On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

TL;DR

This paper establishes upper bounds for the third Hankel determinants across various subclasses of analytic functions and explores new results on close-to-convex harmonic mappings, linking these findings to existing literature.

Contribution

It provides new upper bounds for third Hankel determinants for multiple subclasses and introduces results on a novel subclass of close-to-convex harmonic mappings.

Findings

Upper bounds for third Hankel determinants for starlike, convex, and bounded turning functions.

New results on a subclass of close-to-convex harmonic mappings.

Connections to existing literature are discussed.

Abstract

In this paper, we obtain the upper bounds to the third Hankel determinants for starlike functions of order $\alpha$, convex functions of order $\alpha$ and bounded turning functions of order $\alpha$. Furthermore, several relevant results on a new subclass of close-to-convex harmonic mappings are obtained. Connections of the results presented here to those that can be found in the literature are also discussed.

Equations

$$f(z)=z+\sum _{k=2}^{\infty }{a}_k{z}^k.$$

$$\Re\bigg{(}\frac{zf^{\prime}(z)}{f(z)}\bigg{)}>\alpha\quad(z\in\mbox{$\mathbb{D}$}).$$

$$\Re\bigg{(}1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{)}>\alpha\quad(-1/2\leq\alpha<1;\ z\in\mbox{$\mathbb{D}$}).$$

$$\Re\big{(}f^{\prime}(z)\big{)}>\alpha\quad(z\in\mbox{$\mathbb{D}$}).$$

$$f(z)=h(z)+\overline{g(z)}=z+\sum_{k=2}^{\infty}a_{k}z^{k}+\sum_{k=1}^{\infty}\overline{b_{k}z^{k}}\quad(|b_{1}|<1;\ z\in\mbox{$\mathbb{D}$}).$$

$$\Re\bigg{(}1+\frac{zh^{\prime\prime}(z)}{h^{\prime}(z)}\bigg{)}>\alpha\quad{\rm and}\quad g^{\prime}(z)=zh^{\prime}(z)\quad\big{(}z\in\mbox{$\mathbb{D}$}\big{)}.$$

$$H_{q,n}(f)=\left|\begin{array}[]{cccc}a_{n}&a_{n+1}&\cdots&a_{n+q-1}\\ a_{n+1}&a_{n+2}&\cdots&a_{n+q}\\ \vdots&\vdots&\vdots&\vdots\\ a_{n+q-1}&a_{n+q}&\cdots&a_{n+2(q-1)}\\ \end{array}\right|\quad(q,n\in\mathbb{N}).$$

$$H_{3,1}(f)=\left|\begin{array}[]{ccc}a_{1}&a_{2}&a_{3}\\ a_{2}&a_{3}&a_{4}\\ a_{3}&a_{4}&a_{5}\\ \end{array}\right|.$$

$$H_{3,1}(f)=-a_2^2{a}_5+2a_2{a}_3{a}_4-a_3^3+a_3{a}_5-a_4^2.$$

$$\big{|}H_{3,1}(f)\big{|}\leq 16,\qquad\big{|}H_{3,1}(h)\big{|}\leq\frac{32+33\sqrt{3}}{72\sqrt{3}}\approx 0.714,$$

$$\big{|}H_{3,1}(g)\big{|}\leq\frac{2736\sqrt{3}+675\sqrt{5}}{4860\sqrt{3}}\approx 0.742.$$

$$\big{|}H_{3,1}(f)\big{|}\leq 1,\qquad\big{|}H_{3,1}(h)\big{|}\leq\frac{49}{540}\approx 0.090,\qquad\big{|}H_{3,1}(g)\big{|}\leq\frac{41}{60}\approx 0.683.$$

$$\big{|}H_{3,1}(h)\big{|}\leq\frac{180+69\sqrt{15}}{32\sqrt{15}}\approx 3.609,\qquad\big{|}H_{3,1}(g)\big{|}\leq\frac{439}{540}\approx 0.813.$$

$$\big{|}H_{3,1}(g)\big{|}\leq\frac{(1-\alpha)^{2}}{3}\bigg{[}\frac{8(1-\alpha)}{9}+\frac{1}{4}\bigg{(}\frac{5-4\alpha}{3}\bigg{)}^{\frac{3}{2}}+\frac{4}{5}\bigg{]}.$$

$$p(z)=1+\sum_{n=1}^{\infty}p_{n}z^{n}\quad{\rm and}\quad\Re\big{(}p(z)\big{)}>0\quad(z\in\mbox{$\mathbb{D}$}).$$

$$|p_n|\le 2(n\in \mathbb{N}).$$

$$\varphi (z)=\frac{1+z}{1-z}=1+2\sum _{n=1}^{\infty }{z}^n.$$

$$|p_n-p_k{p}_{n-k}|\le 2(n,k\in \mathbb{N},n>k).$$

$$|p_n-\mu p_k{p}_{n-k}|\le 2(n,k\in \mathbb{N},n>k;0\le \mu \le 1).$$

$$2p_2=p_1^2+(4-p_1^2)x,$$

$$4p_3=p_1^3+2p_1(4-p_1^2)x-p_1(4-p_1^2)x^2+2(4-p_1^2)(1-|x{|}^2)z.$$

$$f(z)=z+\sum _{k=2}^{\infty }{a}_k{z}^k\in {\mathcal{S}}^{\ast }(\alpha ),h(z)=z+\sum _{k=2}^{\infty }{b}_k{z}^k\in \mathcal{K}(\alpha ),g(z)=z+\sum _{k=2}^{\infty }{c}_k{z}^k\in \mathcal{R}(\alpha ).$$

$$\big{|}H_{3,1}(f)\big{|}\leq\frac{1}{18}(1-\alpha)^{2}(18-\alpha),$$

$$\big{|}H_{3,1}(h)\big{|}\leq\frac{1}{540}(1-\alpha)^{2}(49-16\alpha),$$

$$\big{|}H_{3,1}(g)\big{|}\leq\frac{1}{60}(1-\alpha)^{2}(36-20\alpha+5|1-4\alpha|).$$

$$p(z)=\frac{1}{1-\alpha}\left(\frac{zf^{\prime}(z)}{f(z)}-\alpha\right)\quad(0\leq\alpha<1;\ z\in\mbox{$\mathbb{D}$}),$$

$$p(z)=1+\frac{1}{1-\alpha}\big{(}a_{2}z+(2a_{3}-a_{2}^{2})z^{2}+\cdots\big{)}=1+p_{1}z+p_{2}z^{2}+\ldots.$$

$$\left\{\begin{array}[]{rcl}a_{2}&=&(1-\alpha)p_{1},\\ a_{3}&=&\frac{1}{2}(1-\alpha)\big{[}(1-\alpha)p_{1}^{2}+p_{2}\big{]},\\ a_{4}&=&\frac{1}{6}(1-\alpha)\big{[}(1-\alpha)^{2}p_{1}^{3}+3(1-\alpha)p_{1}p_{2}+2p_{3}\big{]},\\ a_{5}&=&\frac{1}{24}(1-\alpha)\big{[}(1-\alpha)^{3}p_{1}^{4}+6(1-\alpha)^{2}p_{1}^{2}p_{2}+8(1-\alpha)p_{1}p_{3}+3(1-\alpha)p_{2}^{2}+6p_{4}\big{]}.\end{array}\right.$$

$$\begin{split}H_{3,1}(f)&=\frac{1}{144}(1-\alpha)^{2}\bigg{\{}-(1-\alpha)^{4}p_{1}^{6}+3(1-\alpha)^{3}p_{1}^{4}p_{2}+8(1-\alpha)^{2}p_{1}^{3}p_{3}-9(1-\alpha)^{2}p_{1}^{2}p_{2}^{2}\\ &\ \ \ \ -18(1-\alpha)p_{1}^{2}p_{4}+24(1-\alpha)p_{1}p_{2}p_{3}-9(1-\alpha)p_{2}^{3}+18p_{2}p_{4}-16p_{3}^{2}\bigg{\}}.\end{split}$$

$$\begin{split}H_{3,1}(f)&=\frac{1}{144}(1-\alpha)^{2}\bigg{\{}(1-\alpha)\big{[}p_{2}-(1-\alpha)p_{1}^{2}\big{]}^{3}-16\big{[}p_{3}-(1-\alpha)p_{1}p_{2}\big{]}^{2}\\ &\ \ \ \ +8\big{[}p_{2}-(1-\alpha)p_{1}^{2}\big{]}\big{[}p_{4}-(1-\alpha)p_{1}p_{3}\big{]}+10\big{[}p_{2}-(1-\alpha)p_{1}^{2}\big{]}\big{[}p_{4}-(1-\alpha)p_{2}^{2}\big{]}\bigg{\}}.\end{split}$$

Full text

**On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

**

**Yong Sun1, Zhi-Gang Wang2 and Antti Rasila3

**

1School of Science, Hunan Institute of Engineering,

Xiangtan, 411104, Hunan, People’s Republic of China

**E-Mail: yongsun2008@foxmail.com

**

2School of Mathematics and Computing Science, Hunan First Normal University,

Changsha, 410205, Hunan, People’s Republic of China

**E-Mail: wangmath@163.com

**

3Department of Mathematics and Systems Analysis, Aalto University,

P. O. Box 11100, FI-00076 Aalto, Finland

**E-Mail: antti.rasila@iki.fi

**

Abstract

In this paper, we obtain the upper bounds to the third Hankel determinants for starlike functions of order $\alpha$, convex functions of order $\alpha$ and bounded turning functions of order $\alpha$. Furthermore, several relevant results on a new subclass of close-to-convex harmonic mappings are obtained. Connections of the results presented here to those that can be found in the literature are also discussed.

2010 Mathematics Subject Classification. 30C45, 30C50, 58E20, 30C80.

Key Words and Phrases. Univalent function; starlike function; convex function; bounded turning function; close-to-convex function; harmonic mapping; Hankel determinant.

1 Introduction

Let $\mathcal{A}$ be the class of functions analytic in the unit disk \mbox{\mathbb{D}}:=\{z\in\mathbb{C}:\,|z|<1\} of the form

[TABLE]

We denote by $\mathcal{S}$ the subclass of $\mathcal{A}$ consisting of univalent functions.

A function $f\in\mathcal{A}$ is said to be starlike of order $\alpha\ (0\leq\alpha<1)$, if it satisfies the following condition:

[TABLE]

We denote by $\mathcal{S}^{*}(\alpha)$ the class of starlike functions of order $\alpha$.

Denote by $\mathcal{K}(\alpha)$ the class of functions $f\in\mathcal{A}$ such that

[TABLE]

In particular, functions in $\mathcal{K}(-1/2)$ are known to be close-to-convex but are not necessarily starlike in $\mathbb{D}$. For $0\leq\alpha<1$, functions in $\mathcal{K}(\alpha)$ are known to be convex of order $\alpha$ in $\mathbb{D}$.

A function $f\in\mathcal{A}$ is said to be in the class $\mathcal{R}(\alpha)$, consisting of functions whose derivative have a positive real part of $\alpha\ (0\leq\alpha<1)$, if it satisfies the following condition:

[TABLE]

Choosing $\alpha=0$, we denote the $\mathcal{S}:=\mathcal{S}^{*}(0)$, $\mathcal{K}:=\mathcal{K}(0)$ and $\mathcal{R}:=\mathcal{R}(0)$, the classes of starlike, convex and bounded turning functions, respectively.

Let $\mathcal{H}$ denote the class of all complex-valued harmonic mappings $f$ in $\mathbb{D}$ normalized by the condition $f(0)=f_{z}(0)-1=0$. It is well-known that such functions can be written as $f=h+\overline{g}$, where $h$ and $g$ are analytic functions in $\mathbb{D}$. We call $h$ the analytic part and $g$ the co-analytic part of $f$, respectively. Let $\mathcal{S}_{H}$ be the subclass of $\mathcal{H}$ consisting of univalent and sense-preserving mappings. Such mappings can be written in the form

[TABLE]

Harmonic mapping $f$ is called locally univalent and sense-preserving in $\mathbb{D}$ if and only if $|h^{\prime}(z)|>|g^{\prime}(z)|$ holds for z\in\mbox{\mathbb{D}}. Observe that $\mathcal{S}_{H}$ reduces to $\mathcal{S}$, the class of normalized univalent analytic functions, if the co-analytic part $g$ vanishes. The family of all functions $f\in\mathcal{S}_{H}$ with the additional property that $f_{\overline{z}}(0)=0$ is denoted by $\mathcal{S}^{0}_{H}$. For further information about planar harmonic mappings, see e.g. [14, 11, 35].

Recall that a function $f\in\mathcal{H}$ is close-to-convex in $\mathbb{D}$ if it is univalent and the range f(\mbox{\mathbb{D}}) is a close-to-convex domain, i.e., the complement of f(\mbox{\mathbb{D}}) can be written as the union of nonintersecting half-lines. A normalized analytic function $f$ in $\mathbb{D}$ is close-to-convex in $\mathbb{D}$ if there exists a convex analytic function in $\mathbb{D}$, not necessarily normalized, $\phi$ such that \Re\big{(}f^{\prime}(z)/\phi^{\prime}(z)\big{)}>0. In particular, if $\phi(z)=z$, then for any $f\in\mathcal{A}$, \Re\big{(}f^{\prime}(z)\big{)}>0 implies $f$ is close-to-convex in $\mathbb{D}$, see [39]. We refer to [7, 21, 31, 36, 37] for discussion and basic results on close-to-convex harmonic mappings.

For a harmonic mapping $f=h+\overline{g}$ in $\mathbb{D}$, a basic result in [30] (see also [29]) shows that if at least one of the analytic functions $h$ and $g$ is convex, then $f$ is univalent whenever it is locally univalent in $\mathbb{D}$. It is natural to study the univalence of $f=h+\overline{g}$ in $\mathbb{D}$ if it is locally univalent and sense-preserving, and analytic function $h$ is univalent and close-to-convex. Motivated by this idea, we next consider the following subclass of $\mathcal{H}$.

Definition 1**.**

For $\alpha\in\mathbb{C}$ with $-1/2\leq\alpha<1$, let $\mathcal{M}(\alpha)$ denote the class of harmonic mapping $f$ in $\mathbb{D}$ of the form (1.2), with $h^{\prime}(0)\neq 0$, which satisfy

[TABLE]

By making use of the similar arguments to those in the proof of [8, Theorem 1], one can easily obtain the close-to-convexity of the class $\mathcal{M}(\alpha)$. For special values of $\alpha$, many authors have studied the class of close-to-convex harmonic mappings, see e.g. [30, 10, 40, 6, 31].

Pommerenke (see [33, 34]) defined the Hankel determinant $H_{q,n}(f)$ as

[TABLE]

Problems involving Hankel determinants $H_{q,n}(f)$ in geometric function theory originate from the work of, e.g., Hadamard, Polya and Edrei (see [12, 15]), who used them in study of singularities of meromorphic functions. For example, they can be used in showing that a function of bounded characteristic in $\mathbb{D}$, i.e., a function which is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational [9]. Pommerenke [33] proved that the Hankel determinants of univalent functions satisfy the inequality $|H_{q,n}(f)|<Kn^{-(\frac{1}{2}+\beta)q+\frac{3}{2}}$, where $\beta>1/4000$ and $K$ depends only on $q$. Furthermore, Hayman [18] has proved a stronger result for areally mean univalent functions, i.e., the estimate $H_{2,n}(f)<An^{1/2}$, where $A$ is an absolute constant.

We note that $H_{2,1}(f)$ is the well-known Fekete-Szegő functional, see [16, 22, 23]. The sharp upper bounds on $H_{2,2}(f)$ were obtained by the authors of articles [4, 19, 20, 24] for various classes of functions.

By the definition, $H_{3,1}(f)$ is given by

[TABLE]

Note that for $f\in\mathcal{A}$, $a_{1}=1$ so that

[TABLE]

Obviously, the case of the upper bounds on $H_{3,1}(f)$ it is much more difficult than the cases of $H_{2,1}(f)$ and $H_{2,2}(f)$. In 2010, Babalola [3] has studied the $\max|H_{3,1}(f)|$ for the classes of starlike, convex and bounded turning functions.

Theorem A.

• Let $f\in\mathcal{S}^{*}$, $h\in\mathcal{K}$ and $g\in\mathcal{R}$, respectively. Then*

[TABLE]

and

[TABLE]

Recently, Zaprawa [42] proved that

Theorem B.

• Let $f\in\mathcal{S}^{*}$, $h\in\mathcal{K}$ and $g\in\mathcal{R}$, respectively. Then*

[TABLE]

Raza and Malik [38] have obtained the upper bound on $|H_{3,1}(f)|$ for a class of analytic functions that is related to the lemniscate of Bernoulli. Also, Bansal et al. [5] obtained the following results

Theorem C.

• Let $h\in\mathcal{K}(-1/2)$ and $g\in\mathcal{R}$, respectively. Then*

[TABLE]

For the class $\mathcal{R}(\alpha)$, Vamshee Krishna et al. [41] proved that

Theorem D.

• Let $g\in\mathcal{R}(\alpha)$ with $0\leq\alpha\leq\frac{1}{4}$. Then*

[TABLE]

In the present investigation, our goal is to discuss the upper bounds to the third Hankel determinants for the subclasses of univalent functions: $\mathcal{S}^{*}(\alpha)$, $\mathcal{K}(\alpha)$ and $\mathcal{R}(\alpha)$. Furthermore, we develop similar results on the Hankel determinants $|H_{3,1}(h)|$ and $|H_{3,1}(g)|$ in the context the close-to-convex harmonic mappings $f=h+\overline{g}\in\mathcal{M}(\alpha)$.

2 Preliminary results

Denote by $\mathcal{P}$ the class of Carathéodory functions $p$ normalized by

[TABLE]

Following results are the well known for functions belonging to the class $\mathcal{P}$.

Lemma 1**.**

[13]* If $p\in\mathcal{P}$ is of the form (2.1), then*

[TABLE]

The inequality (2.2) is sharp and the equality holds for the function

[TABLE]

Lemma 2**.**

[28]* If $p\in\mathcal{P}$ is of the form (2.1), then holds the sharp estimate*

[TABLE]

Lemma 3**.**

[17]* If $p\in\mathcal{P}$ is of the form (2.1), then holds the sharp estimate*

[TABLE]

Lemma 4**.**

[26, 27]* If $p\in\mathcal{P}$ is of the form (2.1), then there exist $x$, $z$ such that $|x|\leq 1$ and $|z|\leq 1$,*

[TABLE]

and

[TABLE]

3 Bounds of Hankel determinants for $\mathcal{S}^{*}(\alpha)$, $\mathcal{K}(\alpha)$ and $\mathcal{R}(\alpha)$

In this section, we assume that

[TABLE]

Theorem 1**.**

Let $f\in\mathcal{S}^{*}(\alpha)$, $h\in\mathcal{K}(\alpha)$ and $g\in\mathcal{R}(\alpha)$ with $0\leq\alpha<1$, respectively. Then

[TABLE]

[TABLE]

and

[TABLE]

Proof.

Let

[TABLE]

then, we have \Re\big{(}p(z)\big{)}>0, and by elementary calculations, we obtain

[TABLE]

It follows from (3.4) that

[TABLE]

Hence, by using the above values of $a_{2}$, $a_{3}$, $a_{4}$ and $a_{5}$ from (3.5), and by a routine computation, we obtain

[TABLE]

From (3.6), we have

[TABLE]

We note that

[TABLE]

by triangle inequality and Lemma 3, we obtain the estimate (3.1) of $H_{3,1}(f)$.

Next, we consider $H_{3,1}(h)$. According to the Alexander relation, $b_{k}=ka_{k}\ (k\in\mathbb{N})$. Putting it into the definition of $H_{3,1}(h)$ and applying the formula (3.5), we have

[TABLE]

From (3.7), we have

[TABLE]

We observe that for $0\leq\alpha<1$ holds

[TABLE]

By using Lemma 1 and Lemma 3 and triangle inequality, it easy to get the estimate (3.2) of $H_{3,1}(g)$.

Finally, for $H_{3,1}(g)$. Let

[TABLE]

If $g\in\mathcal{R}(\alpha)$, then

[TABLE]

Putting it into the definition of $H_{3,1}(g)$ and by the same way, we have

[TABLE]

Hence, it is easy to obtain the bound of $H_{3,1}(g)$. This completes the proof. ∎

Remark 1**.**

By setting $\alpha=0$ in Theorem 1, we obtain the known results of Theorem B, and they are much better than Theorem A. Furthermore, the bounds of $H_{3,1}(g)$ in (3.3) improved and extended the result of the Theorem D. **

In 1960, Lawrence Zalcman posed a conjecture that the coefficients of $\mathcal{S}$ satisfy the sharp inequality

[TABLE]

with equality only for the Koebe function $k(z)=z/(1-z)^{2}$ and its rotations. We call $J_{n}(f)=a_{n}^{2}-a_{2n-1}$ the Zalcman functional for $f\in\mathcal{S}$.

We observe that $H_{3,1}(f)\ (f\in\mathcal{A})$ can be written in the form

[TABLE]

and equivalently,

[TABLE]

An analogous calculation can be applied to the Zalcman functional $J_{n}(f)$ for the classes of starlike, convex and bounded turning functions of order $\alpha$.

Theorem 2**.**

The following estimates hold for $J_{n}(f)$:

If $f\in\mathcal{S}^{*}(\alpha)\ (0\leq\alpha<1)$, then $J_{3}(f)\leq\frac{1}{2}(1-\alpha)(8-7\alpha)$. 2. 2.

If $h\in\mathcal{K}(\alpha)\ (-1/2\leq\alpha<1)$, then $J_{3}(h)\leq\frac{1}{360}(1-\alpha)(127-109\alpha)$. 3. 3.

If $g\in\mathcal{R}(\alpha)\ (0\leq\alpha<1)$, then $J_{n}(g)\leq\frac{2}{2n-1}(1-\alpha)\qquad(n\geq 2)$.

Proof.

Let $f\in\mathcal{S}^{*}(\alpha)$, from (3.5), it follow that

[TABLE]

By using Lemma 1 and Lemma 3, we obtain the above bound for the Zalcman functional $J_{3}(f)$.

Combining the Alexander relation, $b_{k}=ka_{k}$, and the formula (3.5), yields

[TABLE]

Again, by using Lemma 1 and Lemma 3, we obtain the bound for the Zalcman functional $J_{3}(h)$.

For $g\in\mathcal{R}(\alpha)$, according to the formula (3.8), we have

[TABLE]

In view of

[TABLE]

and, by Lemma 3, we have the desired bound of the Zalcman functional $J_{n}(g)$. This completes the proof. ∎

Remark 2**.**

By setting $\alpha=-1/2$ for the class $\mathcal{K}(\alpha)$ in Theorem 2, we obtain the known results [2, Theorem 2.3]. Furthermore, using the similar argument in Theorem 2, we may obtain the bounds of the Zalcman functional $J_{2}(f)$ and $J_{2}(h)$: If $f\in\mathcal{S}^{*}(\alpha)\ (0\leq\alpha<1)$, then $J_{2}(f)\leq 1-\alpha$. If $h\in\mathcal{K}(\alpha)\ (-1/2\leq\alpha<1)$, then $J_{2}(h)\leq\frac{1}{3}(1-\alpha)$ . **

4 Bounds of Hankel determinants for $\mathcal{M}(\alpha)$

In this section, we obtain upper bounds for the Hankel determinants $|H_{3,1}(h)|$ and $|H_{3,1}(g)|$ of close-to-convex harmonic mappings $f=h+\overline{g}\in\mathcal{M}(\alpha)$.

Theorem 3**.**

Let $f=h+\overline{g}\in\mathcal{M}(\alpha)$ be of the form (1.2). Then

[TABLE]

and

[TABLE]

Proof.

Let

[TABLE]

Using the same method of Theorem 1, we get the expression of $H_{3,1}(h)$ is the formula (3.7). We give another decomposition for functional $H_{3,1}(h)$ as follows

[TABLE]

We note that

[TABLE]

by triangle inequality and Lemmas 1-3, we can obtain the estimate of $H_{3,1}(h)$.

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

[TABLE]

which yields

[TABLE]

Then, by using (2.5) and (2.6) in Lemma 4, we obtain that for some $x$ and $z$ such that $|x|\leq 1$ and $|z|\leq 1$,

[TABLE]

By Lemma 1, we may assume that $|p_{1}|=c\in[0,2]$. By applying the triangle inequality in above relation with $\mu=|x|$, we obtain

[TABLE]

We note that

[TABLE]

Hence, we have

[TABLE]

Let

[TABLE]

Then, we obtain

[TABLE]

and

[TABLE]

Solving the equation $\chi^{\prime}(c)=0$, we get the critical points are $c=0$ and

[TABLE]

We observe that

[TABLE]

and

[TABLE]

Hence, we get

[TABLE]

Thus, we obtain the following bound

[TABLE]

∎

Remark 3**.**

In order to obtain the bounds of $H_{3,1}(h)$, we give two kinds of decomposition for formula (3.7) in Theorem 1 and Theorem 3, respectively. Hence, it is a natural question: Whether there is an optimal decomposition for the similar formulae. **

Remark 4**.**

For $H_{3,1}(g)$ in Theorem 3, if we apply the method in Theorem 1, then

[TABLE]

By using Lemmas 1 and 3, we have

[TABLE]

obviously,

[TABLE]

Hence, we choose the bound of in $H_{3,1}(g)$ in Theorem 3. **

Corollary 1**.**

Let $f=h+\overline{g}\in\mathcal{M}(-1/2)$ be of the form (1.2). Then

[TABLE]

Remark 5**.**

The result of $H_{3,1}(h)$ in Corollary 1 is much better than Theorem C (see [5, Theorem 2.7]). From the upper bounds of $H_{3,1}(h)$ and $H_{3,1}(g)$, we note that the former is much larger than the latter, this implies that the analytic part $h$ accounts for absolute advantage than the co-analytic part $g$ for the harmonic mappings $f=h+\overline{g}\in\mathcal{M}(\alpha)$. **

Acknowledgements

The present investigation was supported by the Natural Science Foundation of Hunan Province under Grant no. 2016JJ2036, and the Foundation of Educational Committee of Hunan Province under Grant no. 15C1089.