1}{2}+\frac{g_1+g_2}{2}exp\left(-\frac{2(g_1+g_2)}{g_1^2}+2\right).$$

The above estimates are sharp.

3. Difference of the Moduli of Successive Coefficients for the First Elements of G-Loewner Chains

Another coefficient problem which has attracted considerable attention is to estimate the difference of the moduli of successive coefficients

$$\left|\right|a_3|-|a_2\left|\right|,$$

for normalized univalent functions $f\left(\zeta \right)=\zeta +a_2{\zeta }^2+a_3{\zeta }^3+\cdots$ on the unit disc.

For $f\in S$ with $f\left(\zeta \right)=\zeta +a_2{\zeta }^2+a_3{\zeta }^3+\cdots$, the following result is known (see [32] [Theorem 3.11])

$$-1\le |a_3|-|a_2|\le \frac{3}{4}+e^{{\lambda }_1}(2e^{-{\lambda }_1}-1)=1.029\cdots ,$$

(10)

where ${\lambda }_1$ is the unique solution for the equation $4\lambda e^{-\lambda }=1$ in $(0,1)$. Both bounds are sharp.

The following lemma is well known.

Lemma 1. 

Let $p\left(\zeta \right)=1+b_1\zeta +b_2{\zeta }^2+\cdots$ be a holomorphic function on $\mathbb{D}$ which is subordinate to a holomorphic function g on $\mathbb{D}$ with $g\left(0\right)=1$. Then, it holds that

$$|b_1|\le |g^{\prime }\left(0\right)|.$$

The following lemma has been obtained by Hamada, Kohr and Kohr [5] [Lemma 2.8].

Lemma 2. 

Let $p\left(\zeta \right)=1+b_1\zeta +b_2{\zeta }^2+\cdots$ and $g\left(\zeta \right)=1+g_1\zeta +g_2{\zeta }^2+\cdots$ be holomorphic functions on $\mathbb{D}$ such that $g_1,g_2\in \mathbb{R}$ with $g_1>0$, $g_2\le g_1$. Assume that p is subordinate to g. Then it holds that

$$g_1+\Re b_2\ge \frac{g_1+g_2}{g_1^2}{(\Re b_1)}^2.$$

(11)

Furthermore,

(i) 

if $g_2=g_1$ and $|b_1|=g_1$, then the equality in estimate $\left(11\right)$ holds;

(ii) 

if $g_2<g_1$ and the equality in estimate $\left(11\right)$ holds, then $b_1,b_2\in \mathbb{R}$;

(iii) 

if $g_2<g_1$ and $|b_1|=g_1$, then the equality in estimate $\left(11\right)$ holds if and only if $b_1,b_2\in \mathbb{R}$ and

$$p\left(\zeta \right)=g(\pm \zeta ),\zeta \in \mathbb{D};$$

(iv) 

if $g_2<g_1$ and $|b_1|<g_1$, then the equality in estimate $\left(11\right)$ holds if and only if $b_1,b_2\in \mathbb{R}$ and

$$p\left(\zeta \right)=g\left(\frac{c\zeta -{\zeta }^2}{1-c\zeta }\right),\zeta \in \mathbb{D},$$

where $c=b_1/g_1$.

We use the following Valiron–Landau lemma (see [32] [p. 104]).

Lemma 3. 

Let $\Phi \left(t\right)$ be real-valued and continuous and $|\Phi \left(t\right)|\le e^{-t}$ for $t\ge 0$ with

$${\int }_0^{\infty }{(\Phi \left(t\right))}^{2}dt=\left(\lambda +\frac{1}{2}\right)e^{-2\lambda },0\le \lambda <\infty .$$

Then

$$\left|{\int }_0^{\infty }\Phi \left(t\right)dt\right|\le (\lambda +1)e^{-\lambda },$$

with equality occurring if and only if $\Phi \left(t\right)=\pm \Psi \left(t\right)$, where

$$\Psi \left(t\right)=\left\{\begin{array}{cc}{e}^{-\lambda },\hfill & 0\le t\le \lambda ,\hfill \\ e^{-t},\hfill & \lambda <t<\infty .\hfill \end{array}\right.$$

We obtain the following theorem which generalizes the inequality (10) for $f\in S$ to the first elements of g-Loewner chains on $\mathbb{D}$.

Theorem 3. 

Let $g\left(\zeta \right)=1+g_1\zeta +g_2{\zeta }^2+\cdots$ satisfy Assumption 1 and let $f\left(\zeta \right)=\zeta +a_2{\zeta }^2+a_3{\zeta }^3+\cdots \in S_g$ be the first element of a g-Loewner chain $f(\zeta ,t)$ defined on $\mathbb{D}\times [0,\infty )$. Assume that $g_1,g_2\in \mathbb{R}$, $g_1>0$, $g_2\le g_1$ and $g_1^2\le g_1+g_2$. Then, we have

(i) 

if $g_1\le 1$, then

$$-g_1\le |a_3|-|a_2|\le \frac{g_1}{2};$$

(ii) 

if $g_1\in (1,2]$ and $2g_1\le exp((g_1+g_2)/g_1^2)$, then

$$-1\le |a_3|-|a_2|\le \frac{g_1}{2};$$

(iii) 

if $g_1\in (1,2]$ and $2g_1>exp((g_1+g_2)/g_1^2)$, then

$$\begin{array}{ccc}\hfill -1& \le & |a_3|-|a_2|\hfill \\ \hfill & \le & max\left\{\frac{g_1}{2},\frac{g_1}{2}-\frac{1}{4}+\frac{g_1+g_2}{2}{e}^{-{\lambda }_1}\left(e^{-{\lambda }_1}-\frac{1}{g_1}\right)\right\},\hfill \end{array}$$

(12)

where ${\lambda }_1$ is the unique solution of the equation

$$g_1-2\{g_1^2(\lambda +1)-(g_1+g_2)\}{e}^{-\lambda }=0,\lambda \in \left(0,\frac{g_1+g_2}{g_1^2}\right).$$

Proof. 

Since $f(\zeta ,t)$ is a g-Loewner chain on $\mathbb{D}\times [0,\infty )$, there exists a Herglotz function $p(\zeta ,t)$ with $p(\mathbb{D},t)\subset g\left(\mathbb{D}\right)$ for a.e. $t\ge 0$ such that

$$\frac{\partial f}{\partial t}(\zeta ,t)=\zeta f^{\prime }(\zeta ,t)p(\zeta ,t),\zeta \in \mathbb{D},a.e.t\ge 0.$$

Let

$$p(\zeta ,t)=1+\sum _{k=1}^{\infty }{b}_k\left(t\right){\zeta }^k,\zeta \in \mathbb{D},t\ge 0$$

be the Taylor expansion of $p(·,t)$. Then, we have (see the proof of [5] [Theorem 3.1])

$$a_2=-{\int }_0^{\infty }{e}^{-t}{b}_1\left(t\right)dt$$

(13)

and

$$a_3={\left({\int }_0^{\infty }{e}^{-t}{b}_1\left(t\right)dt\right)}^2-{\int }_0^{\infty }{e}^{-2t}{b}_2\left(t\right)dt.$$

(14)

Therefore, by Lemma 1, we have

$$|a_2|\le {\int }_0^{\infty }{e}^{-t}\left|b_1\left(t\right)\right|dt\le {\int }_0^{\infty }{e}^{-t}{g}_{1}dt=g_1\le 2.$$

(15)

First, we prove the lower bound. If $g_1\in (0,1]$, then by estimate (15), we have

$$|a_2|-|a_3|\le |a_2|\le g_1.$$

If $g_1\in (1,2]$ and $|a_2|\ge 1$, then, by letting $\lambda \to 1^{-}$ in Proposition 2, we have

$$|a_2|-|a_3|\le |a_2{|}^2-|a_3|\le |a_3-a_2^2|\le \frac{g_1}{2}\le 1.$$

If $g_1\in (1,2]$ and $|a_2|<1$, then we have

$$|a_2|-|a_3|\le |a_2|<1.$$

Therefore, we have $|a_2|-|a_3|\le 1$ in the case $g_1\in (1,2]$.

Next, we prove the upper bound. By considering the transformation $f_{\theta }(\zeta ,t)=e^{-i\theta }f(e^{i\theta }\zeta ,t)$, $\theta \in \mathbb{R}$, we may assume that $\Re a_3=\left|a_3\right|$. Let $a_2=u+iv$, where $u,v\in \mathbb{R}$. Since $p(·,t)\prec g$ for a.e. $t\ge 0$, by relations (11), (13) and (14), we have

$$\begin{array}{ccc}\hfill \Re a_3& =& u^2-v^2-{\int }_0^{\infty }{e}^{-2t}\Re b_2\left(t\right)dt\hfill \\ & \le & u^2-v^2+{\int }_0^{\infty }{e}^{-2t}(g_1-c_g{(\Re b_1\left(t\right))}^2)dt,\hfill \end{array}$$

where

$$c_g=\frac{g_1+g_2}{g_1^2}\ge 1.$$

Therefore, we have

$$\begin{array}{ccc}\hfill \Re a_3-\left|a_2\right|& \le & u^2-v^2+{\int }_0^{\infty }{e}^{-2t}(g_1-c_g{(\Re b_1\left(t\right))}^2)dt-\sqrt{u^2+v^2}\hfill \\ \hfill & \le & \frac{g_1}{2}+u^2-\left|u\right|-c_g{\int }_0^{\infty }{e}^{-2t}{(\Re b_1\left(t\right))}^{2}dt\hfill \\ \hfill & =& \frac{g_1}{2}+u^2-\left|u\right|-(g_1+g_2){\int }_0^{\infty }{e}^{-2t}{\left(\frac{\Re b_1\left(t\right)}{g_1}\right)}^{2}dt.\hfill \end{array}$$

(16)

If $\left|u\right|\le 1$, then we have $u^2-\left|u\right|\le 0$ and

$$\Re a_3-\left|a_2\right|\le \frac{g_1}{2}.$$

(17)

This implies that if $g_1\le 1$ or $g_1\in (1,2]$ and $\left|u\right|\le 1$, then the inequality (17) holds.

We consider the case $g_1\in (1,2]$ and $\left|u\right|>1$. In this case, we have

$$1<\left|u\right|\le |a_2|\le g_1.$$

Since $|b_1\left(t\right)|\le g_1$ by Lemma 1, putting

$${\int }_0^{\infty }{e}^{-2t}{\left(\frac{\Re b_1\left(t\right)}{g_1}\right)}^{2}dt=\left(\lambda +\frac{1}{2}\right)e^{-2\lambda },0\le \lambda <\infty$$

and applying Lemma 3, we have

$$\left|u\right|\le g_1(\lambda +1)e^{-\lambda }.$$

(18)

Since $g_1>1$ and the function $(\lambda +1)e^{-\lambda }$ is strictly decreasing on $[0,\infty )$, there exist a unique ${\lambda }_0\in (0,\infty )$ such that $g_1({\lambda }_0+1)e^{-{\lambda }_0}=1$. Since $u^2-\left|u\right|$ is increasing for $\left|u\right|>1/2$, by inequalities (16) and (18), we have

$$\Re a_3-\left|a_2\right|\le \frac{g_1}{2}+g_1^2{(\lambda +1)}^2{e}^{-2\lambda }-g_1(\lambda +1)e^{-\lambda }-(g_1+g_2)\left(\lambda +\frac{1}{2}\right)e^{-2\lambda }.$$

Let

$$\phi \left(\lambda \right)=\frac{g_1}{2}+g_1^2{(\lambda +1)}^2{e}^{-2\lambda }-g_1(\lambda +1)e^{-\lambda }-(g_1+g_2)\left(\lambda +\frac{1}{2}\right)e^{-2\lambda }.$$

Since $\left|u\right|>1$ and inequality (18) holds, it suffices to consider the upper bound of $\phi \left(\lambda \right)$ in the interval $[0,{\lambda }_0)$. We have

$$\begin{array}{ccc}\hfill {\phi }^{\prime }\left(\lambda \right)& =& g_1^2(-2{\lambda }^2-2\lambda )e^{-2\lambda }+g_1\lambda e^{-\lambda }+2(g_1+g_2)\lambda e^{-2\lambda }\hfill \\ & =& \lambda e^{-\lambda }(g_1-2(g_1^2(\lambda +1)-(g_1+g_2))e^{-\lambda }).\hfill \end{array}$$

Let

$$\psi \left(\lambda \right)=g_1-2(g_1^2(\lambda +1)-(g_1+g_2))e^{-\lambda }.$$

Then

$${\psi }^{\prime }\left(\lambda \right)=-2(-g_1^2\lambda +(g_1+g_2))e^{-\lambda }.$$

Then ${\psi }^{\prime }\left(\lambda \right)=0$ if and only if $\lambda =c_g$ and

$$\underset{\lambda \in [0,\infty )}{min}\psi \left(\lambda \right)=\psi \left(c_g\right)=g_1(1-2g_1{e}^{-c_g}).$$

If $\psi \left(c_g\right)\ge 0$, then we have $\psi \left(\lambda \right)\ge \psi \left(c_g\right)\ge 0$ and it implies that ${\phi }^{\prime }\left(\lambda \right)\ge 0$. Therefore, we have

$$\phi \left(\lambda \right)\le \phi \left({\lambda }_0\right)<\frac{g_1}{2}.$$

If $\psi \left(c_g\right)<0$, then, by the conditions, ${\psi }^{\prime }\left(\lambda \right)<0$ on $(0,c_g)$, ${\psi }^{\prime }\left(\lambda \right)>0$ on $(c_g,\infty )$, $\psi \left(0\right)\ge g_1>0$ and $\psi (\infty )=g_1>0$, we obtain that the solutions of the equation ${\phi }^{\prime }\left(\lambda \right)=0$ on $(0,\infty )$ are ${\lambda }_1\in (0,c_g)$ and ${\lambda }_2\in (c_g,\infty )$. Since $\psi \left(c_g\right)=g_1(1-2g_1{e}^{-c_g})<0$, we have

$$g_1(c_g+1)e^{-c_g}>\frac{c_g+1}{2}\ge 1,$$

which implies that ${\lambda }_0>c_g>{\lambda }_1$. Since ${\phi }^{\prime }\left(\lambda \right)>0$ on $(0,{\lambda }_1)$, ${\phi }^{\prime }\left(\lambda \right)<0$ on $({\lambda }_1,{\lambda }_2)$ and ${\phi }^{\prime }\left(\lambda \right)>0$ on $({\lambda }_2,\infty )$, we have

$$\underset{0\leqq \lambda <{\lambda }_0}{sup}\phi \left(\lambda \right)=max\{\phi \left({\lambda }_0\right),\phi \left({\lambda }_1\right)\}.$$

Also, since $\phi \left({\lambda }_0\right)\le \frac{g_1}{2}$, we have

$$\underset{0\leqq \lambda <{\lambda }_0}{sup}\phi \left(\lambda \right)\le max\left\{\frac{g_1}{2},\phi \left({\lambda }_1\right)\right\}.$$

Since

$$\phi \left({\lambda }_1\right)=\frac{g_1}{2}-\frac{1}{4}+\frac{g_1+g_2}{2}{e}^{-{\lambda }_1}\left(e^{-{\lambda }_1}-\frac{1}{g_1}\right),$$

we obtain the upper bound in inequality (12), as desired. This completes the proof. □

We next consider the case when g is a linear fractional transformation of the form

$$g\left(\zeta \right)=\frac{1+a\zeta }{1-b\zeta },\zeta \in \mathbb{D}$$

with $a,b\in [-1,1]$ and $a+b>0$. Then, g satisfies the assumptions of Theorem 3 with $g_1=a+b$, $g_2=b(a+b)$. So, we obtain the following corollary.

Corollary 4. 

Let

$$g\left(\zeta \right)=\frac{1+a\zeta }{1-b\zeta },\zeta \in \mathbb{D}$$

be a linear fractional transformation with $a,b\in [-1,1]$ and $a+b>0$ and let $f\left(\zeta \right)=\zeta +a_2{\zeta }^2+a_3{\zeta }^3+\cdots \in S_g$ be the first element of a g-Loewner chain defined on $\mathbb{D}\times [0,\infty )$. Then, the following assertions hold:

(i) 

if $a+b\le 1$, then

$$-(a+b)\le |a_3|-|a_2|\le \frac{a+b}{2};$$

(ii) 

if $a+b\in (1,2]$ and $2(a+b)\le exp\left(\right(1+b)/(a+b\left)\right)$, then

$$-1\le |a_3|-|a_2|\le \frac{a+b}{2};$$

(iii) 

if $a+b\in (1,2]$ and $2(a+b)>exp\left(\right(1+b)/(a+b\left)\right)$, then

$$\begin{array}{ccc}\hfill -1& \le & |a_3|-|a_2|\hfill \\ & \le & max\left\{\frac{a+b}{2},\frac{a+b}{2}-\frac{1}{4}+\frac{1+b}{2}{e}^{-{\lambda }_1}\left((a+b)e^{-{\lambda }_1}-1\right)\right\},\hfill \end{array}$$

where ${\lambda }_1$ is the unique solution of the equation

$$1-2\{(a+b)(\lambda +1)-(1+b)\}{e}^{-\lambda }=0,\lambda \in \left(0,\frac{1+b}{a+b}\right).$$

In particular, when $g\left(\zeta \right)=(1+k\zeta )/(1-k\zeta )$, where $k\in (0,1]$ is fixed, we have $g_1=2k$ and $g_2=2k^2$. So, we obtain the following result.

Corollary 5. 

Let $k\in (0,1]$ and

$$g\left(\zeta \right)=\frac{1+k\zeta }{1-k\zeta },\zeta \in \mathbb{D}.$$

Let $f(z,t)$ be a Loewner chain defined on $\mathbb{D}\times [0,\infty )$ whose Herglotz function p satisfies

$$p(\mathbb{D},t)\subset U\left(k\right)=\{\zeta \in \mathbb{C}:|(\zeta -1)/(\zeta +1)|<k\},a.e.t\ge 0.$$

Then, we have

(i) 

if $k\le 1/2$, then

$$-2k\le |a_3|-|a_2|\le k;$$

(ii) 

if $k\in (1/2,1]$ and $4k\le exp\left(\right(1+k)/(2k\left)\right)$, then

$$-1\le |a_3|-|a_2|\le k;$$

(iii) 

if $k\in (1/2,1]$ and $4k>exp\left(\right(1+k)/(2k\left)\right)$, then

$$\begin{array}{ccc}\hfill -1& \le & |a_3|-|a_2|\hfill \\ & \le & max\left\{k,k-\frac{1}{4}+\frac{k+1}{2}{e}^{-{\lambda }_1}\left(2k{e}^{-{\lambda }_1}-1\right)\right\},\hfill \end{array}$$

where ${\lambda }_1$ is the unique solution of the equation

$$1-2\{2k(\lambda +1)-k-1\}{e}^{-\lambda }=0,\lambda \in \left(0,\frac{1+k}{2k}\right).$$

By combining Proposition 2 and the argument similar to that in the proof of Theorem 3, we obtain the following theorem.

Theorem 4. 

Let $g\left(\zeta \right)=1+g_1\zeta +g_2{\zeta }^2+\cdots$ satisfy Assumption 1 and let $F\in S_g^0\left(\mathbb{B}\right)$ be the first element of a g-Loewner chain $F(z,t)$ defined on $\mathbb{B}\times [0,\infty )$ such that

$$\frac{1}{2!}{D}^{2}F(0,t)\left(z^2\right)=L_F(z,t)z,z\in X,t\ge 0,$$

where $L_F(·,t)\in L(X,\mathbb{C})$ for each $t\ge 0$. For $\parallel z_0\parallel =1$ and ${\ell }_{z_0}\in T\left(z_0\right)$, let

$$a_3=\frac{1}{3!}{\ell }_{z_0}\left(D^{3}F\left(0\right)\left(z_0^3\right)\right)$$

and

$$a_2=\frac{1}{2!}{\ell }_{z_0}\left(D^{2}F\left(0\right)\left(z_0^2\right)\right).$$

Assume that $g_1,g_2\in \mathbb{R}$, $g_1>0$, $g_2\le g_1$ and $g_1^2\le g_1+g_2$. Then, we have

(i) 

if $g_1\le 1$, then

$$-g_1\le |a_3|-|a_2|\le \frac{g_1}{2}$$

(ii) 

if $g_1\in (1,2]$ and $2g_1\le exp((g_1+g_2)/g_1^2)$, then

$$-1\le |a_3|-|a_2|\le \frac{g_1}{2};$$

(iii) 

if $g_1\in (1,2]$ and $2g_1>exp((g_1+g_2)/g_1^2)$, then

$$\begin{array}{ccc}\hfill -1& \le & |a_3|-|a_2|\hfill \\ & \le & max\left\{\frac{g_1}{2},\frac{g_1}{2}-\frac{1}{4}+\frac{g_1+g_2}{2}{e}^{-{\lambda }_1}\left(e^{-{\lambda }_1}-\frac{1}{g_1}\right)\right\},\hfill \end{array}$$

where ${\lambda }_1$ is the unique solution of the equation

$$g_1-2\{g_1^2(\lambda +1)-(g_1+g_2)\}{e}^{-\lambda }=0,\lambda \in \left(0,\frac{g_1+g_2}{g_1^2}\right).$$

In particular, we have the following corollary.

Corollary 6. 

Let $g\left(\zeta \right)=1+g_1\zeta +g_2{\zeta }^2+\cdots$ satisfy Assumption 1 and let $F\in S_g^0\left(\mathbb{B}\right)$ be the first element of a g-Loewner chain $F(z,t)$ defined on $\mathbb{B}\times [0,\infty )$ such that

$$\frac{1}{2!}{D}^{2}F(0,t)\left(z^2\right)=L_F(z,t)z,z\in X,t\ge 0,$$

where $L_F(·,t)\in L(X,\mathbb{C})$ for each $t\ge 0$ and

$$\frac{1}{3!}{D}^{3}F\left(0\right)\left(z^3\right)=Q_F\left(z\right)z,z\in X,$$

where $Q_F\left(z\right)$ is a homogeneous polynomial of degree 2 with values in $\mathbb{C}$. Assume that $g_1,g_2\in \mathbb{R}$, $g_1>0$, $g_2\le g_1$ and $g_1^2\le g_1+g_2$. Let $\parallel z_0\parallel =1$. Then, we have

(i) 

if $g_1\le 1$, then

$$-g_1\le ∥\frac{1}{3!}{D}^{3}F\left(0\right)\left(z_0^3\right)∥-∥\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)∥\le \frac{g_1}{2};$$

(ii) 

if $g_1\in (1,2]$ and $2g_1\le exp((g_1+g_2)/g_1^2)$, then

$$-1\le ∥\frac{1}{3!}{D}^{3}F\left(0\right)\left(z_0^3\right)∥-∥\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)∥\le \frac{g_1}{2};$$

(iii) 

if $g_1\in (1,2]$ and $2g_1>exp((g_1+g_2)/g_1^2)$, then

$$\begin{array}{ccc}\hfill -1& \le & ∥\frac{1}{3!}{D}^{3}F\left(0\right)\left(z_0^3\right)∥-∥\frac{1}{2!}{D}^{2}F\left(0\right)\left(z_0^2\right)∥\hfill \\ & \le & max\left\{\frac{g_1}{2},\frac{g_1}{2}-\frac{1}{4}+\frac{g_1+g_2}{2}{e}^{-{\lambda }_1}\left(e^{-{\lambda }_1}-\frac{1}{g_1}\right)\right\},\hfill \end{array}$$

where ${\lambda }_1$ is the unique solution of the equation

$$g_1-2\{g_1^2(\lambda +1)-(g_1+g_2)\}{e}^{-\lambda }=0,\lambda \in \left(0,\frac{g_1+g_2}{g_1^2}\right).$$