Some properties of certain close-to-convex harmonic mappings

Abstract

In this paper, we determine the sharp estimates for Toeplitz determinants of a subclass of close-to-convex harmonic mappings. Moreover, we obtain an improved version of Bohr’s inequalities for a subclass of close-to-convex harmonic mappings, whose analytic parts are Ma-Minda convex functions.

1 Introduction

A complex-valued function f in the unit disk $\mathbb {D}=\{z:|z|<1\}$ is called a harmonic mapping if $\Delta f=4 f_{z \overline{z}}=0$. Let $\mathcal {H}$ denote the class of sense-preserving harmonic mappings $f=h +\overline{g}$ in $\mathbb {D}$, where

$$\begin{aligned} h(z)=z+\sum _{n=2}^{\infty } a_{n} z^{n}\ \text {and}\ g(z)=\sum _{n=1}^{\infty } b_{n} z^{n} \end{aligned}$$

(1.1)

are analytic functions in $\mathbb {D}$. Let $\mathcal {S}_{\mathcal {H}}$ be the subclass of $\mathcal {H}$ consisting of univalent mappings. We observe that $\mathcal {S}_\mathcal {H}$ reduces to the class $\mathcal {S}$ of normalized univalent analytic functions, if the co-analytic part $g\equiv 0$. Denote by $\mathcal {K}_{\mathcal {H}}$ the close-to-convex subclass of $\mathcal {S}_{\mathcal {H}}$. If $b_1 = 0$, then the class $\mathcal {K}_{\mathcal {H}}$ reduces to $\mathcal {K}^{0}_{\mathcal {H}}$.

Lewy [37] proved that $f=h+\overline{g}$ is locally univalent in $\mathbb {D}$ if and only if the Jacobian $J_{f}=\left| h^{\prime }\right| ^{2}-\left| g^{\prime }\right| ^{2}\ne 0$ in $\mathbb {D}$. Noting that the harmonic mapping f is sense-preserving, i.e. $J_{f}>0$ or $\left| h^{\prime }\right| >\left| g^{\prime }\right|$ in $\mathbb {D}$. At this point, its dilatation $\omega _{f}=g^{\prime } / h^{\prime }$ has the property $\left| \omega _{f}\right| <1$ in $\mathbb {D}$. The reader can find much information about planar harmonic mappings from [18, 22, 46].

Let $\mathcal {P}$ denote the class of analytic functions p in $\mathbb {D}$ of the form

$$\begin{aligned} p(z)=1+\sum _{n=1}^{\infty } p_{n} z^{n} \end{aligned}$$

(1.2)

such that ${\text {Re}} (p(z))>0$ in $\mathbb {D} .$

Denote by $\mathcal {A}$ the class of analytic functions in $\mathbb {D}$ with $f(0)=f'(0)-1=0$, and $\mathcal {K}(\alpha )$ denotes the class of functions $f \in \mathcal {A}$ such that

$$\begin{aligned} \mathrm{Re}\left( 1+\frac{z f^{\prime \prime }(z)}{f^{\prime }(z)}\right) >\alpha \quad \left( -\frac{1}{2}\le \alpha <1;\ z \in \mathbb {D} \right) . \end{aligned}$$

(1.3)

Particularly, the elements in $\mathcal {K}(-1 / 2)$ are close-to-convex but are not necessarily starlike in $\mathbb {D}$. For $0 \le \alpha <1$, the elements in $\mathcal {K}(\alpha )$ are known to be convex functions of order $\alpha$ in $\mathbb {D}$. For more properties of starlike and convex functions, the reader can refer to the monographs [23, 53].

By making use of the subordination in analytic functions, Ma and Minda [42] introduced a more general class $\mathcal {C}(\phi )$, consisting of functions in $\mathcal {S}$ for which

$$\begin{aligned} 1+ \frac{zf''(z)}{f'(z)} \prec \phi (z). \end{aligned}$$

Here the function $\phi :\mathbb {D} \rightarrow \mathbb {C}$, called Ma-Minda function, is analytic and univalent in $\mathbb {D}$ such that $\phi (\mathbb {D})$ has positive real part, symmetric with respect to the real axis, starlike with respect to $\phi (0)=1$ and $\phi '(0)>0$ (for more details, see [50, 57]). A Ma-Minda function has the form

$$\phi (z)=1+ \sum _{n=1}^{\infty } B_{n}z^{n}.$$

The extremal function K for the class $\mathcal {C}(\phi )$ is given by

$$\begin{aligned} K(z)=\int _{0}^{z}\exp \left( \int _{0}^{\zeta } \frac{\phi (t)-1}{t} d t\right) d \zeta \quad (z \in \mathbb {D}), \end{aligned}$$

(1.4)

which satisfies the condition

$$1+\frac{zK''(z)}{K'(z)}=\phi (z).$$

We recall the following natural class of close-to-convex harmonic mappings $\mathcal {M}(\alpha ,\zeta ,n)$, due to Wang et al. [56] (see also [47, 55]).

Definition 1.1

A harmonic mapping $f=h+\overline{g}\in \mathcal {H}$ is said to be in the class $\mathcal {M}(\alpha ,\zeta ,n)$ if $h \in \mathcal {K}(\alpha )$, for some $\alpha \in \left[ -{1}/{2},1\right)$, given by (1.3) and g satisfies the condition

$$\begin{aligned} g'(z)=\zeta z^nh'(z) \quad \left( \zeta \in \mathbb {C}\ \mathrm{with}\ \left| \zeta \right| \le \frac{1}{2n-1};\, n\in {\mathbb {N}}:=\{1,2,3,\cdots \}\right) . \end{aligned}$$

(1.5)

For $n=1$, $\alpha =-{1}/{2}$ and $|\zeta |=1$, the class $\mathcal {M}(-{1}/{2},\zeta ,1)$ was introduced by Bharanedhar and Ponnusamy [12]. For $n=1$, the class $\mathcal {M}(\alpha ,\zeta ,1)$ was studied in [9, 52].

In 2021, Allu and Halder [9] introduced and investigated the following subclass $\mathcal {H C}(\phi )$ of close-to-convex harmonic mappings.

Definition 1.2

For $\zeta \in \mathbb {C}$ with $|\zeta | \le 1$, let $\mathcal {H C}(\phi )$ denote the class of harmonic mappings $f=h+\overline{g}$ in $\mathbb {D}$ of the form (1.1), whose analytic part h belongs to $\mathcal {C}(\phi )$ and $h^{\prime }(0) \ne 0$, along with the condition $g^{\prime }(z)=\zeta z h^{\prime }(z)$.

Motivated essentially by the classes $\mathcal {M}(\alpha ,\zeta ,n)$ and $\mathcal {H C}(\phi )$, we define a new subclass $\mathcal {HC}_n(\phi )$ of close-to-convex harmonic mappings as follows:

Definition 1.3

A harmonic mapping $f=h+\overline{g}\in \mathcal {H}$ is said to be in the class $\mathcal {HC}_n(\phi )$ if $h\in \mathcal {C}(\phi )$ and g satisfies the condition (1.5).

In 2019, Sun et al. [51] investigated upper bounds of the third Hankel determinants for the class $\mathcal {M}(\alpha ,1,1)$ of close-to-convex harmonic mappings. In recent years, the Toeplitz determinants and Hankel determinants of functions in the class $\mathcal {S}$ or its subclasses have attracted many researchers’ attention (see [11, 17, 19, 20, 28, 29, 33,34,35,36]). Among them, the symmetric Toeplitz determinant $|T_q(n)|$ for subclasses of $\mathcal {S}$ with small values of n and q, are investigated by [2, 7, 10, 49, 54, 58].

The symmetric Toeplitz determinant $T_q(n)$ for analytic functions f is defined as follows:

$$\begin{aligned} T_{q}(n)[f]:= \begin{vmatrix} a_n&a_{n+1}&\cdots&a_{n+q-1}\\ a_{n+1}&a_n&\cdots&a_{n+q-2}\\ \vdots&\vdots&\vdots&\vdots&\\ a_{n+q-1}&a_{n+q-2}&\cdots&a_{n} \end{vmatrix}, \end{aligned}$$

where $n,q\in {\mathbb {N}}$ and $a_1=1$. In particular, for functions in starlike and convex classes, $T_{2}(2)[f],\, T_{3}(1)[f]$ and $T_{3}(2)[f]$ were studied by Ali $et\ al.$ [7].

Let $\mathcal {B}$ be the class of analytic functions f in $\mathbb {D}$ such that $|f(z)|<1$ for all $z\in \mathbb {D}$, and let $\mathcal {B}_0=\{f\in \mathcal {B} : f(0)=0\}$. In 1914, Bohr [16] proved that if $f\in \mathcal {B}$ is of the form $f(z)=\sum _{n=0}^{\infty }a_nz^n$, then the majorant series $M_f(r)= \sum _{n=0}^{\infty }|a_n||z|^n$ of f satisfies

$$\begin{aligned} M_{f_0}(r)=\sum _{n=1}^{\infty }|a_n||z|^n\le 1-|a_0|=d(f(0),\partial f(\mathbb {D})) \end{aligned}$$

(1.6)

for all $z\in \mathbb {D}$ with $|z|=r\le 1/3$, where $f_0(z)=f(z)-f(0)$. Bohr actually obtained the inequality (1.6) for $|z|\le 1/6$. Moreover, Wiener, Riesz and Schur, independently, established the Bohr inequality (1.6) for $|z|\le 1/3$ (known as Bohr radius for the class $\mathcal {B}$) and proved that 1/3 is the best possible.

The Bohr phenomenon was reappeared in the 1990s due to Dixon [21]. Furthermore, Boas and Khavinson [15] found bounds for Bohr’s radius in any complete Reinhard domains. Other works one can see [3, 4, 14, 44, 45]. In recent years, Bohr inequality and Bohr radius have become an active research field in geometric function theory (see [6, 8, 27, 31, 38, 40, 43]). Furthermore, initiated by the work of [32], the Bohr’s phenomenon for the complex-valued harmonic mappings have been widely studied (see [1, 9, 25, 26, 30, 41]).

In this paper, we aim at determining the sharp estimates for Toeplitz determinants of the class $\mathcal {M}(\alpha ,\zeta ,n)$. Moreover, we will derive an improved version of Bohr’s inequalities for the class $\mathcal {HC}_n(\phi )$.

2 Preliminary results

To prove our main results, we need the following lemmas.

Lemma 2.1

([23, p. 41]) For a function $p \in \mathcal {P}$ of the form (1.2), the sharp inequality $\left| p_{n}\right| \le 2$ holds for each $n \ge 1 .$ Equality holds for the function

$$p(z)=\frac{1+z}{1-z}.$$

Lemma 2.2

([24, Theorem 1]) Let $p \in \mathcal {P}$ be of the form (1.2) and $\mu \in \mathbb {C}$. Then

$$\begin{aligned} \left| p_{n}-\mu p_{k} p_{n-k}\right| \le 2 \max \{1,\,|2 \mu -1|\} \quad (1 \le k \le n-1). \end{aligned}$$

If $|2 \mu -1| \ge 1$, then the inequality is sharp for the function

$$\begin{aligned} p(z)=\frac{1+z}{1-z} \end{aligned}$$

or its rotations. If $|2 \mu -1|<1$, then the inequality is sharp for

$$\begin{aligned} p(z)=\frac{1+z^{n}}{1-z^{n}} \end{aligned}$$

or its rotations.

Lemma 2.3

([56]) Let $f=h+\overline{g}\in \mathcal {M}(\alpha ,\zeta ,n)$. Then the coefficients $a_k\ (k\in {\mathbb {N}}\setminus \{1\})$ of h satisfy

$$\begin{aligned} \left| a_k\right| \le \frac{1}{k!}\prod _{j=2}^{k}(j-2\alpha )\ (k\in {\mathbb {N}}\setminus \{1\}). \end{aligned}$$

(2.1)

Moreover, the coefficients $b_k\ (k=n+1, n+2, \ldots ; n\in {\mathbb {N}})$ of g satisfy

$$\begin{aligned} \left| b_{n+1}\right| \le \frac{\left| \zeta \right| }{n+1}\ \text { and }\ \left| b_{k+n}\right| \le \frac{\left| \zeta \right| }{(k+n)(k-1)!}\prod _{j=2}^{k}(j-2\alpha )\ (k\in {\mathbb {N}}\setminus \{1\};\, n\in {\mathbb {N}}). \end{aligned}$$

(2.2)

The bounds are sharp for the extremal function given by

$$\begin{aligned} f(z)=\int _0^z\frac{dt}{(1-\delta t)^{2-2\alpha }}+\overline{\int _0^z\frac{\zeta t^n}{(1-\delta t)^{2-2\alpha }}dt}\quad (\left| \delta \right| =1;\, z\in {\mathbb {D}}). \end{aligned}$$

(2.3)

Lemma 2.4

([56]) Let $f\in \mathcal {M}(\alpha ,\zeta ,n)$ with $0\le \alpha <1$ and $0\le \zeta <\frac{1}{2n-1}\ (n\in {\mathbb {N}})$. Then

$$\begin{aligned} \Phi (r;\alpha ,\zeta ,n)\le \left| f(z)\right| \le \Psi (r;\alpha ,\zeta ,n)\quad (r=|z|<1), \end{aligned}$$

(2.4)

where

$$\begin{aligned}&\Phi (r;\alpha ,\zeta ,n) \\&\quad =\left\{ \begin{array}{ll} \log (1+r)-\displaystyle \frac{\zeta \, r^{n+1} \, _2F_1(1,\, n+1;\, n+2;\, -r)}{n+1} &{}(\alpha =1/2),\\ \\ \displaystyle \frac{(1+r)^{2 \alpha -1}-1}{2 \alpha -1}-\frac{\zeta \, r^{n+1} \, _2F_1(n+1,\, 2-2 \alpha ;\, n+2;\, -r)}{n+1} &{}(\alpha \ne 1/2), \end{array}\right. \end{aligned}$$

and

$$\begin{aligned}&\Psi (r;\alpha ,\zeta ,n)\\&\quad =\left\{ \begin{array}{ll}-\log (1-r)+\displaystyle \frac{\zeta \, r^{n+1} \, _2F_1(1,\, n+1;\, n+2;\, r)}{n+1} &{} (\alpha =1/2), \\ \\ \displaystyle \frac{1-(1-r)^{2 \alpha -1}}{2 \alpha -1}+\frac{\zeta \, r^{n+1} \, _2F_1(n+1,\, 2-2 \alpha ;\, n+2;\, r)}{n+1} &{} (\alpha \ne 1/2). \end{array}\right. \end{aligned}$$

All these bounds are sharp, the extremal function is $f_{\alpha ,\zeta ,n}=h_{\alpha }+\overline{g_{\alpha ,\zeta ,n}}$ or its rotations, where

$$\begin{aligned} f_{\alpha ,\zeta ,n}(z) =\left\{ \begin{array}{ll} -\log (1-z)+\displaystyle \overline{\frac{\zeta \, z^{n+1} \, _2F_1(1,\, n+1;\, n+2;\, z)}{n+1}} &{}(\alpha =1/2), \\ \\ \displaystyle \frac{1-(1-z)^{2\alpha -1}}{2\alpha -1} +\overline{\frac{\zeta \, z^{n+1} \, _2F_1(n+1,\, 2-2 \alpha ;\, n+2;\, z)}{n+1}} &{}(\alpha \ne 1/2). \end{array}\right. \end{aligned}$$

(2.5)

The following two results are due to Ma and Minda [42].

Lemma 2.5

Let $f \in \mathcal {C}(\phi )$. Then $zf''(z)/f'(z) \prec zK''(z)/K'(z)$ and $f'(z) \prec K'(z)$, where K is given by (1.4).

Lemma 2.6

Assume that $f \in \mathcal {C}(\phi )$ and $|z|=r<1$. Then

$$\begin{aligned} K'(-r) \le |f'(z)| \le K'(r), \end{aligned}$$

(2.6)

where K is given by (1.4). Equality holds for some $z \ne 0$ if and only if f is a rotation of K.

Lemma 2.7

([13]) Let $f(z)=\sum _{n=0}^{\infty } a_{n}z^{n}$ and $g(z)=\sum _{n=0}^{\infty } b_{n}z^{n}$ be two analytic functions in $\mathbb {D}$ and $g \prec f$. Then

$$\begin{aligned} \sum _{n=0}^{\infty } |b_{n}| r^{n} \le \sum _{n=0}^{\infty } |a_{n}| r^{n} \end{aligned}$$

(2.7)

for $|z|=r \le 1/3.$

Remark 2.1

Lemma 2.7 continues to hold for quasi-subordination (cf. [5]). Moreover, the bound 1/3 is optimal as shown by [48, Lemma 1].

3 Toeplitz determinants for the class $\mathcal {M}(\alpha ,\zeta ,n)$

In this section, we will give several sharp estimates for Toeplitz determinants $|T_{q}(n)[\cdot ]|$ of functions in the class $\mathcal {M}(\alpha ,\zeta ,n)$.

Theorem 3.1

Let $f\in \mathcal {M}(\alpha ,\zeta ,n)$. Then

$$\begin{aligned} |T_{2}(n)[h]|\le \left( \frac{1}{n!}\prod _{j=2}^{n}(j-2\alpha )\right) ^2+ \left( \frac{1}{(n+1)!}\prod _{j=2}^{n+1}(j-2\alpha )\right) ^2\quad (n\in \mathbb {N}\backslash \{1\}), \end{aligned}$$

(3.1)

and

$$\begin{aligned} |T_{2}(n)[g]|\le \frac{1}{[(2n-1)(n+1)]^2}. \end{aligned}$$

(3.2)

The inequalities in (3.1) and (3.2) are sharp for the extremal function given by (2.3).

Proof

Suppose that $f\in \mathcal {M}(\alpha ,\zeta ,n)$. By Lemma 2.3, we see that

$$\begin{aligned} |T_{2}(n)[h]|=|a_n^2-a_{n+1}^{2}|\le |a_n^2| +|a_{n+1}^2| \end{aligned}$$

(3.3)

yields (3.1). Equality in (3.3) holds for the function h given by

$$\begin{aligned} \begin{aligned} h(z)&= \int _0^z\frac{dt}{(1-\delta t)^{2-2\alpha }}\\&= z+\frac{1}{2} (2-2 \alpha ) \delta ^2 z^2+\frac{1}{6} (2-2\alpha ) (3-2 \alpha ) \delta ^3z^3\\&\qquad +\frac{1}{24} (2-2 \alpha )(3-2 \alpha )(4-2 \alpha )\delta ^4 z^4 \\&\qquad +\frac{1}{120} (2-2 \alpha )(3-2 \alpha )(4-2 \alpha )(5-2 \alpha )\delta ^5 z^5+ \cdots \quad (\left| \delta \right| =1;\, z\in {\mathbb {D}}). \end{aligned} \end{aligned}$$

(3.4)

By virtue of (2.2), we get the assertion (3.2). The proof of Theorem 3.1 is thus completed. $\square$

Corollary 3.1

Let $f\in \mathcal {M}(\alpha ,\zeta ,2)$. Then

$$\begin{aligned} |T_{2}(2)[h]|\le \frac{2}{9} (1-\alpha )^2 \left( 2 \alpha ^2-6 \alpha +9\right) , \end{aligned}$$

(3.5)

and

$$\begin{aligned} |T_{2}(2)[g]|\le \frac{1}{81}. \end{aligned}$$

(3.6)

The inequalities in (3.5) and (3.6) are sharp for the extremal function given by (2.3) with $n=2$.

Theorem 3.2

Let $f\in \mathcal {M}(\alpha ,\zeta ,1)$. Then

$$\begin{aligned} \left| T_{3}(1)[h]\right| \le {\left\{ \begin{array}{ll} {\frac{1}{9} \big (8 \alpha ^4-34 \alpha ^3+71 \alpha ^2-72 \alpha +36\big ) \quad \ (-\frac{1}{2}\le \alpha \le \frac{1}{2}}), \\ {\frac{1}{9} \left( -2 \alpha ^3+25 \alpha ^2-44 \alpha +30\right) \quad \quad \quad \quad \ (\frac{1}{2}\le \alpha < 1}), \end{array}\right. } \end{aligned}$$

(3.7)

and

$$\begin{aligned} |T_{3}(1)[g]|\le \frac{1}{3}(1-\alpha ). \end{aligned}$$

(3.8)

The inequality in (3.7) is sharp for the function h given by (3.4), and the inequality in (3.8) is sharp for the function g defined by

$$\begin{aligned} g(z)={\int _0^z\frac{\zeta t}{(1-\delta t)^{2-2\alpha }}dt} \quad (\left| \delta \right| =1;\, \left| \zeta \right| \le 1;\, z\in {\mathbb {D}}). \end{aligned}$$

(3.9)

Proof

For $f\in \mathcal {M}(\alpha ,\zeta ,1)$, we see that

$$\begin{aligned} p(z)=\frac{1}{1-\alpha }\left( 1+\frac{z h^{\prime \prime }(z)}{h^{\prime }(z)}-\alpha \right) \in \mathcal {P} \quad \left( -\frac{1}{2}\le \alpha <1;\, z \in \mathbb {D}\right) . \end{aligned}$$

It follows that

$$\begin{aligned} n(n-1) a_{n}=(1-\alpha ) \sum _{k=1}^{n-1} k a_{k} p_{n-k} \quad (n \ge 2). \end{aligned}$$

(3.10)

From (3.10), we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} a_{2}=\frac{1}{2}(1-\alpha ) p_{1}, \\ a_{3}=\frac{1}{6}(1-\alpha )\left[ (1-\alpha ) p_{1}^{2}+p_{2}\right] , \\ a_{4}=\frac{1}{24}(1-\alpha )\left[ (1-\alpha )^{2} p_{1}^{3}+3(1-\alpha ) p_{1} p_{2}+2 p_{3}\right] . \\ \end{array}\right. } \end{aligned}$$

(3.11)

By virtue of Lemma 2.2 and (3.11), we get

$$\begin{aligned} \begin{aligned} \left| T_{3}(1)[h]\right|&=\left| 1-2 a_{2}^{2}+2 a_{2}^{2} a_{3}-a_{3}^{2}\right| \\&\le 1+2\left| a_{2}^{2}\right| +\left| a_{3}\right| \left| a_{3}-2 a_{2}^{2}\right| \\&\le 1+ \frac{1}{2}(1-\alpha )^2p_1^2+ \frac{1}{36} (1-\alpha )^2|(1-\alpha )p_1^2+p_2||p_2-2(1-\alpha )p_1^2| \\&\le {\left\{ \begin{array}{ll} {\frac{1}{9} \big (8 \alpha ^4-34 \alpha ^3+71 \alpha ^2-72 \alpha +36\big ) \quad (-\frac{1}{2}\le \alpha \le \frac{1}{2}}), \\ {\frac{1}{9} \big (-2 \alpha ^3+25 \alpha ^2-44 \alpha +30\big )\qquad \quad \ \ (\frac{1}{2}\le \alpha < 1}). \end{array}\right. } \end{aligned} \end{aligned}$$

(3.12)

In what follows, we shall prove that the equality in (3.12) holds for the function h given by (3.4). It follows from (3.4) that

$$\begin{aligned} {\left\{ \begin{array}{ll} |a_{2}|= 1-\alpha , \\ |a_{3}|=\frac{1}{3}(1-\alpha )(3-2\alpha ). \\ \end{array}\right. } \end{aligned}$$

(3.13)

Therefore, we obtain

$$\begin{aligned} \begin{aligned} \left| T_{3}(1)[h]\right|&=\left| 1-2 a_{2}^{2}+2 a_{2}^{2} a_{3}-a_{3}^{2}\right| \\&\le 1+2\left| a_{2}^{2}\right| +\left| a_{3}\right| \left| a_{3}-2 a_{2}^{2}\right| \\&= 1+ 2(1-\alpha )^2 + \frac{1}{3} (1-\alpha ) (3-2\alpha ) \left| \frac{1}{3} (1-\alpha ) (3-2\alpha )-2(1-\alpha )^2 \right| \\&={\left\{ \begin{array}{ll} {\frac{1}{9} \big (8 \alpha ^4-34 \alpha ^3+71 \alpha ^2-72 \alpha +36\big ) \quad \ \, (-\frac{1}{2}\le \alpha \le \frac{1}{2}}), \\ {\frac{1}{9} \big (-2 \alpha ^3+25 \alpha ^2-44 \alpha +30\big )\qquad \quad \ \ (\frac{1}{2}\le \alpha < 1}). \end{array}\right. } \end{aligned} \end{aligned}$$

(3.14)

The graph of $|T_{3}(1)[h]|$ with $\alpha \in$ $[-1/2, 1)$ is presented as Fig. 1.

By the power series representations of h and g for $f=h+\overline{g} \in \mathcal {M}(\alpha ,\zeta ,1)$, we find that

$$\begin{aligned} (k+1) b_{k+1}=\zeta k a_{k} \quad (k\in \mathbb {N};\, |\zeta |\le 1;\, a_1=1), \end{aligned}$$

which implies that

$$\begin{aligned} {\left\{ \begin{array}{ll} b_{2} = \frac{1}{2}\zeta a_{1},\\ b_{3} = \frac{2}{3}\zeta a_{2}.\\ \end{array}\right. } \end{aligned}$$

(3.15)

Thus, by Lemma 2.1, (3.11) and (3.15), we deduce that the assertion (3.8) of Theorem 3.2 is true. The sharpness of (3.8) follows from (3.7). $\square$

Fig. 1

The left graph is $|T_{3}(1)[h]|$ of $\frac{1}{9} \big (8 \alpha ^4-34 \alpha ^3+71 \alpha ^2-72 \alpha +36\big )$ and $\frac{1}{9} \big (-2 \alpha ^3+25 \alpha ^2-44 \alpha +30\big )$, respectively; The right graph is a locally enlarged version

Theorem 3.3

Let $f\in \mathcal {M}(\alpha ,\zeta ,2)$. Then

$$\begin{aligned} |T_{3}(2)[h]|\le {\left\{ \begin{array}{ll} {\frac{1}{108} \left( 1-\alpha )^3 (2 \alpha ^2-7 \alpha +12 ) (10 \alpha ^2-27 \alpha +36 \right) \quad (-\frac{1}{2}\le \alpha \le \frac{1}{7}}), \\ {\frac{5}{108} \left( 1-\alpha )^3 (2 \alpha ^2-7 \alpha +12 ) (2 \alpha ^2-4 \alpha +7 \right) \qquad \ \, (\frac{1}{7}\le \alpha < 1}), \end{array}\right. } \end{aligned}$$

(3.16)

and

$$\begin{aligned} |T_{3}(2)[g]|=|2b_3^2b_4|\le \frac{1}{243}\left( 1-\alpha \right) . \end{aligned}$$

(3.17)

The inequality in (3.16) is sharp for the function h given by (3.4), and the inequality in (3.17) is sharp for the function g defined by

$$\begin{aligned} g(z)={\int _0^z\frac{\zeta t^2}{(1-\delta t)^{2-2\alpha }}dt} \quad \left( \left| \delta \right| =1;\, \left| \zeta \right| \le \frac{1}{3};\, z\in {\mathbb {D}}\right) . \end{aligned}$$

(3.18)

.

Proof

Suppose that $f\in \mathcal {M}(\alpha ,\zeta ,2)$. It follows that

$$\begin{aligned} T_{3}(2)[h]=\left( a_{2}-a_{4}\right) \left( a_{2}^{2}-2 a_{3}^{2}+a_{2} a_{4}\right) . \end{aligned}$$

In view of (3.11) and Lemma 2.1, we find that

$$\begin{aligned} \begin{aligned} \left| a_{2}-a_{4}\right|&\le \left| a_{2}\right| +\left| a_{4}\right| \\&\le \left| \frac{1}{2}(1-\alpha ) p_{1}\right| + \left| \frac{1}{24}(1-\alpha )\left[ (1-\alpha )^{2} p_{1}^{3}+3(1-\alpha ) p_{1} p_{2}+2 p_{3}\right] \right| \\&\le \frac{1}{6}\left( 1-\alpha )(2\alpha ^2-7\alpha +12\right) . \end{aligned} \end{aligned}$$

(3.19)

Next, we shall maximize $\left| a_{2}^{2}-2 a_{3}^{2}+a_{2} a_{4}\right|$. With the help of (3.11), Lemma 2.1 and Lemma 2.2, we get

$$\begin{aligned} \begin{aligned}&|a_{2}^{2}-2 a_{3}^{2}+a_{2} a_{4} |\\&\quad =\frac{(1-\alpha )^2}{144} |-5(1-\alpha )^2p_1^4+36p_1^2-7(1-\alpha )p_1^2p_2-8p_2^2+6p_1p_3|\\&\quad \le \frac{(1-\alpha )^2}{144} \left[ 5(1-\alpha )^2|p_1|^4+36|p_1|^2+8|p_2|^2+6|p_1|\left| p_3-\frac{7}{6}(1-\alpha )p_1p_2\right| \right] \\&\quad \le {\left\{ \begin{array}{ll} {\frac{1}{18}\big (1-\alpha )^2 \big (10 \alpha ^2-27 \alpha +36\big )\quad \ \, (-\frac{1}{2}\le \alpha \le \frac{1}{7}}),\\ {\frac{5}{18} \big (1-\alpha )^2 \big (2 \alpha ^2-4 \alpha +7\big ) \quad \qquad (\frac{1}{7}\le \alpha < 1}). \end{array}\right. } \end{aligned} \end{aligned}$$

(3.20)

Therefore, combining (3.19) with (3.20), we obtain (3.16). By noting that for $f\in \mathcal {M}(\alpha ,\zeta ,2)$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} b_{3} = \frac{1}{3}\zeta a_{1},\\ b_{4} = \frac{1}{2}\zeta a_{2}.\\ \end{array}\right. } \end{aligned}$$

(3.21)

By means of Lemma 2.1, we get the assertion (3.17). The sharpness of (3.16) and (3.17) are similar to that of Theorem 3.2, we choose to omit the details here. $\square$

Remark 3.1

By setting $\alpha =0$ in Corollary 3.1, Theorem 3.2 and Theorem 3.3, respectively, we get $|T_{2}(2)[h]|\le 2$, $|T_{3}(1)[h]|\le 4$ and $|T_{3}(2)[h]|\le 4$. The bounds for convex functions were recently proved by Ali et al. [7, Theorem 2.11].

4 Bohr inequality for the class $\mathcal {HC}_n(\phi )$

In this section, we firstly give the sharp growth estimate for the class $\mathcal {HC}_n(\phi )$.

Proposition 4.1

Let $f \in \mathcal {HC}_n(\phi )$. Then

$$\begin{aligned} L(\zeta ,n,r) \le |f(z)| \le R(\zeta ,n,r), \end{aligned}$$

(4.1)

where

$$\begin{aligned} L(\zeta ,n,r)=-K(-r)-|\zeta |\int _{0}^{r} t^n K'(-t)dt, \end{aligned}$$

(4.2)

and

$$\begin{aligned} R(\zeta ,n,r)=K(r)+|\zeta |\int _{0}^{r}t^n K'(t)dt. \end{aligned}$$

(4.3)

The bounds are sharp for the extremal function $f_{\zeta }=h_{\zeta }+\overline{g_{\zeta }}$ with $h_{\zeta }=K$, where K satisfies (1.4) or its rotations and $g_{\zeta }$ satisfies $g'_{\zeta }=\zeta z^n h'_{\zeta }$.

Proof

Let $f=h+\overline{g} \in \mathcal {HC}_n(\phi )$. By Lemma 2.6, we know that

$$\begin{aligned} K'(-r) \le |h'(z)| \le K'(r) \quad (|z|=r). \end{aligned}$$

(4.4)

Let $\gamma$ be the linear segment joining 0 to z in $\mathbb {D}$. Then, we see that

$$\begin{aligned} |f(z)|&= \left| \int _{\gamma }\frac{\partial f}{\partial \theta }\,\, d\theta +\frac{\partial f}{\partial \overline{\theta }}\,\, d\overline{\theta } \right| \le \int _{\gamma } \left( |h'(\theta )|+|g'(\theta )|\right) \, |d\theta | \nonumber \\&=\int _{\gamma } \left( 1+|\zeta ||\theta |^n\right) |h'(\theta )|\, |d\theta |. \end{aligned}$$

(4.5)

Combining (4.4) and (4.5), we obtain

$$\begin{aligned} |f(z)|\le \int _{0}^{r} \left( 1+|\zeta |t^n\right) K'(t) \, dt=K(r)+|\zeta |\int _{0}^{r}t^nK'(t)\,\,dt= R(\zeta ,n,r). \end{aligned}$$

(4.6)

Let $\Gamma$ be the preimage of the line segment joining 0 to f(z) under the function f. It follows that

$$\begin{aligned} \begin{aligned} |f(z)|=\left| \int _{\Gamma }\frac{\partial f}{\partial \theta }\, d\theta +\frac{\partial f}{\partial \overline{\theta }}\,\, d\overline{\theta } \right|&\ge \int _{\Gamma } \left( |h'(\theta )|-|g'(\theta )|\right) \, |d\theta | \\&=\int _{\Gamma } \left( 1-|\zeta ||\theta |^n\right) |h'(\theta )|\, |d\theta |. \end{aligned} \end{aligned}$$

(4.7)

From (4.4) and (4.7), we have

$$\begin{aligned} |f(z)|\ge \int _{0}^{r} \left( 1-|\zeta |t^n\right) K'(-t) \, dt=-K(-r)-|\zeta |\int _{0}^{r}t^nK'(-t)\, dt=L(\zeta ,n,r). \end{aligned}$$

(4.8)

In view of (4.6) and (4.8), we deduce that

$$\begin{aligned} L(\zeta ,n,r) \le |f(z)| \le R(\zeta ,n,r). \end{aligned}$$

(4.9)

To show the sharpness, we consider the function $f_{\zeta }=h_{\zeta }+\overline{g_{\zeta }}$ with $h_{\zeta }=K$ or its rotations. It is easy to see that $h_{\zeta }=K \in \mathcal {C}(\phi )$ and $g_{\zeta }$ satisfies $g'_{\zeta }(z)=\zeta z^n h'_{\zeta }(z)$, which shows that $f_{\zeta }\in \mathcal {HC}_n(\phi )$. The equality holds on both sides of (4.4) for suitable rotations of K. For $0\le \zeta < 1/{(2n-1)}$, we see that $f_{\zeta }(r)=R(\zeta ,n,r)$ and $f_{\zeta }(-r)=-L(\zeta ,n,r)$. Hence $|f_{\zeta }(r)|=R(\zeta ,n,r)$ and $|f_{\zeta }(-r)|=L(\zeta ,n,r)$. This completes the proof of Proposition 4.1. $\square$

Proposition 4.2

Let $f \in \mathcal {HC}_n(\phi )$ and $S_{r}$ be the area of the image $f(\mathbb {D}_{r})$ $\mathrm{(}\mathbb {D}_{r}:=\{z\in \mathbb {C}:|z|< r\le 1\}\mathrm{)}$. Then

$$\begin{aligned} 2\pi \int _{0}^{r}t\left( 1-|\zeta |^{2}t^{2n}\right) (K'(-t))^{2}\,dt \le S_{r} \le 2\pi \int _{0}^{r}t\left( 1-|\zeta |^{2}t^{2n}\right) (K'(t))^{2}\, dt. \end{aligned}$$

(4.10)

Proof

Let $f=h+\overline{g} \in \mathcal {HC}_n(\phi )$. Then, the area of image of $\mathbb {D}_{r}$ under a harmonic mapping f is given by

$$\begin{aligned} S_{r}=\iint _{\mathbb {D}_{r}} \left( |h'(z)|^{2}-|g'(z)|^{2}\right) \, dx dy= \iint _{\mathbb {D}_{r}} \left( 1-|\zeta |^{2}|z|^{2n}\right) |h'(z)|^{2}dxdy. \end{aligned}$$

(4.11)

Since $h\in \mathcal {C}(\phi )$, in view of (4.4) and (4.11), we have

$$\begin{aligned} \int _{0}^{r} \int _{0}^{2\pi }t\left( 1-|\alpha |^{2}t^{2}\right) (K'(-t))^{2}d \theta dt \le S_{r} \le \int _{0}^{r}\int _{0}^{2\pi }t\left( 1-|\alpha |^{2}t^{2}\right) (K'(t))^{2}d\theta dt. \end{aligned}$$

(4.12)

Therefore, the assertion (4.10) of Proposition 4.2 follows directly from (4.12). $\square$

In what follows, we derive the Bohr inequality for the class $\mathcal {HC}_n(\phi )$.

Theorem 4.1

Let $f \in \mathcal {HC}_n(\phi )$. Then the majorant series of f satisfies the inequality

$$\begin{aligned} |z|+\sum _{n=2}^{\infty } (|a_{n}|+|b_{n}|)|z|^{n} \le d(f(0),\partial f(\mathbb {D})) \end{aligned}$$

(4.13)

for $|z|=r\le \min \{1/3,r_{f}\}$, where $r_{f}$ is the smallest positive root in (0, 1) of

$$\begin{aligned} L(\zeta ,n,1)=M_{K}(r)+|\zeta |\int _{0}^{r} t^n M_{K'}(t)\,dt, \end{aligned}$$

and $L(\zeta ,n,1)$ is given by (4.2) with $r=1$.

Proof

Let $f=h+\overline{g} \in \mathcal {HC}_n(\phi )$. Since $h \in \mathcal {C}(\phi )$, from Lemma 2.5, we know that

$$\begin{aligned} h' \prec K'. \end{aligned}$$

(4.14)

Let $K(z)=z+\sum \limits _{n=2}^{\infty } k_{n}z^{n}$. In view of Lemma 2.7 and (4.14), we have

$$\begin{aligned} 1+\sum \limits _{n=2}^{\infty } n|a_{n}|r^{n-1}=M_{h'}(r) \le M_{K'}(r)=1+\sum \limits _{n=2}^{\infty } n|k_{n}|r^{n-1} \end{aligned}$$

(4.15)

for $|z|=r\le 1/3$. By integrating (4.15) with respect to r from 0 to r, we get

$$\begin{aligned} M_{h}(r)=r+\sum \limits _{n=2}^{\infty } |a_{n}|r^{n} \le r+\sum \limits _{n=2}^{\infty } |k_{n}|r^{n}=M_{K}(r)\quad (r\le 1/3). \end{aligned}$$

(4.16)

From the definition of $\mathcal {HC}_n(\phi )$, we know that

$$\begin{aligned} g'(z)=\zeta z^n h'(z). \end{aligned}$$

This relationship along with (4.15) yields

$$\begin{aligned} \sum \limits _{n=2}^{\infty } n|b_{n}|r^{n-1}=M_{g'}(r)=|\zeta |r^n M_{h'}(r) \le |\zeta |r^n M_{K'}(r)\quad (r\le 1/3). \end{aligned}$$

(4.17)

By integrating (4.17) with respect to r from 0 to r, it follows that

$$\begin{aligned} M_{g}(r)=\sum \limits _{n=2}^{\infty } |b_{n}|r^{n} \le |\zeta | \int _{0}^{r} t^n M_{K'}(t)dt\quad (r\le 1/3). \end{aligned}$$

(4.18)

Therefore, for $|z|=r\le 1/3$, from (4.16) and (4.18), we obtain

$$\begin{aligned} M_{f}(r)=|z|+\sum _{n=2}^{\infty } (|a_{n}|+|b_{n}|)r^{n} \le M_{K}(r)+ |\zeta | \int _{0}^{r} t^n M_{K'}(t)dt=R_{\mathcal {C}}(n,r). \end{aligned}$$

(4.19)

In view of (4.1), it is evident that the Euclidean distance between f(0) and the boundary of $f(\mathbb {D})$ is given by

$$\begin{aligned} d(f(0), \partial f(\mathbb {D}))= \liminf \limits _{|z|\rightarrow 1} |f(z)-f(0)| \ge L(\zeta ,n,1). \end{aligned}$$

(4.20)

We note that $R_{\mathcal {C}}(n,r) \le L(\zeta ,n,1)$ whenever $r \le r_{f}$, where $r_{f}$ is the smallest positive root of $R_{\mathcal {C}}(n,r)=L(\zeta ,n,1)$ in (0, 1). Let

$$\begin{aligned} H_{1}(n,r)=R_{\mathcal {C}}(n,r)-L(\zeta ,n,1). \end{aligned}$$

Then $H_{1}(n,r)$ is a continuous function in [0, 1]. Since

$$\begin{aligned} M_{K}(r) \ge K(r)>-K(-r), \end{aligned}$$

it follows that

$$\begin{aligned} \begin{aligned} H_{1}(n,1)&=R_{\mathcal {C}}(n,1)-L(\zeta ,n,1)\\&= M_{K}(1)+K(-1)+|\zeta |\int _{0}^{r} t^n \left( M_{K'}(t)+K'(t)\right) dt \\&\ge K(1)+K(-1)+|\zeta |\int _{0}^{r} t^n \left( M_{K'}(t)+K'(t)\right) dt>0. \end{aligned} \end{aligned}$$

(4.21)

On the other hand,

$$\begin{aligned} H_{1}(n,0)=-L(\zeta ,n,1)=K(-1)(1-|\zeta |)+n|\zeta |\int _{0}^{1}t^{n-1}K(-t)\, dt<0. \end{aligned}$$

(4.22)

Therefore, $H_{1}$ has a root in (0, 1). Let $r_{f}$ be the smallest root of $H_{1}$ in (0, 1). Then $R_{\mathcal {C}}(n,r)\le L(\zeta ,n,1)$ for $r\le r_{f}$. Now, in view of the inequalities (4.19) and (4.20) with the relationship $R_{\mathcal {C}}(n,r)\le L(\zeta ,n,1)$ for $r\le r_{f}$, we obtain

$$\begin{aligned} |z|+\sum _{n=2}^{\infty } (|a_{n}|+|b_{n}|)r^{n} \le d(f(0), \partial f(\mathbb {D})) \end{aligned}$$

for $|z|=r\le \min \{1/3, r_{f}\}$. $\square$

For a particular choice of $\phi$ in Theorem 4.1, we get the following result.

Corollary 4.1

Let $f \in \mathcal {M}(\alpha ,\zeta ,n)$ with $0\le \alpha < 1$ and $0 \le \zeta <1/(2n-1)$. Then the inequality (4.13) holds for $|z|=r \le r_{f}$, where $r_{f}$ is the smallest root in (0, 1) of

$$F_{n}(r):=R(\alpha ,\zeta ,n,r)-L(\alpha ,\zeta ,n,1)=0.$$

The radius $r_{f}$ is sharp.

Proof

From Lemma 2.4, the Euclidean distance between f(0) and the boundary of $f(\mathbb {D})$ shows that

$$\begin{aligned} d(f(0), \partial f(\mathbb {D}))= \liminf \limits _{|z|\rightarrow 1} |f(z)-f(0)|\ge L(\alpha ,\zeta ,n,1). \end{aligned}$$

(4.23)

We note that $r_{f}$ is the root of the equation $R(\alpha ,\zeta ,n,r)=L(\alpha ,\zeta ,n,1)$ in (0, 1). The existence of the root is ensured by the relation $R(\alpha ,\zeta ,n,1) >L(\alpha ,\zeta ,n,1)$ with (2.4). For $0<r\le r_{f}$, it is evident that $R(\alpha ,\zeta ,n,r)\le L(\alpha ,\zeta ,n,1)$. In view of Lemma 2.3 and (4.23), for $|z|=r\le r_{f}$, we have

$$\begin{aligned} |z|+\sum \limits _{n=2}^{\infty } (|a_{n}|+|b_{n}|)|z|^{n}&\le r_{f} + (|a_{2}|+|b_{2}|)r_{f}^{2}+\sum \limits _{n=3}^{\infty } (|a_{n}|+|b_{n}|)r_{f}^{n}\\&=R(\alpha ,\zeta ,n,r_{f}) \le L(\alpha ,\zeta ,n,1) \le d(f(0), \partial f(\mathbb {D})). \end{aligned}$$

To show the sharpness of the radius $r_{f}$, we consider the function $f=f_{\alpha ,\zeta ,n}$, which is defined in Lemma 2.4. We see that $f_{\alpha ,\zeta ,n}$ belongs to $\mathcal {M}(\alpha ,\zeta ,n)$. Since the left side of the growth inequality in Lemma 2.4 holds for $f=f_{\alpha ,\zeta ,n}$ or its rotations, we have $d(f(0), \partial f(\mathbb {D}))=L(\alpha ,\zeta ,n,1)$. Therefore, the function $f=f_{\alpha ,\zeta ,n}$ for $|z|=r_{f}$ gives

$$\begin{aligned} |z|+\sum \limits _{n=2}^{\infty } (|a_{n}|+|b_{n}|)|z|^{n}&= r_{f} + (|a_{2}|+|b_{2}|)r_{f}^{2}+\sum \limits _{n=3}^{\infty } (|a_{n}|+|b_{n}|)r_{f}^{n}\\&=R(\alpha ,\zeta ,n,r_{f}) = L(\alpha ,\zeta ,n,1) =d(f(0), \partial f(\mathbb {D})), \end{aligned}$$

which reveals that the radius $r_{f}$ is the best possible. $\square$

The roots $r_{f}$ of $F_{n}(r)=0$ for different values of $\alpha$, $\zeta$ and n have been shown in Tables 1, 2 and Fig. 2, respectively.

Fig. 2

The graphs of $F_{n}(r)$, respectively, for $\alpha =0.5$ and $\alpha =0.9$ when $n=1,2,3,4,5$

Table 1 The roots $r_{f}$ of $F_{n}(r)=0$ for different values of $\zeta$ when $\alpha =0.5$

Table 2 The roots $r_{f}$ of $F_{n}(r)=0$ for different values of $\zeta$ when $\alpha =0.9$

Remark 4.1

For $n=1$ and $\left| \zeta \right| \le 1$, $r_{f}$ can be found in [9]. For $\alpha =0.5$, when $n\rightarrow \infty$, the sharp radius is 0.500000. For $\alpha =0.9$, when $n\rightarrow \infty$, the sharp radius is 0.815323. For $n=1$, when $\alpha \rightarrow 1$, the sharp radius is 0.645751.

Now, we give an improved version of Bohr inequality for the class $\mathcal {HC}_n(\phi )$. By adding area quantity ${S_{r}}/{(2 \pi )}$ with the majorant series of $f \in \mathcal {HC}_n(\phi )$, the sum is still less than $d(f(0),\partial f(\mathbb {D}))$ for some radius $r \le \min \{1/3,\widetilde{r}_{f}\}<1$.

Note that the additional term such as ${S_{r}}/{(2 \pi )}$ to the majorant sum was first mooted by Kayumov and Ponnusamy [31] to refine and improve the Bohr inequality. This variation of Bohr inequality was proved for harmonic mappings in [25]. Subsequently, several extensions were made by many authors (cf. [39]).

Theorem 4.2

Let $f \in \mathcal {HC}_n(\phi )$ and $S_{r}$ be the area of the image $f(\mathbb {D}_{r})$. Then the inequality

$$\begin{aligned} M_{f}(r)+ \frac{S_{r}}{2\pi } \le d(f(0),\partial f(\mathbb {D})) \end{aligned}$$

holds for $|z|=r\le \min \{1/3,\widetilde{r}_{f}\}$, where $\widetilde{r}_{f}$ is the smallest positive root in (0, 1) of

$$\begin{aligned} L(\zeta ,n,1) =M_{K}(r)+|\zeta |\int _{0}^{r} t^n M_{K'}(t)\, dt + \int _{0}^{r}t\left( 1-|\zeta |^{2}t^{2n}\right) (K'(t))^{2} dt, \end{aligned}$$

and $L(\zeta ,n,1)$ is given by (4.2) with $r=1$.

Proof

Let $f \in \mathcal {HC}_n(\phi )$ be of the form (1.1). Then, from the right hand inequality in (4.10) and (4.19), we obtain

$$\begin{aligned} \begin{aligned} M_{f}(r)+ \frac{S_{r}}{2\pi }&\le M_{K}(r)+|\zeta |\int _{0}^{r} t^n M_{K'}(t)\, dt + \int _{0}^{r}t\left( 1-|\zeta |^{2}t^{2n}\right) (K'(t))^{2}\, dt \\&= R_{\mathcal {C}}(n,r) +\int _{0}^{r}t\left( 1-|\zeta |^{2n}t^{2}\right) (K'(t))^{2}\, dt =\widetilde{R}_{f}(n,r) \end{aligned} \end{aligned}$$

(4.24)

for $r\le 1/3$. Suppose that $H_{2}(n, r)=\widetilde{R}_{f}(n,r)-L(\zeta ,n,1)$. Then $H_{2}(n,r)$ is a continuous function in [0, 1]. The inequality (4.22) yields that $H_{2}(n,0)=-L(\zeta ,n,1)<0$. By virtue of (4.21), we get

$$\begin{aligned} R_{\mathcal {C}}(n,1)-L(\zeta ,n,1)>0. \end{aligned}$$

(4.25)

For $|\zeta |\le 1/(2n-1)$, we observe that

$$\begin{aligned} t\left( 1-|\zeta |^{2}t^{2n}\right) (K'(t))^{2}\ge 0, \end{aligned}$$

and hence

$$\begin{aligned} \int _{0}^{r}t\left( 1-|\zeta |^{2}t^{2n}\right) (K'(t))^{2}dt\ge 0. \end{aligned}$$

(4.26)

From (4.24) and (4.25), we obtain

$$\begin{aligned} H_{2}(n,1)=R_{\mathcal {C}}(n,1)-L(\zeta ,n,1) + \int _{0}^{1}t\left( 1-|\zeta |^{2}t^{2n}\right) (K'(t))^{2}\, dt>0. \end{aligned}$$

Since $H_{2}(n,0)<0$ and $H_{2}(n,1)>0$, $H_{2}$ has a root in (0, 1) and choose $\widetilde{r}_{f}$ to be the smallest root in (0, 1), we know that $\widetilde{R}_{f}(n,r)\le L(\zeta ,n,1)$ for $r \le \widetilde{r}_{f}$. Therefore, by virtue of (4.20) and (4.24), we conclude that

$$\begin{aligned} M_{f}(r)+ \frac{S_{r}}{2\pi }\le d(f(0), \partial f(\mathbb {D})) \end{aligned}$$

for $r \le \min \{1/3, \widetilde{r}_{f}\}$. $\square$

Remark 4.2

By setting $n=1$ in Theorems 4.1 and 4.2, we get the corresponding results obtained in [9].