Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function

Abstract

The aim of this paper is to introduce two new subclasses $\mathcal{R}_{\sin }^{m}(\Im )$ and $\mathcal{R}_{\sin }(\Im )$ of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function $\Im (v)=\frac{2}{1+e^{-v}}$, $v\geq 0$ in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results.

1 Introduction

Let $\mathcal{A}$ denote the class of functions satisfying the following series form:

$$ \chi (z)=z+\sum_{n=2}^{\infty }d_{n}z^{n}, $$

(1.1)

which are analytic in the open unit disc $E= \{ z:z\in {\mathbb{C} } \text{{ and }} \vert z \vert <1 \}$. The functions χ having the series form (1.1) are called univalent in E and denoted by $\mathcal{S}$, if it is one to one, that is, for all $z_{1},z_{2}\in E$, if $\chi (z_{1})=\chi (z_{2})$ implies $z_{1}=z_{2}$.

In [1] the author defined the class $\mathcal{R}_{\alpha }$ ($\alpha \geq 0$) having the functions that satisfy the condition

$$ \operatorname{Re} \bigl\{ \chi ^{{\prime }}(z)+\alpha z\chi ^{{\prime \prime }}(z) \bigr\} >0,\quad (z\in E). $$

They also proved that if $\chi \in \mathcal{R}_{\alpha }$ then χ is univalent in E. Singh and Singh [2] showed that if $\chi \in \mathcal{R}$ ($\alpha =1$) then χ is starlike in E. Furthermore, they proved in [2] that the class $\mathcal{R}$ is closed under convolution, that is, if $\chi ,g\in \mathcal{R}$ then $(\chi \ast g)\in \mathcal{R}$, where ∗ stands for convolution. Also in [3], Krzyz showed by an example that the class $\mathcal{R}$ is not a subset of the convex functions class C. In 2015, Noor et al. [4] generalized the class $\mathcal{R}$ by using the idea of multivalent functions and conic regions. Khan et al. [5] generalized it further in 2021.

An analytic function w under the condition $w(0)=0$ and $\vert w(z) \vert <1$ is known as a Schwarz function. Consider the functions χ, $g\in \mathcal{A}$, we say that χ is subordinate to g (indicated by $\chi \prec g$) if $\chi (z)=g(w(z))$. Further, if g is univalent in E, then $\chi \prec g\Leftrightarrow \chi (0)=g(0)$ and $\chi (E)\subset g(E)$. The class $\mathcal{P}$ denotes the well-known class of Caratheodory functions [6], which satisfy the conditions $p(0)=1$ and $\operatorname{Re}(p(z)>0$, where $z\in E$. Every $p\in \mathcal{P}$, having the series form

$$ p(z)=1+\sum_{n=1}^{\infty }c_{n}z^{n}. $$

(1.2)

Since the nineteenth century, when the geometric function theory was established, coefficient bounds have been continuously important. The Bieberbach conjecture [7], which came to be in 1916, offered an innovative field of study for this field of research. He conjectured that for every $\chi \in \mathcal{S}$ having the series form (1.1), $\vert d_{n} \vert \leq n$, $n\geq 4$. This conjecture was attempted to be proved for a long time by mathematicians until de-Branges [8] proved it in 1985.

In 1992, the authors contributed in their article [9] by revealing the basic structure of families of univalent functions $\mathcal{S}^{\ast }(\varphi )$, where φ is an analytic function satisfying $\varphi (0)>0$ and $\operatorname{Re}(\varphi (z))>0$ in E. When we fix $\varphi (z)=\frac{1+z}{1-z}$, then $\mathcal{S}^{\ast }(\varphi )\approxeq \mathcal{S}^{\ast }$. Several subfamilies of generalized analytic functions have been studied recently as a specific case of $\mathcal{S}^{\ast }(\varphi )$.

For example, Janowski [10] examined the class of Janowski starlike functions $\mathcal{S}^{\ast }[L,M]$ ($-1\leq M< L\leq 1$). Furthermore, by choosing $L= ( 1-2\alpha )$ and $M=-1$, we get $\mathcal{S}^{\ast }(\alpha )$ ($0\leq \alpha <1$). Sokól and Stankiewicz [11] set $\varphi (z)=\sqrt{1+z}$ and defined the family of class $S_{\mathcal{L}}^{\ast }$

$$ S_{\mathcal{L}}^{\ast }= \biggl\{ \chi \in \mathcal{A}: \frac{z\chi ^{{\prime }}(z)}{\chi (z)} \prec \sqrt{1+z} \biggr\} . $$

Recently, the authors [12, 13] chose $\varphi (z)=1+\sin z$ and $\varphi (z)=1+\frac{4}{3}z+\frac{2}{3}z^{2}$ and defined the following classes of convex, starlike, and bounded turning functions:

$$\begin{aligned}& \mathcal{C}_{\sin } = \biggl\{ \chi \in \mathcal{S}:1+ \frac{z\chi ^{{^{\prime \prime }}}(z)}{\chi ^{{\prime }}(z)}\prec 1+ \sin z \biggr\} , \\& \mathcal{S}_{\sin }^{\ast } = \biggl\{ \chi \in \mathcal{A}: \frac{z\chi ^{{\prime }}(z)}{\chi (z)}\prec 1+\sin z \biggr\} , \\& \mathcal{R}_{\sin } = \bigl\{ \chi \in \mathcal{A}:\chi ^{{\prime }}(z)+z \chi ^{\prime \prime }(z)\prec 1+\sin z \bigr\} , \\& \mathcal{R}_{\mathrm{card}} = \biggl\{ \chi \in \mathcal{A}:\chi ^{{\prime }}(z)+z \chi ^{\prime \prime }(z)\prec 1+\frac{4}{3}z+ \frac{2}{3}z^{2} \biggr\} . \end{aligned}$$

Authors investigated initial bounds, Fekete–Szego problems, and third Hankel determinant for the above mentioned classes. In [13, 14] authors introduced the class of starlike functions whose image under an open unit has a cardioid form. In [15], Mendiratta et al. studied the function class $\mathcal{S}_{e}^{\ast }\equiv \mathcal{S}^{\ast } ( e^{z} )$ of starlike functions by using the idea of exponential functions and subordination technique. This class was recently generalized by Srivastava et al. [16], who also found an upper bound for the third-order Hankel determinant.

In 1966 Pommerenke [17] explored research on the Hankel determinants for univalent functions, which were further investigated by Noonan and Thomas [18]. For $\chi \in \mathcal{A}$, the jth Hankel determinant is defined by

$$ \mathcal{H}_{j,n}(\chi )= \begin{vmatrix} d_{n} & d_{n+1} & \cdots & d_{n+j-1} \\ d_{n+1} & d_{n+2} & \cdots & d_{n+j} \\ \vdots & \vdots & \ddots & \vdots \\ d_{n+j-1} & d_{n+j-2} & \ldots & d_{n+2j-2}\end{vmatrix}, $$

(1.3)

where $n,j\in \mathbb{N}$ and $d_{1}=1$.

For different values of j and n, the jth Hankel determinant $\mathcal{H}_{j,n}(\chi )$ has a different form. For example, Fekete–Szego functional that is

$$ \mathcal{H}_{2,1}(\chi )= \bigl\vert d_{3}-d_{2}^{2} \bigr\vert \quad \text{for }j=2\text{ and }n=1, $$

and its modified form is $\vert d_{3}-\mu d_{2}^{2} \vert$, where $\mu \in \mathbb{R}$ (or $\mathbb{C}$) (see [19]). The second Hankel determinant was similarly provided by Janteng [20] in the following form:

$$ \mathcal{H}_{2,2}(\chi )= \begin{vmatrix} d_{2} & d_{3} \\ d_{3} & d_{4}\end{vmatrix}= \bigl( d_{2}d_{4}-d_{3}^{2} \bigr), $$

and a number of scholars then looked into it for a few other classes of analytic functions. Further, the third Hankel determinant form is indicated below:

$$ \mathcal{H}_{3,1}(\chi )=d_{3} \bigl( d_{2}d_{4}-d_{3}^{2} \bigr) -d_{4} ( d_{4}-d_{2}d_{3} ) +d_{5} \bigl( d_{3}-d_{2}^{2} \bigr) \quad \text{for }j=3\text{ and }n=1. $$

(1.4)

In 2021, for $\chi \in \mathcal{S}$, the authors in [21] found Hankel determinants of second and third order

$$ \bigl\vert \mathcal{H}_{2,2}(\chi ) \bigr\vert \leq \lambda ,\qquad 1 \leq \lambda \leq \frac{11}{3} $$

and

$$ \bigl\vert \mathcal{H}_{3,1}(\chi ) \bigr\vert \leq \lambda ,\qquad \frac{4}{9}\leq \lambda \leq \frac{32+\sqrt{285}}{15}. $$

Recently, different researchers are active to find the sharp bounds of $\mathcal{H}_{j,n}(\chi )$ for a different family of functions. For instance, Cho et al. [22, 23] computed bounds of the second Hankel determinant of the classes of convex, starlike, and bounded turning. Compared to the second and third Hankel determinants, the mathematical computation of the fourth Hankel determinant is significantly more difficult. For specific classes of univalent functions, Babalola [24] calculated the third Hankel determinant in 2010. See the following articles for more details [25–28].

After that, a number of researchers examined the third Hankel determinant for different subclasses of analytic and bi-univalent functions using the same methodology. Zaprawa [29] in 2017 investigated third Hankel determinant for two basic subclasses of univalent functions class as follows:

$$|{\mathcal{H}}_{3,1}(\chi )|\le \left\{\begin{array}{cc}1& \text{if }\chi \in {\mathcal{S}}^{\ast }\text{ (class of starlike functions)}\\ \frac{49}{540}& \text{if }\chi \in \mathcal{K}\text{ (class of convex functions)}\end{array}\right\}.$$

But in [30] authors improved the above result in the year 2018 and proved that $\vert \mathcal{H}_{3,1}(\chi ) \vert \leq \frac{8}{9}$, ($\chi \in \mathcal{S}^{\ast }$). In 2021, Zaprawa et al. [31] again improved the result of [30] as follows:

$$ \bigl\vert \mathcal{H}_{3,1}(\chi ) \bigr\vert \leq \frac{5}{9},\quad \chi \in \mathcal{S}^{\ast }. $$

For the analysis of power series with integral coefficients and singularities, $\mathcal{H}_{j,n}(\chi )$ is very helpful. Numerous technological studies have made use of Hankel determinants, particularly those that depend mainly on mathematical approaches. Readers interested in understanding how the solutions to the above listed problems make use of Hankel determinants ought to read [32–40].

Let $\mathcal{A}_{\Im }$ denote the class of sigmoid functions having the form (see [41])

$$ \chi _{\Im }(z)=z+\sum_{n=2}^{\infty }\Im ( v ) d_{n}z^{n}, $$

(1.5)

where

$$ \Im ( v ) =\frac{2}{1+e^{-v}},\quad v\geq 0. $$

(1.6)

From (1.6) we see that $\Im ( 0 ) =1$ and $\mathcal{A}_{1}=\mathcal{A}$.

Definition 1.1

[42]. The Sălăgean type differential operator $S^{m}:\mathcal{A}_{\Im }\rightarrow \mathcal{A}_{\Im }$ is defined by

$$\begin{aligned}& S_{q}^{0}\chi _{\Im }(z) = \chi _{\Im }(z),\qquad S^{1}\chi (z)=z \chi _{\Im }^{{\prime }}(z),\quad \ldots , \\& S^{m}\chi _{\Im }(z) = zS \bigl( S^{m-1}\chi _{\Im }(z) \bigr) , \end{aligned}$$

(1.7)

where $\chi _{\Im }(z)\in \mathcal{A}_{\Im }$, $m\in \mathbb{N} \cup \{0\}$. It is easy to prove that if

$$ \chi _{\Im }(z)=z+\sum_{n=2}^{\infty }\Im ( v ) d_{n}z^{n} \in \mathcal{A}_{\Im }, $$

then

$$ S^{m}\chi _{\Im }(z)=z+\sum_{n=2}^{\infty }n^{m} \Im ( v ) d_{n}z^{n}. $$

Remark 1.2

When $\Im ( v ) =1$, we have the Sălăgean differential operator [43].

Here we define a new class of bounded turning functions connected with modified sigmoid function and sine functions as follows.

Definition 1.3

A function $\chi _{\Im }\in \mathcal{R}_{\sin }^{m}(\Im )$, where $\chi _{\Im }$ is of the form (1.5), if

$$ \bigl( S^{m}\chi _{\Im }(z) \bigr) ^{{\prime }}\prec 1+\sin ( z ) , $$

(1.8)

or it can be defined as

$$ \frac{S^{m+1}\chi _{\Im }(z)}{z}\prec 1+\sin ( z ) . $$

When $v=0$ and $m=0$ in Definition 1.3, we get a known class proved in [44].

When $m=0$ in Definition 1.3, we get the following function class.

Definition 1.4

A function $\chi _{\Im }\in \mathcal{R}_{\sin } ( \Im )$, where $\chi _{\Im }$ is of the form (1.5), if

$$ \chi _{\Im }^{{\prime }}(z)\prec 1+\sin ( z ) . $$

(1.9)

2 Set of lemmas

Lemma 2.1

Let the function p be of the form (1.2), then

$$\begin{aligned}& \vert c_{n} \vert \leq 2,\quad n\geq 1, \end{aligned}$$

(2.1)

$$\begin{aligned}& \vert c_{n+k}-\mu c_{n}c_{k} \vert < 2, \quad \textit{for } 0 \leq \mu \leq 1 \end{aligned}$$

(2.2)

$$\begin{aligned}& \vert c_{m}c_{n}-c_{k}c_{l} \vert \leq 4 \quad \textit{for } n+m=k+l, \end{aligned}$$

(2.3)

$$\begin{aligned}& \bigl\vert c_{n+2k}-\mu c_{n}c_{k}^{2} \bigr\vert \leq 2(1+2\mu ) \quad \textit{for } \mu \in \mathbb{R} . \end{aligned}$$

(2.4)

$$\begin{aligned}& \biggl\vert c_{2}-\frac{c_{l}^{2}}{2} \biggr\vert \leq 2- \frac{ \vert c_{l}^{2} \vert }{2}, \end{aligned}$$

(2.5)

for complex number μ, we have

$$ \bigl\vert c_{2}-\mu c_{2}^{2} \bigr\vert \leq 2\max \bigl\{ 1, \vert 2\mu -1 \vert \bigr\} . $$

(2.6)

For the results in (2.1), (2.5), (2.2), (2.4), (2.3), see [45]. Also, see [46] for inequality (2.6)

Lemma 2.2

[47]. Let the function $p\in \mathcal{P}$ be given by (1.2)), then

$$ \bigl\vert c_{3}-2Bc_{1}c_{2}+Dc_{1}^{3} \bigr\vert \leq 2, $$

if

$$ 0\leq B\leq 1,\quad \textit{and}\quad B(2B-1)\leq D\leq B. $$

Lemma 2.3

[48, 49]. Let the function $p\in \mathcal{P}$ be given by (1.2)), then

$$ 2c_{2}=c_{1}^{2}+x\bigl(4-c_{1}^{2} \bigr) $$

and

$$ 4c_{3}=c_{1}^{3}+2\bigl(4-c_{1}^{2} \bigr)c_{1}x-\bigl(4-c_{1}^{2} \bigr)c_{1}x^{2}+2\bigl(4-c_{1}^{2} \bigr) \bigl(1- \bigl\vert x^{2} \bigr\vert \bigr)z, $$

where $x,z\in \mathbb{C}$ with $\vert z \vert \leq 1$ and $\vert x \vert \leq 1$.

Lemma 2.4

[50]. Consider the function $p\in \mathcal{P}$ of the form (1.2), $0<\gamma <1$, $0<\alpha <1$, and

$$\begin{aligned}& 8\gamma (1-\gamma ) \bigl\{ ( \alpha \beta -2\lambda ) ^{2}+ \bigl( \alpha ( \gamma +\alpha ) -\beta \bigr) ^{2} \bigr\} \\& \qquad {}+\alpha ( 1-\alpha ) ( \beta -2\gamma \alpha ) ^{2} \\& \quad \leq 4\alpha ^{2}\gamma ( 1-\alpha ) ^{2}(1-\gamma ). \end{aligned}$$

(2.7)

Then

$$ \biggl\vert \lambda b_{1}^{4}+\gamma b_{2}^{2}+2\alpha b_{1}b_{3}- \frac{3}{2}\beta b_{1}^{2}b_{2}-b_{4} \biggr\vert \leq 2. $$

(2.8)

We divided our paper into four parts. In Sect. 1, we give some basic definitions including some subclasses of analytic functions, such as starlike, bounded turning, and convex functions, also we give the definitions of the Hankel determinant, the modified sigmoid function, and the sine function. Inspired by the above mentioned work, we create new classes of analytic functions related to modified sigmoid function and sine functions. Also we consider the Salagean type of differential operator and define a new class of analytic functions. In Sect. 2, we use known lemmas to prove our article’s main results. For functions χ in the classes $\mathcal{R}_{\sin }^{m}(\Im )$ and $\mathcal{R}_{\sin }(\Im )$, we first calculate the initial coefficient bounds, Fekete–Szego problem, and the second, third, and fourth Hankel determinants in Sect. 3 and highlight some known results.

3 Main results

Main findings for the function class $\mathcal{R}_{\sin }^{m}(\Im )$.

Theorem 3.1

If $\chi _{\Im }\in \mathcal{R}_{\sin }^{m}(\Im )$ where $\chi _{\Im }$ has the form (1.5), then

$$\begin{aligned}& \vert d_{2} \vert \leq \frac{1}{2 ( 2^{m}\Im (v) ) }, \end{aligned}$$

(3.1)

$$\begin{aligned}& \vert d_{3} \vert \leq \frac{1}{3 ( 3^{m}\Im (v) ) }, \end{aligned}$$

(3.2)

$$\begin{aligned}& \vert d_{4} \vert \leq \frac{1}{4 ( 4^{m}\Im (v) ) }, \end{aligned}$$

(3.3)

$$\begin{aligned}& \vert d_{5} \vert \leq \frac{1}{5 ( 5^{m}\Im (v) ) }, \end{aligned}$$

(3.4)

where $\Im (v)$ is given by (1.6). The results are sharp.

Proof

Let $\chi _{\Im }\in \mathcal{R}_{\sin }^{m}(\Im )$, then from relation (1.8) we have

$$ \bigl( S^{m}\chi _{\Im }(z) \bigr) ^{{\prime }}=1+\sin \bigl(w(z)\bigr), $$

(3.5)

where w is a Schwarz function.

Now consider a function p such that

$$ p(z)=\frac{1+w(z)}{1-w(z)}=1+c_{1}z+c_{2}z^{2}+c_{3}z^{3}+ \cdots, $$

(3.6)

then $p\in \mathcal{P}$. From (3.6), a simple computation yields

$$ w(z)= \frac{c_{1}z+c_{2}z^{2}+c_{3}z^{3}+\cdots}{2+c_{1}z+c_{2}z^{2}+c_{3}z^{3}+\cdots}. $$

From (1.8), we can write

$$\begin{aligned} \bigl( S^{m}\chi _{\Im }(z) \bigr) ^{{\prime }} =&1+2 \bigl( 2^{m} \Im (v) \bigr) d_{2}z+3 \bigl( 3^{m}\Im (v) \bigr) d_{3}z^{2} \\ &{}+4 \bigl( 4^{m}\Im (v) \bigr) d_{4}z^{3}+5 \bigl( 5^{m}\Im (v) \bigr) d_{5}z^{4}+\cdots. \end{aligned}$$

(3.7)

By using the above values and after simplification, we get

$$\begin{aligned} 1+\sin \bigl(w(z)\bigr) =&1+\frac{1}{2}c_{1}z+ \biggl( \frac{c_{2}}{2}- \frac{c_{1}^{2}}{4} \biggr) z^{2}+ \biggl( \frac{5c_{1}^{3}}{48}-\frac{c_{1}c_{2}}{2}+ \frac{c_{3}}{2} \biggr) z^{3} \\ &{}+ \biggl( -\frac{c_{1}^{4}}{32}+\frac{5c_{1}^{2}c_{2}}{16}- \frac{c_{1}c_{3}}{2}- \frac{c_{2}^{2}}{2}+\frac{c_{4}}{2} \biggr) z^{4}+\cdots. \end{aligned}$$

(3.8)

From (3.5), (3.7), and (3.8), it follows that

$$\begin{aligned}& d_{2}=\frac{c_{1}}{4 ( 2^{m}\Im (v) ) }, \end{aligned}$$

(3.9)

$$\begin{aligned}& d_{3}=\frac{1}{3 ( 3^{m}\Im (v) ) } \biggl( \frac{c_{2}}{2}- \frac{c_{1}^{2}}{4} \biggr) , \end{aligned}$$

(3.10)

$$\begin{aligned}& d_{4}=\frac{1}{4 ( 4^{m}\Im (v) ) } \biggl( \frac{5c_{1}^{3}}{48}-\frac{c_{1}c_{2}}{2}+\frac{c_{3}}{2} \biggr) , \end{aligned}$$

(3.11)

and

$$ d_{5}=\frac{1}{5 ( 5^{m}\Im (v) ) } \biggl( \frac{-1}{32}c_{1}^{4}-\frac{c_{2}^{2}}{4}-\frac{c_{1}c_{3}}{2}+\frac{5c_{1}^{2}c_{2}}{16}+ \frac{c_{4}}{2} \biggr) . $$

(3.12)

Applying relation (2.1) in (3.9), we obtain

$$ \vert d_{2} \vert \leq \frac{1}{2 ( 2^{m}\Im (v) ) }. $$

Using relations (2.1) and (2.5) on (3.10), we obtain

$$\begin{aligned} \vert d_{3} \vert =& \frac{1}{6 ( 3^{m}\Im (v) ) } \biggl\vert c_{2}- \frac{c_{1}^{2}}{2} \biggr\vert \\ \leq &\frac{1}{6 ( 3^{m}\Im (v) ) } \biggl( 2- \frac{ \vert c_{1} \vert ^{2}}{2} \biggr) \\ =&\frac{1}{3 ( 3^{m}\Im (v) ) }. \end{aligned}$$

Rearranging (3.11) gives

$$\begin{aligned} \vert d_{4} \vert =&\frac{1}{4 ( 4^{m}\Im (v) ) } \biggl\vert \frac{c_{3}}{2}-\frac{c_{1}c_{2}}{2}+\frac{5}{24}c_{1}^{3} \biggr\vert . \\ =&\frac{1}{8 ( 4^{m}\Im (v) ) } \bigl\vert c_{3}-2Bc_{1}c_{2}+Dc_{1}^{3} \bigr\vert . \end{aligned}$$

Let

$$ B=\frac{1}{2}\quad \text{and}\quad D=\frac{5}{24}. $$

We can observe that $0< B$ and $B ( 2B-1 ) < D< B$. Therefore, using Lemma 2.2 leads us to

$$ \vert d_{4} \vert \leq \frac{1}{4 ( 4^{m}\Im (v) ) }. $$

From (3.12) it follows that

$$ \vert d_{5} \vert = \frac{1}{5 ( 5^{m}\Im (v) ) } \biggl\vert \frac{-1}{32}c_{1}^{4}- \frac{c_{2}^{2}}{4}- \frac{c_{1}c_{3}}{2}+\frac{5c_{1}^{2}c_{2}}{16}+\frac{c_{4}}{2} \biggr\vert $$

or

$$\begin{aligned} \vert d_{5} \vert =& \frac{1}{10 ( 5^{m}\Im (v) ) } \biggl\vert \frac{1}{16}c_{1}^{4}+\frac{c_{2}^{2}}{2}+2 \biggl( \frac{c_{1}c_{3}}{2} \biggr) -\frac{3}{2} \biggl( \frac{5c_{1}^{2}c_{2}}{24} \biggr) -c_{4} \biggr\vert \\ =&\frac{1}{10 ( 5^{m}\Im (v) ) } \biggl\vert \lambda c_{1}^{4}+ \gamma c_{2}^{2}+2\alpha c_{1}c_{3}- \frac{3}{2}\beta c_{1}^{2}c_{2}-c_{4} \biggr\vert , \end{aligned}$$

where

$$ \lambda =\frac{1}{16}, \qquad \gamma =\frac{1}{2},\qquad \alpha = \frac{1}{2}, \qquad \beta =\frac{5}{24}. $$

We see that $0<\gamma <1$ and $0<\alpha <1$. Now we calculate inequality (3.1), we see that

$$\begin{aligned}& 8\gamma (1-\gamma ) \bigl\{ ( \alpha \beta -2\lambda ) ^{2}+ \bigl( \alpha ( \gamma +\alpha ) -\beta \bigr) ^{2} \bigr\} +\alpha ( 1-\alpha ) ( \beta -2\gamma \alpha ) ^{2} \\& \quad \leq 4\alpha ^{2}\gamma ( 1-\alpha ) ^{2}(1-\gamma ), \end{aligned}$$

where

$$\begin{aligned}& 8\gamma (1-\gamma ) = 2, \qquad ( \alpha \beta -2\lambda ) ^{2}= \frac{1}{2304}, \qquad \bigl( \alpha ( \gamma + \alpha ) -\beta \bigr) ^{2}=\frac{49}{576}, \\& \alpha ( 1-\alpha ) ( \beta -2\gamma \alpha ) ^{2} = \frac{49}{2304}, \qquad 4\alpha ^{2}\gamma ( 1-\alpha ) ^{2}(1- \gamma )=\frac{1}{16}. \end{aligned}$$

By using Lemma 2.4, we get

$$ \vert d_{5} \vert \leq \frac{1}{5 ( 5^{m}\Im (v) ) }. $$

For $n=2,3,4,5$, we take the function $\chi _{n}(z)=z+\cdots$ , such that

$$ \bigl( S^{m}\chi _{n,\Im }(z) \bigr) ^{{\prime }}=1+\sin \bigl(z^{n-1}\bigr), \quad z\in E, $$

then

$$ \bigl( S^{m}\chi _{\Im }(z) \bigr) ^{{\prime }}\prec 1+\sin (z) $$

and $\chi _{n,\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$, where

$$\begin{aligned}& \chi _{2,\Im }(z)=z+\frac{1}{2 ( 2^{m}\Im (v) ) }z^{2}+\cdots, \quad z\in E, \end{aligned}$$

(3.13)

$$\begin{aligned}& \chi _{3,\Im }(z)=z+\frac{1}{3 ( 3^{m}\Im (v) ) }z^{3}+\cdots, \quad z\in E, \end{aligned}$$

(3.14)

$$\begin{aligned}& \chi _{4,\Im }(z)=z+\frac{1}{4 ( 4^{m}\Im (v) ) }z^{4}+\cdots,\quad z\in E, \end{aligned}$$

(3.15)

$$\begin{aligned}& \chi _{5,\Im }(z)=z+\frac{1}{5 ( 5^{m}\Im (v) ) }z^{5}+\cdots, \quad z\in E, \end{aligned}$$

(3.16)

which shows that the bounds are sharp. □

For $m=0$ and $v=0$, we get the known sharp result proved in [44].

Conjecture 3.2

If a function $\chi _{\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$ is of the form (1.8), then

$$ \vert d_{n} \vert \leq \frac{1}{n ( n^{m}\Im (v) ) }\quad \textit{for }n\geq 6, $$

(3.17)

where $\Im (v)$ is given by (1.6).

Theorem 3.3

If a function $\chi _{\Im }$ given in (1.5) belongs to the class $\mathcal{R}_{\sin }(\Im )$, then

$$ \vert d_{2} \vert \leq \frac{1}{2\Im (v)}, \qquad \vert d_{3} \vert \leq \frac{1}{3\Im (v)}, \qquad \vert d_{4} \vert \leq \frac{1}{4\Im (v)}, \qquad \vert d_{5} \vert \leq \frac{1}{5\Im (v)}, $$

where $\Im (v)$ is given by (1.6). The results are sharp for functions (3.13) to (3.16).

Proof

Using the same procedure as we adopted in Theorem 3.2, we obtain the required result of Theorem 3.3. □

Conjecture 3.4

If a function $\chi _{\Im }\in \mathcal{R}_{\sin } ( \Im )$ is of the form (1.9), then

$$ \vert d_{n} \vert \leq \frac{1}{n\Im (v)},\quad n\geq 6, $$

where $\Im (v)$ is given by (1.6).

Theorem 3.5

Let $\chi _{\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$. Then, for a complex number ρ,

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert \leq \frac{1}{3 ( 3^{m}\Im (v) ) } \max \biggl\{ 1, \biggl\vert \frac{3\rho ( 3^{m}\Im (v) ) }{4 ( 2^{m}\Im (v) ) ^{2}} \biggr\vert \biggr\} , $$

(3.18)

where $\Im (v)$ is given by (1.6). The result is sharp.

Proof

Using (3.9) and (3.10), one may write

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert = \frac{1}{6 ( 3^{m}\Im (v) ) } \biggl\vert c_{2}- \biggl( \frac{4 ( 2^{m}\Im (v) ) ^{2}+3\rho ( 3^{m}\Im (v) ) }{8 ( 2^{m}\Im (v) ) ^{2}} \biggr) c_{1}^{2} \biggr\vert . $$

Application of relation (2.6) gives

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert \leq \frac{1}{3 ( 3^{m}\Im (v) ) } \max \biggl\{ 1, \biggl\vert \frac{3\rho ( 3^{m}\Im (v) ) }{4 ( 2^{m}\Im (v) ) ^{2}} \biggr\vert \biggr\} . $$

For the sharpness of (3.18), we consider (3.13) with $n=2$

$$ \chi _{2,\Im }(z)=z+\frac{1}{2 ( 2^{m}\Im (v) ) }z^{2}+\cdots, \quad z\in E, $$

which gives equality in (3.18) when $\vert \rho \vert \geq \frac{4 ( 2^{m}\Im (v) ) ^{2}}{3 ( 3^{m}\Im (v) ) }$, namely

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert = \bigl\vert \rho d_{2}^{2} \bigr\vert = \frac{ \vert \rho \vert }{4 ( 2^{m}\Im (v) ) ^{2}}. $$

For $\vert \rho \vert \leq \frac{4 ( 2^{m}\Im (v) ) ^{2}}{3 ( 3^{m}\Im (v) ) }$, consider

$$ \chi _{3}(z)=z+\frac{1}{3 ( 3^{m}\Im (v) ) }z^{3}+\cdots, \quad z \in E, $$

which gives

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert = \vert d_{3} \vert = \frac{1}{3 ( 3^{m}\Im (v) ) }= \frac{1}{3 ( 3^{m}\Im (v) ) }\max \biggl\{ 1, \biggl\vert \frac{3\rho ( 3^{m}\Im (v) ) }{4 ( 2^{m}\Im (v) ) ^{2}} \biggr\vert \biggr\} . $$

 □

Corollary 3.6

If $\chi _{\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$, then for a complex number $\vert \rho \vert \leq \frac{4 ( 2^{m}\Im (v) ) ^{2}}{3 ( 3^{m}\Im (v) ) }$, we have

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert \leq \frac{1}{3 ( 3^{m}\Im (v) ) }, $$

(3.19)

where $\Im (v)$ is given by (1.6). This inequality is sharp.

Corollary 3.7

[44]. Let $\chi \in \mathcal{R}_{\sin }$. Then, for a complex number ρ,

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert \leq \frac{1}{3}\max \biggl\{ 1, \biggl\vert \frac{3\rho }{4} \biggr\vert \biggr\} . $$

Theorem 3.8

Let $\chi _{\Im }\in \mathcal{R}_{\sin } ( \Im )$. Then, for a complex number ρ,

$$ \bigl\vert d_{3}-\rho d_{2}^{2} \bigr\vert \leq \frac{1}{3\Im (v)}\max \biggl\{ 1, \frac{\rho ( 3\Im (v) ) }{ ( 2\Im (v) ) ^{2}} \biggr\} , $$

where $\Im (v)$ is given by (1.6). The result is sharp.

Proof

Using the same procedure as we adopted in Theorem 3.5, we get the required result. □

Theorem 3.9

Let $\chi _{\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$. Then

$$ \bigl\vert d_{3}-d_{2}^{2} \bigr\vert \leq \frac{1}{3 ( 3^{m}\Im (v) ) }, $$

(3.20)

where $\Im (v)$ is given by (1.6). This inequality is sharp for the function

$$ \chi _{3}(z)=z+\frac{1}{3 ( 3^{m}\Im (v) ) }z^{3}+\cdots,\quad z \in E. $$

Proof

From (3.9) and (3.10), we have

$$\begin{aligned} \bigl\vert d_{3}-d_{2}^{2} \bigr\vert =& \frac{1}{6 ( 3^{m}\Im (v) ) } \biggl\vert c_{2}-\frac{1}{2} \biggl( 1+ \frac{3 ( 3^{m}\Im (v) ) }{ ( 2^{m}\Im (v) ) ^{2}} \biggr) c_{1}^{2} \biggr\vert , \\ =&\frac{1}{6 ( 3^{m}\Im (v) ) } \bigl\vert c_{2}-\phi c_{1}^{2} \bigr\vert , \end{aligned}$$

where $\phi =\frac{1}{2} ( 1+ \frac{3 ( 3^{m}\Im (v) ) }{ ( 2^{m}\Im (v) ) ^{2}} )$ since $v\geq 0$ and $0<\phi <1$. Now, by using (2.2) for $n=2$ and $k=1$, we obtain (3.20). □

Theorem 3.10

Let $\chi _{\Im }\in \mathcal{R}_{\sin } ( \Im )$. Then

$$ \bigl\vert d_{3}-d_{2}^{2} \bigr\vert \leq \frac{1}{3\Im (v)}, $$

where $\Im (v)$ is given by (1.6). This inequality is sharp for

$$ \chi _{3}(z)=z+\frac{1}{3\Im (v)}z^{3}+\cdots, \quad z\in E. $$

Proof

Using the same procedure as we adopted in Theorem 3.9, we obtain the required result of Theorem 3.10. □

Theorem 3.11

Let $\chi _{\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$. Then

$$ \vert d_{2}d_{3}-d_{4} \vert \leq \frac{1}{4 ( 4^{m}\Im (v) ) }. $$

(3.21)

This inequality is sharp for

$$ \chi _{4,\Im }(z)=z+\frac{1}{4 ( 4^{m}\Im (v) ) }z^{4}+\cdots, \quad z\in E, $$

where $\Im (v)$ is given by (1.6).

Proof

From (3.9), (3.10), and (3.11), we have

$$ \vert d_{2}d_{3}-d_{4} \vert = \left \vert \begin{gathered} \biggl( \frac{4 ( 4^{m}\Im (v) ) +5 ( 2^{m}\Im (v) ) ( 3^{m}\Im (v) ) }{192 ( 2^{m}\Im (v) ) ( 3^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \biggr) c_{1}^{3} \\ - \biggl( \frac{ ( 4^{m}\Im (v) ) +3 ( 2^{m}\Im (v) ) ( 3^{m}\Im (v) ) }{24 ( 2^{m}\Im (v) ) ( 3^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \biggr) c_{1}c_{2}+ \biggl( \frac{1}{8 ( 4^{m}4^{m}\Im (v) ) } \biggr) c_{3}\end{gathered} \right \vert . $$

By using the applications of Lemma 2.3, and after some simple calculations, we get

$$ \vert d_{2}d_{3}-d_{4} \vert \leq \frac{1}{4 ( 4^{m}\Im (v) ) }. $$

This completes the proof. □

Corollary 3.12

[44]. Let $\chi \in \mathcal{R}_{\sin }$ ($m=0$ and $v=0$). Then

$$ \vert d_{2}d_{3}-d_{4} \vert \leq \frac{1}{4}. $$

This inequality is sharp for

$$ \chi _{4}(z)=z+\frac{1}{4}z^{4}+\cdots. $$

Theorem 3.13

Let $\chi _{\Im }\in \mathcal{R}_{\sin } ( \Im )$. Then

$$ \vert d_{2}d_{3}-d_{4} \vert \leq \frac{1}{4\Im (v)}, $$

where $\Im (v)$ is given by (1.6). This inequality is sharp for

$$ \chi _{4,\Im }(z)=z+\frac{1}{4\Im (v)}z^{4}+\cdots,\quad z\in E. $$

Proof

Using the same procedure of Theorem 3.11, we obtain the result of Theorem 3.13. □

Remark 3.14

For $v=0$, in Theorem 3.13, we get known result proved in [44].

Theorem 3.15

Let $\chi _{\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$. Then

$$ \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert \leq \frac{1}{9 ( 3^{m}\Im (v) ) ^{2}}. $$

(3.22)

This inequality is sharp for

$$ \chi _{3,\Im }(z)=z+\frac{1}{3 ( 3^{m}\Im (v) ) }z^{3}+\cdots, \quad z\in E, $$

where $\Im (v)$ is given by (1.6).

Proof

From (3.9), (3.10), and (3.11), we have

$$ \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert = \bigl\vert C ( m,v ) c_{1}c_{3}-B(m,v)c_{1}^{2}c_{2}-D(m,v)c_{2}^{2}-A(m,v)c_{1}^{4} \bigr\vert , $$

where

$$\begin{aligned} A(m,v) =& \frac{5}{ ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) },\qquad D(m,v)=\frac{1}{32 ( 3^{m}\Im (v) ) ^{2}} \\ C(m,v) =& \frac{1}{32 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }, \\ B(m,v) =& \frac{9 ( 3^{m}\Im (v) ) ^{2}-8 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }{288 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) ( 3^{m}\Im (v) ) ^{2}}. \end{aligned}$$

Using Lemma 2.3, we obtain

$$ \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert = \left \vert \begin{gathered} T_{1}(m,v)c_{1}^{4}+T_{2}(m,v)c_{1}^{2} \bigl( 4-c_{1}^{2} \bigr) x- \bigl( 3^{m}\Im (v) \bigr) T_{3}(m,v)c_{1}^{2}x \\ - \frac{ ( 4-c_{1}^{2} ) c_{1}^{2}x^{2}}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }- \frac{ ( 4-c_{1}^{2} ) ^{2}x^{2}}{144 ( 3^{m}\Im (v) ) ^{2}}+ \frac{c ( 4-c_{1}^{2} ) ( 1- \vert x \vert ^{2} ) }{144 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }z\end{gathered} \right \vert , $$

where

$$\begin{aligned}& T_{1}(m,v) = \frac{1}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }-T_{3}(q) \\& \hphantom{T_{1}(m,v) =} {}+ \biggl( \frac{15 ( 3^{m}\Im (v) ) ^{2}-16 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }{2304 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) ( 3^{m}\Im (v) ) ^{2}} \biggr) -\frac{1}{144 ( 3^{m}\Im (v) ) ^{2}}, \\& T_{2}(m,v) = \bigl( \bigl( 3^{m}\Im (v) \bigr) ^{2}- \bigl( 2^{m} \Im (v) \bigr) \bigl( 4^{m}\Im (v) \bigr) \bigr) , \\& T_{3}(m,v) = \frac{ ( 9 ( 3^{m}\Im (v) ) ^{2}-8 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) ) }{576 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) ( 3^{m}\Im (v) ) ^{2}}. \end{aligned}$$

Let $\vert z \vert =1$, $\vert x \vert =t$, $t\in {}[ 0;1]$, $\vert c_{1} \vert =c\in {}[ 0;2]$. Then, using the triangle inequality, we get

$$\begin{aligned}& \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert \\& \quad \leq \bigl\vert T_{1}(m,v) \bigr\vert c^{4}+ \bigl\vert T_{2}(m,v) \bigr\vert c^{2} \bigl( 4-c^{2} \bigr) t+ \bigl( 3^{m}\Im (v) \bigr) T_{3}(m,v)c_{1}^{2}t \\& \qquad {}+ \frac{ ( 4-c_{1}^{2} ) c_{1}^{2}t^{2}}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }+ \frac{ ( 4-c_{1}^{2} ) ^{2}t^{2}}{144 ( 3^{m}\Im (v) ) ^{2}}+ \frac{c ( 4-c^{2} ) ( 1- \vert t \vert ^{2} ) }{144 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \\& \quad = H(c,t). \end{aligned}$$

Taking the derivative of $H(c,t)$ w.r.t., t we get

$$\begin{aligned} \frac{\partial H(c,t)}{\partial t} =& \bigl\vert T_{2}(m,v) \bigr\vert c^{2} \bigl( 4-c^{2} \bigr) + \bigl( 3^{m}\Im (v) \bigr) \bigl\vert T_{3}(m,v) \bigr\vert c_{1}^{2} \\ &{}+ \frac{ ( 4-c_{1}^{2} ) ^{2}t}{72 ( 3^{m}\Im (v) ) ^{2}} \\ >&0, \end{aligned}$$

which shows that $H(c,t)$ increases on $[0,1]$ with respect to t. That is, $H(c,t)$ has a maximum value at $t=1$, which is

$$\begin{aligned} \max H(c,t) =&H(c,1) \\ =& \bigl\vert T_{1}(m,v) \bigr\vert c^{4}+ \bigl\vert T_{2}(m,v) \bigr\vert c^{2} \bigl( 4-c^{2} \bigr) + \bigl( 3^{m} \Im (v) \bigr) T_{3}(m,v)c^{2} \\ &{}+ \frac{ ( 4-c^{2} ) c^{2}}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }+ \frac{ ( 4-c^{2} ) ^{2}}{144 ( 3^{m}\Im (v) ) ^{2}} \\ =&G(c). \end{aligned}$$

Differentiation gives

$$\begin{aligned} G^{{\prime }}(c) =&4 \bigl\vert T_{1}(m,v) \bigr\vert c^{3}+ \bigl\vert T_{2}(m,v) \bigr\vert 8c+2 \bigl( 3^{m}\Im (v) \bigr) \bigl\vert T_{3}(m,v) \bigr\vert c \\ &{}+ \frac{8c-2c^{3}-2c^{3}}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }- \frac{4c ( 4-c^{2} ) }{144 ( 3^{m}\Im (v) ) ^{2}} \\ =& \biggl( \bigl\vert T_{2}(m,v) \bigr\vert 8+2 \bigl( 3^{m}\Im (v) \bigr) \bigl\vert T_{3}(m,v) \bigr\vert \\ &{}- \frac{16}{144 ( 3^{m}\Im (v) ) ^{2}}+ \frac{8}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \biggr) c \\ &{}+ \biggl( \bigl\vert T_{1}(m,v) \bigr\vert - \frac{1}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }+ \frac{1}{144 ( 3^{m}\Im (v) ) ^{2}} \biggr) c^{3}. \end{aligned}$$

If $G^{{\prime }}(c)=0$, then the root is $c=0$ and

$$ c^{2}= \frac{- ( \vert T_{2}(m,v) \vert 8+2 ( 3^{m}\Im (v) ) \vert T_{3}(m,v) \vert -\frac{16}{144 ( 3^{m}\Im (v) ) ^{2}}+\frac{8}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) } ) }{4 ( \vert T_{1}(m,v) \vert -\frac{1}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }+\frac{1}{144 ( 3^{m}\Im (v) ) ^{2}} ) }. $$

Again, taking the derivative of $G^{{\prime }}(c)$, we have

$$\begin{aligned} G^{{\prime \prime }}(0) =& \bigl\vert T_{2}(m,v) \bigr\vert 8+2 \bigl( 3^{m}\Im (v) \bigr) \bigl\vert T_{3}(m,v) \bigr\vert \\ &{}-\frac{16}{144 ( 3^{m}\Im (v) ) ^{2}}+ \frac{8}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \\ &{}+3 \biggl( \bigl\vert T_{1}(m,v) \bigr\vert - \frac{1}{128 ( 2^{m}\Im (v) ) ( 4^{m}\Im (v) ) }+ \frac{1}{144 ( 3^{m}\Im (v) ) ^{2}} \biggr) c^{2}. \end{aligned}$$

We see that

$$ G^{{\prime \prime }}(0)< 0, $$

so the function $G(c)$ can attain the maximum value at $c=0$; which is

$$ \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert \leq \frac{1}{9 ( 3^{m}\Im (v) ) ^{2}}. $$

This completes the result. □

Theorem 3.16

Let $\chi _{\Im }\in \mathcal{R}_{\sin } ( \Im )$. Then

$$ \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert \leq \frac{1}{9 ( \Im (v) ) ^{2}}, $$

where $\Im (v)$ is given by (1.6). This inequality is sharp for

$$ \chi _{3,\Im }(z)=z+\frac{1}{3\Im (v)}z^{3}+\cdots,\quad z\in E. $$

Proof

Using the same procedure as we adopted in Theorem 3.15, we obtain the result of Theorem 3.16. □

Theorem 3.17

Let $\chi _{\Im }\in \mathcal{R}_{\sin }^{m} ( \Im )$. Then

$$\begin{aligned} \bigl\vert H_{3,1}(\chi ) \bigr\vert \leq & \frac{1}{27 ( 3^{m}\Im (v) ) ( 3^{m}\Im (v) ) ^{2}}+ \frac{1}{16 ( 4^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \\ &{}+ \frac{1}{15 ( 5^{m}\Im (v) ) ( 3^{m}\Im (v) ) }, \end{aligned}$$

where $\Im (v)$ is given by (1.6).

Proof

From (1.4), it is easy to see that

$$ H_{3,1}(\chi )=d_{3} \bigl( d_{2}d_{4}-d_{3}^{2} \bigr) +d_{4} ( d_{2}d_{3}-d_{4} ) +d_{5} \bigl( d_{3}-d_{2}^{2} \bigr) , $$

where $d_{1}=1$. This implies that

$$ \bigl\vert H_{3,1}(\chi ) \bigr\vert \leq \vert d_{3} \vert \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert + \vert d_{4} \vert \vert d_{2}d_{3}-d_{4} \vert + \vert d_{5} \vert \bigl\vert d_{3}-d_{2}^{2} \bigr\vert . $$

By using (3.1), (3.2), (3.3), (3.4), (3.19), (3.21), and (3.22), we have

$$\begin{aligned} \bigl\vert H_{3,1}(\chi ) \bigr\vert \leq & \frac{1}{27 ( 3^{m}\Im (v) ) ( 3^{m}\Im (v) ) ^{2}}+ \frac{1}{16 ( 4^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \\ &{}+ \frac{1}{15 ( 5^{m}\Im (v) ) ( 3^{m}\Im (v) ) }, \end{aligned}$$

which is our required result. □

Theorem 3.18

Let $\chi _{\Im }\in \mathcal{R}_{\sin } ( \Im )$. Then

$$\begin{aligned} \bigl\vert H_{3,1}(\chi ) \bigr\vert \leq &\frac{1}{27 ( \Im (v) ) ^{3}}+ \frac{1}{16 ( \Im (v) ) ^{2}} \\ &{}+\frac{1}{15 ( \Im (v) ) ^{2}}, \end{aligned}$$

where $\Im (v)$ is given by (1.6).

Proof

Using the same procedure as we adopted in Theorem 3.17, we obtain the result of Theorem 3.18. □

Remark 3.19

For $m=1$, $v=0$, in Theorem 3.18, we get known result proved in [44].

4 Bound of $|H_{4,1}(\chi )|$ for the functions class $R_{\sin }^{m} ( \Im )$ and $R_{\sin } ( \Im )$

First of all we can deduce the form of $H_{4,1}(\chi )$ from (1.3) in the following way:

$$\begin{aligned} H_{4,1}(\chi ) =&d_{7} \bigl( H_{3,1}(\chi ) \bigr) -2d_{5}d_{6} ( d_{2}d_{3}-d_{4} ) -2d_{4}d_{6} \bigl( d_{2}d_{4}-d_{3}^{2} \bigr) \\ &{}-d_{6}^{2} \bigl( d_{3}-d_{2}^{2} \bigr) +d_{5}^{2} \bigl( d_{2}d_{4}+2d_{3}^{2} \bigr) +d_{5}^{2} \bigl( d_{2}d_{4}-d_{3}^{2} \bigr) \\ &{}-d_{5}^{3}+d_{4}^{4}-3d_{3}d_{4}^{2}d_{5}. \end{aligned}$$

(4.1)

We need the following simple result for the function class $\mathcal{R}_{\sin }^{m} ( \Im )$, that is, if $\chi \in \mathcal{R}_{\sin }^{m} ( \Im )$ of the form (1.8), then

$$\begin{aligned} \bigl\vert d_{2}d_{4}+2d_{3}^{2} \bigr\vert \leq & \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert +3 \vert d_{3} \vert ^{2} \\ \leq &\frac{4}{9 ( 3^{m}\Im (v) ) ^{2}}. \end{aligned}$$

(4.2)

Now we move towards the forth-order Hankel determinant.

Theorem 4.1

Let $\chi _{\varphi }\in \mathcal{R}_{\sin }^{m} ( \Im )$. Then

$$\begin{aligned} \bigl\vert H_{4,1}(\chi ) \bigr\vert \leq & \frac{1}{7 ( 7^{m}\Im (v) ) } \biggl( \frac{1}{27 ( 3^{m}\Im (v) ) ( 3^{m}\Im (v) ) ^{2}} \\ &{} + \frac{1}{16 ( 4^{m}\Im (v) ) ( 4^{m}\Im (v) ) }+\frac{1}{15 ( 5^{m}\Im (v) ) ( 3^{m}\Im (v) ) } \biggr) \\ &{}+ \frac{1}{108 ( 4^{m}\Im (v) ) ( 6^{m}\Im (v) ) ( 3^{m}\Im (v) ) ^{2}} \\ &{}+ \frac{1}{120 ( 5^{m}\Im (v) ) ( 6^{m}\Im (v) ) ( 4^{m}\Im (v) ) } \\ &{}+ \frac{1}{108 ( 3^{m}\Im (v) ) ( 6^{m}\Im (v) ) ^{2}}+\frac{1}{225 ( 3^{m}\Im (v) ) ^{2} ( 5^{m}\Im (v) ) ^{2}} \\ &{}+ \frac{1}{54 ( 3^{m}\Im (v) ) ^{2} ( 5^{m}\Im (v) ) ^{2}} \\ &{}-\frac{1}{125 ( 5^{m}\Im (v) ) ^{3}}+ \frac{1}{256 ( 4^{m}\Im (v) ) ^{4}} \\ &{}- \frac{1}{80 ( 3^{m}\Im (v) ) ( 4^{m}\Im (v) ) ^{2} ( 5^{m}\Im (v) ) }. \end{aligned}$$

where $\Im (v)$ is given by (1.6).

Proof

Taking modulus on both sides of (4.1) and then applying the triangle inequality, we obtain

$$\begin{aligned} \bigl\vert H_{4,1}(\chi ) \bigr\vert \leq & \vert d_{7} \vert \bigl\vert H_{3,1}(\chi ) \bigr\vert +2 \vert d_{4} \vert \vert d_{6} \vert \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert +2 \vert d_{5} \vert \vert d_{6} \vert \vert d_{2}d_{3}-d_{4} \vert \\ &{}+ \vert d_{6} \vert ^{2} \bigl\vert d_{3}-d_{2}^{2} \bigr\vert + \vert d_{5} \vert ^{2} \bigl\vert d_{2}d_{4}-d_{3}^{2} \bigr\vert + \vert d_{5} \vert ^{2} \bigl\vert d_{2}d_{4}+2d_{3}^{2} \bigr\vert \\ &{}+ \vert d_{5} \vert ^{3}+ \vert d_{4} \vert ^{4}+3 \vert d_{3} \vert \vert d_{4} \vert ^{2} \vert d_{5} \vert . \end{aligned}$$

Now, by using (3.1), (3.2),(3.3),(3.4),(3.17), (3.20), (3.21), (3.22), and (4.2), we get the required result. □

Theorem 4.2

Let $\chi _{\varphi }\in \mathcal{R}_{\sin } ( \Im )$. Then

$$\begin{aligned} \bigl\vert H_{4,1}(\chi ) \bigr\vert \leq &\frac{1}{7\Im (v)} \biggl( \frac{1}{27 ( \Im (v) ) ^{3}}+ \frac{1}{16 ( \Im (v) ) ^{2}}+\frac{1}{15 ( \Im (v) ) ^{2}} \biggr) \\ &{}+\frac{1}{108 ( \Im (v) ) ^{4}}+ \frac{1}{120 ( \Im (v) ) ^{3}} \\ &{}+\frac{1}{108 ( \Im (v) ) ^{3}}+ \frac{1}{225 ( \Im (v) ) ^{4}}+\frac{1}{54 ( \Im (v) ) ^{4}} \\ &{}-\frac{1}{125 ( \Im (v) ) ^{3}}+ \frac{1}{256 ( \Im (v) ) ^{4}}-\frac{1}{80 ( \Im (v) ) ^{4}}, \end{aligned}$$

where $\Im (v)$ is given by (1.6).

Proof

By using a similar method as we adopted in the above theorem, we get the required result. □