Upper bound for the second and third Hankel determinants of analytic functions associated with the error function and q-convolution combination

Abstract

Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class $\mathcal{R}_{\alpha ,\Upsilon}^{\lambda ,q}(\delta ,\eta )$ of analytic functions was studied:

$$\begin{aligned} &\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta ) \\ &\quad := \biggl\{ \mathcal{ F}: {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta}+(\delta -2\eta ) \bigl(\mathcal{H} _{\Upsilon}^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{ \prime}}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime}}} \biggr) \biggr\} \\ &\quad >\alpha \quad (0\leqq \alpha < 1). \end{aligned}$$

For these general analytic functions $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon}^{\lambda ,q}(\delta , \eta )$, we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.

1 Introduction and preliminaries

The error function $\operatorname{erf}(\zeta )$ defined by

$$ \operatorname{erf}(\zeta )=\frac{2}{\sqrt{\pi }} \int _{0}^{\zeta }\exp \bigl( -t^{2} \bigr) \,dt =\frac{2}{\sqrt{\pi }}\sum_{j=0}^{ \infty } \frac{ (-1)^{j}}{j! (2j+1)} \zeta ^{2j+1} $$

(1.1)

has recently been an area of considerable study and applications. Several properties and inequalities of the error function are given in [3, 7, 13, 14]. The complementary error function $\mathrm{erfc}(\zeta )$ is also used extensively in many fields of applied mathematics and mathematical physics, including data analysis [16] and probability and statistics [6]. The authors of [11, 12] investigated many their properties. The inverse of the error function $\operatorname{erf}(\zeta )$, which was first considered by Carlitz [5], is widely used in mathematics and natural sciences. Examples include concentration-dependent diffusion problems, solutions to Einstein’s scalar-field problems, and the heat conduction problem [6, 20].

Let $\mathcal{A}$ denote the class of analytic functions of the form

$$ f(\zeta ):=\zeta +\sum_{j=2}^{\infty }c_{j} \zeta ^{j}\quad \bigl(\zeta \in \Lambda :=\bigl\{ \zeta : \zeta \in \mathbb{C} \text{ and } \vert \zeta \vert < 1\bigr\} \bigr). $$

(1.2)

Also, let $\mathcal{S}\subset \mathcal{A}$ consist of functions in Λ that are univalent in Λ.

Let the function $\Upsilon \in \mathcal{A}$ be given by

$$ \Upsilon (\zeta ):=\zeta +\sum_{j=2}^{\infty }b_{j} \zeta ^{j} \quad (\zeta \in \Lambda ). $$

(1.3)

The Hadamard product (or convolution) of the functions f and ϒ is defined by

$$ (f\ast \Upsilon ) (\zeta ):=\zeta +\sum_{j=2}^{\infty }c_{j}b_{j} \zeta ^{j}\quad (\zeta \in \Lambda ). $$

We assume that $E_{r}$ is a normalized analytic function in Λ given by

$$ E_{r}(\zeta )=\frac{\sqrt{\pi \zeta }}{2}\operatorname{erf}(\sqrt{\zeta })= \zeta +\sum_{j=2}^{\infty } \frac{(-1)^{j-1}}{(2j-1) ( j-1 ) !} \zeta ^{j}\quad ( \zeta \in \Lambda ). $$

El-Deeb and Cotîrlă [10] defined a family of analytic functions in Λ as follows:

$$ \mathcal{F}(\zeta )=(f\ast E_{r}) (\zeta )=\zeta +\sum _{j=2}^{ \infty} \frac{(-1)^{j-1}}{(2j-1) ( j-1 ) !}c_{j} \zeta ^{j} \quad (\zeta \in \Lambda ). $$

Srivastava [26] systematically presented the definitions and notations and used several kinds of the q-calculus and the fractional q-calculus operators. The definition of the q-shifted factorial for $\vartheta ,q\in \mathbb{C}$ and $n\in \mathbb{N}_{0}=\mathbb{N} \cup \{0\}$ is given as follows:

$$ (\vartheta ;q)_{j}=\textstyle\begin{cases} 1 & (j=0), \\ (1-\vartheta ) (1-\vartheta q ) \cdots ( 1-\vartheta q^{j-1} ) & (j\in \mathbb{N}).\end{cases} $$

We denote the basic (or q-)number $[\nu ] _{q}$ defined as follows:

$$ {}[ \nu ]_{q}:=\textstyle\begin{cases} \frac {1-q^{\nu}}{1-q} & (\nu \in \mathbb{C}), \\ 1+\sum_{k=1}^{j-1}q^{k} & (\nu =k\ (k \in \mathbb{N}) ).\end{cases} $$

(1.4)

Using definition (1.4), we have

• (i) The q-shifted factorial is given for any nonnegative integer j by

  $$ {}[ j]_{q}!:=\textstyle\begin{cases} {{0}} & (j=0), \\ \prod_{n=1}^{j}[n]_{q} & (j\in \mathbb{N}). \end{cases} $$

• (ii) The definition of the generalized q-Pochhammer symbol for all positive integers r is given by

  $$ [r ]_{q,j}:=\textstyle\begin{cases} {{0}} & (j=0), \\ \prod_{n=r}^{r+j-1}[n]_{q} & (j\in \mathbb{N}). \end{cases} $$

The q-derivative operator for a function $\mathcal{F}\ast \Upsilon$ is defined for $0< q<1$ by (see [9, 10, 17])

$$\begin{aligned} \mathcal{D}_{q} ( \mathcal{F}\ast \Upsilon ) (\zeta )&:= \mathcal{D} _{q} \Biggl( \zeta +\sum_{j=2}^{\infty } \frac {(-1)^{j-1}}{(2j-1) ( j-1 ) !}c_{j}b_{j}\zeta ^{j} \Biggr) \\ & = \frac { ( \mathcal{F}\ast \Upsilon ) (\zeta )- ( \mathcal{F} \ast \Upsilon ) (q\zeta )}{\zeta (1-q)} \\ & =1+\sum_{j=2}^{\infty } \frac{(-1)^{j-1}}{(2j-1) ( j-1 ) !}[j]_{q}c_{j}b_{j}\zeta ^{j-1}\quad (\zeta \in \Lambda ), \end{aligned}$$

where

$$ {}[ j]_{q}:=\frac{1-q^{j}}{1-q}=1+\sum _{n=1}^{j-1}q^{n}, \quad \text{and} \quad [ 0 ] _{q}:=0. $$

(1.5)

For $\lambda >-1$ and $0< q<1$, El-Deeb et al. [9] defined the linear operator $\mathcal{H}_{\Upsilon }^{\lambda ,q}:\mathcal{A}\rightarrow \mathcal{A}$ by

$$ \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )\ast \mathcal{M}_{q,\lambda +1}(\zeta )=\zeta \mathcal{D}_{q} ( \mathcal{F} \ast \Upsilon ) (\zeta )\quad (\zeta \in \Lambda ), $$

where

$$ \mathcal{M}_{q,\lambda +1}(\zeta ):=\zeta +\sum_{j=2}^{ \infty } \frac{ [\lambda +1]_{q,j-1}}{[j-1]_{q}!}\zeta ^{j}\quad (\zeta \in \Lambda ). $$

A simple computation shows that

$$\begin{aligned} \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) &:=\zeta + \sum _{j=2}^{\infty } \frac{(-1)^{j-1}[j]_{q}!}{(2j-1) ( j-1 ) ![\lambda +1]_{q,j-1}}c_{j}b_{j} \zeta ^{j} \\ &=\zeta +\sum_{j=2}^{\infty }\rho _{j}c_{j} \zeta ^{j} \quad (\lambda >-1; 0< q< 1; \zeta \in \Lambda ), \end{aligned}$$

(1.6)

where

$$ \rho _{j}= \frac{(-1)^{j-1}[j]_{q}!}{(2j-1) ( j-1 ) ![\lambda +1]_{q,j-1}}b_{j}. $$

(1.7)

From the definition in (1.6) we find the following relations for all $\mathcal{F}\in \mathcal{A}$:

$$ [\lambda +1]_{q}\mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F} ( \zeta )=[\lambda ]_{q}\mathcal{H}_{\Upsilon}^{\lambda +1,q} \mathcal{F} (\zeta )+q^{\lambda } \zeta \mathcal{D}_{q} \bigl( \mathcal{H}_{\Upsilon }^{\lambda +1,q}\mathcal{F}(\zeta ) \bigr) \quad ( \zeta \in \Lambda ) $$

(1.8)

and

$$ \mathcal{I}_{\Upsilon }^{\lambda }\mathcal{F}(\zeta ):=\lim _{q \rightarrow 1{-}}\mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F}( \zeta )=\zeta +\sum_{j=2}^{\infty } \frac {j}{(\lambda +1)_{j-1}} \frac{(-1)^{j-1}}{(2j-1)}c_{j}b_{j} \zeta ^{j}\quad (\zeta \in \Lambda ). $$

(1.9)

Remark 1

Taking different particular values of the coefficients $b_{j}$, we obtain the following particular cases for the operator $\mathcal{H}_{\Upsilon }^{\lambda ,q}$: (i) For $b_{j}=1$, we obtain the operator $\mathcal{E}_{q}^{\lambda}$ defined as (see [28])

$$ \mathcal{E}_{q}^{\lambda }\mathcal{F}(\zeta ):=\zeta +\sum _{j=2}^{ \infty } \frac{(-1)^{j-1}[j]_{q}!}{(2j-1) ( j-1 ) ![\lambda +1]_{q,j-1}}c_{j} \zeta ^{j}\quad ( \lambda >-1; 0< q< 1; \zeta \in \Lambda ); $$

(1.10)

(ii) For $b_{j}= \frac {(-1)^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )}$, $\upsilon >0$, we obtain the operator $\mathcal{E} _{\upsilon ,q}^{\lambda }$ defined as (see [8])

$$\begin{aligned} \mathcal{E}_{\upsilon ,q}^{\lambda }\mathcal{F}(\zeta )&:=\zeta +\sum _{j=2}^{\infty } \frac{(-1)^{2 ( j-1 ) }\Gamma (\upsilon +1)[j]_{q}!}{4^{j-1}(2j-1) ( (j-1)! ) ^{2}\Gamma (j+\upsilon )[\lambda +1]_{q,j-1}}c_{j} \zeta ^{j} \\ &=\zeta +\sum_{j=2}^{\infty }\varkappa _{j}c_{j} \zeta ^{j} \quad ( \upsilon >0; \lambda >-1; 0< q< 1; \zeta \in \Lambda ), \end{aligned}$$

(1.11)

where

$$ \varkappa _{j}:= \frac{(-1)^{2 ( j-1 ) }\Gamma (\upsilon +1)[j]_{q}!}{4^{j-1}(2j-1) ( (j-1)! ) ^{2}\Gamma (j+\upsilon )[\lambda +1]_{q,j-1}}; $$

(1.12)

(iii) For $b_{j}= (\frac {n+1}{n+j} ) ^{r}$, $r>0$, and $n\geqq 0$, we obtain the operator $\mathcal{E}_{n,q}^{\lambda ,r}$ defined as

$$ \mathcal{E}_{n,q}^{\lambda ,r}\mathcal{F}(\zeta ):=\zeta +\sum _{j=2}^{\infty } \frac{(-1)^{j-1} ( n+1 ) ^{r}[j]_{q}!}{ (2j-1) ( n+j ) ^{r} ( j-1 ) ![\lambda +1]_{q,j-1}}c_{j} \zeta ^{j}\quad (\zeta \in \Lambda ); $$

(1.13)

(iv) For $b_{j}=\frac {\rho ^{j-1}}{(j-1)!}e^{-\rho }$ and $\rho >0$, we obtain the operator defined by

$$ \mathcal{E}_{q}^{\lambda ,\rho }\mathcal{F}(\zeta ):=\zeta +\sum _{j=2}^{\infty} \frac{(-1)^{j-1}\rho ^{j-1}[j]_{q}!e^{-\rho }}{(2j-1) ( ( j-1 ) ! ) ^{2}[\lambda +1]_{q,j-1}}c_{j} \zeta ^{j}\quad (\zeta \in \Lambda ). $$

(1.14)

In the same way as that of Ali et al. [2] who defined a class $\mathcal{W}_{\beta }(\delta ,\eta )$, by using the operator $\mathcal{H}_{\Upsilon }^{\lambda ,q}$ we now introduce the new class $\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$ of normalized analytic functions defined in Λ such that the function $\mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$ satisfies the conditions specified below:

$$ {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F}(\zeta )}{\zeta }+(\delta -2\eta ) \bigl( \mathcal{H}_{ \Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{\prime }}+ \eta \zeta \bigl( \mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime }}} \biggr) >\alpha $$

(1.15)

for all $\zeta \in \Lambda$. Here δ, $\eta \geqq 0$ and $0\leqq \alpha <1$. The class $\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$ combines some well-known subclasses of $\mathcal{S}$ for various choices of δ, η, and α:

• I. The class $\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$ reduces to the well-known class $\mathcal{I}_{\Upsilon }^{\lambda ,q}$ for $\delta =1$, $\eta =0$, and $\alpha =0$,

  $$ \mathcal{I}_{\Upsilon }^{\lambda ,q}= \bigl\{ \mathcal{F}\in \mathcal{A}:{\Re} \bigl( \mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F}(\zeta ) \bigr) ^{{\prime }}>0 \bigr\} . $$

  • (1) For $\delta =1+2\eta$ and $\alpha =0$, the class $\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$ becomes the class

    $$ \mathcal{R}_{\Upsilon ,\eta }^{\lambda ,q}= \bigl\{ \mathcal{F}\in \mathcal{A} :{\Re} \bigl( \bigl( \mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F} ( \zeta ) \bigr) ^{{\prime }}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{{\prime \prime }}} \bigr) >0 \bigr\} . $$

    For the class of univalent starlike functions in Λ, see [23].

  • (2) When $\delta =\eta =0$, the class $\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$ becomes the class $\mathcal{T}_{\Upsilon ,\alpha }^{\lambda ,q}$, where

    $$ \mathcal{T}_{\Upsilon ,\alpha }^{\lambda ,q}= \biggl\{ \mathcal{F}\in \mathcal{A}:{\Re} \biggl( \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta } \biggr) >\alpha \biggr\} . $$

  • (3) When $\eta =0$, the class $\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$ becomes the class $\mathrm{P}_{\Upsilon ,\alpha }^{\lambda ,q}(\delta )$, where

    $$ \mathrm{P}_{\Upsilon ,\alpha }^{\lambda ,q}(\delta )= \biggl\{ \mathcal{F}\in \mathcal{A}:{\Re} \biggl( (1-\delta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta }+\delta \bigl( \mathcal{H}_{\Upsilon }^{ \lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{\prime }} \biggr) >\alpha \biggr\} . $$

• II. We get the class $\mathcal{T}_{\alpha }^{\lambda ,q}(\delta ,\eta )$ if $b_{j}=1$. This class represents the functions $\mathcal{F}\in \mathcal{A}$ that satisfy (1.15) for $\mathcal{H}_{\Upsilon }^{\lambda ,q}$ replaced with $\mathcal{E}_{q}^{\lambda }$ as in (1.10).

• III. We obtain the class $\mathcal{P}_{\alpha ,\upsilon }^{\lambda ,q}(\delta ,\eta )$ if

  $$ b_{j}= \frac {(-1)^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )} \quad \text{and} \quad \upsilon >0. $$

  It consists of the functions $\mathcal{F}\in \mathcal{A}$ that satisfy (1.15) with $\mathcal{H}_{\Upsilon }^{\lambda ,q}$ replaced by $\mathcal{E}_{\upsilon ,q}^{\lambda }$ as in (1.11).

• IV. We get the class $\mathcal{R}_{\alpha ,n,r}^{\lambda ,q}(\delta ,\eta )$ if

  $$ b_{j}= \biggl( \frac {n+1}{n+j} \biggr) ^{r},\quad r>0, n\geqq 0. $$

  It consists of the functions $\mathcal{F}\in \mathcal{A}$ that satisfy (1.15) with $\mathcal{H}_{\Upsilon }^{\lambda ,q}$ replaced by $\mathcal{E}_{n,q}^{\lambda ,r}$ as in (1.13).

• V. We obtain the class $\mathcal{D}_{\alpha ,\rho }^{\lambda ,q}(\delta ,\eta )$ if

  $$ b_{j}=\frac {\rho ^{j-1}}{(j-1)!}e^{-\rho }\quad \text{and} \quad \rho >0. $$

  It consists of the functions $\mathcal{F}\in \mathcal{A}$ that satisfy (1.15) with $\mathcal{H}_{\Upsilon }^{\lambda ,q}$ replaced by $\mathcal{E}_{q}^{\lambda ,\rho }$ as in (1.14).

It is known that for the $\mathfrak{q}$th Hankel determinant $H_{\mathfrak{q}}(n)$ of f with $\mathfrak{q}\geqq 1$ and $n\geqq 1$ (see, for example, [22]),

$$H_{\mathfrak{q}}(n)=\left|\begin{array}{cccc}{c}_n& c_{n+1}\cdots & c_{n+\mathfrak{q}-1}& \\ c_{n+1}& \cdots & ⋮& \\ ⋮& & & \\ c_{n+\mathfrak{q}-1}& \cdots & c_{n+2\mathfrak{q}-2}& \end{array}\right|.$$

There has been a lot of interest in the literature to determine upper bounds for the Hankel determinant $H_{\mathfrak{q}}(n)$, whose elements are the coefficients of univalent functions (see [15, 18, 19]).

For a number of subclasses of the normalized univalent analytic function class $\mathcal{S}$, specific bounds for the cases where $q=2$ and $n=2$ have been found. Most of these bounds result from the technique used in [21]. On the other hand, when $f\in \mathcal{S}$, the correct development order for $H_{\mathfrak{q}}(n)$ is unknown (see [24]).

The third Hankel determinant $H_{3}(1)$ is defined as follows (see, for example, [4]):

$$H_3(1)=\left|\begin{array}{ccc}{c}_1& c_2& c_3\\ c_2& c_3& c_4\\ c_3& c_4& c_5\end{array}\right|.$$

For $\mathcal{F}\in \mathcal{A}$, we have

$$ H_{3}(1)=c_{3} \bigl( c_{2}c_{4}-c_{3}^{2} \bigr) +c_{4} ( c_{2}c_{3}-c_{4} ) +c_{5} \bigl( c_{3}-c_{2}^{2} \bigr) \quad (c_{1}=1). $$

(1.16)

Moreover, by the triangle inequality we find that

$$ \bigl\vert H_{3}(1) \bigr\vert \leqq \vert c_{3} \vert \bigl\vert c_{2}c_{4}-c_{3}^{2} \bigr\vert + \vert c_{4} \vert \vert c_{2}c_{3}-c_{4} \vert + \vert c_{5} \vert \bigl\vert c_{3}-c_{2}^{2} \bigr\vert . $$

(1.17)

The second Hankel determinant $H_{2}(2)$ is represented by $|c_{2}c_{4}-c_{3}^{2}|$, whereas the Fekete–Szegö functional is given by $\vert c_{3}-c_{2}^{2} \vert$. The present study is motivated essentially by several recent developments (see, for example, [25, 27, 29, 30]). In this paper, we derive upper bounds on the second Hankel determinant $H_{2}(2)$ and the Fekete–Szegö functional $\vert c_{3}-c_{2}^{2} \vert$, and also on the third Hankel determinant $H_{3}(1)$ for functions $\mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta )$. We will also show how these results on upper bounds reduce to the bounds of various other function classes for particular selections of the parameters involved.

Let $\mathcal{P}$ bet the set of all functions $p(\zeta )$ of the form

$$ p(\zeta )=1+\omega _{1}\zeta +\omega _{2}\zeta ^{2}+\cdots \quad ( \zeta \in \Lambda ) $$

such that

$$ \Re \bigl( p(\zeta ) \bigr) >0 \quad (\zeta \in \Lambda ). $$

Then it is known that

$$ \vert \omega _{j} \vert \leqq 2 \quad (p\in \mathcal{P}). $$

Our main result is proved by using the lemma given by Libera and Złotkiewicz [21].

Lemma 1

Let the function $p(\zeta )$ given by

$$ p(\zeta )=1+\omega _{1}\zeta +\omega _{2}\zeta ^{2}+\cdots \quad ( \zeta \in \Lambda ) $$

belong to the class $\mathcal{P}$. Suppose also that $\omega _{1}\geqq 0$. Then

$$\begin{aligned} 2\omega _{2}={\omega _{1}^{2}}+x \bigl(4-{ \omega _{1}^{2}} \bigr), \end{aligned}$$

and

$$\begin{aligned} 4\omega _{3}=\omega _{1}^{3}+2x\omega {_{1}} \bigl(4-{\omega _{1}^{2}} \bigr)-x^{2} \omega {_{1}} \bigl(4-{\omega _{1}}^{2} \bigr)+2\zeta \bigl(1- \vert x \vert ^{2} \bigr) \bigl(4-\omega{_{1}} ^{2} \bigr) \end{aligned}$$

for some x, ζ such that $|x|\leqq 1$ and $|\zeta |\leqq 1$.

It is also necessary to derive our results by using the next two lemmas due to Ali [1].

Lemma 2

If

$$ p(\zeta )=1+\omega _{1}\zeta +\omega _{2}\zeta ^{2}+\cdots \in \mathcal{P}, $$

then

$$ \bigl\vert \omega _{2}-v\omega _{1}^{2} \bigr\vert \leqq 2\max { \bigl\{ 1, \vert 2v-1 \vert \bigr\} }. $$

Lemma 3

Let

$$ p(\zeta )=1+\omega _{1}\zeta +\omega _{2}\zeta ^{2}+\cdots \in \mathcal{P}. $$

If $0\leqq \mathcal{C}\leqq 1$ and $\mathcal{C}(2 \mathcal{C}-1)\leqq \mathcal{D}\leqq \mathcal{C}$, then

$$ \bigl\vert \omega _{3}-2\mathcal{C}\omega _{1} \omega _{2}+ \mathcal{D}\omega _{1}^{3} \bigr\vert \leqq 2. $$

2 Coefficient inequalities

We apply the notations used in [2]. Assume that $\vartheta \geqq 0$ and $\mu \geqq 0$ satisfy

$$ \mu +\vartheta =\delta -\eta \quad \text{and}\quad \mu \vartheta = \eta . $$

(2.1)

• μ has been chosen to be 0 for $\eta =0$; in this case, $\vartheta =\delta \geqq 0$.

• If $\delta =1+2$, then

  $$ \mu +\vartheta =1+\eta =1+\mu \vartheta \quad \text{or} \quad (\mu -1) (1- \vartheta )=0. $$

  • i. For $\eta >0$, if we choose $\mu =1$, then $\vartheta =\eta$.

  • ii. For $\eta =0$, if we choose $\mu =0$, then $\vartheta =\delta =1$.

Theorem 1

Let $0\leqq \mu \leqq 1$ and $0\leqq \vartheta \leqq 1$ satisfy (2.1). If $\mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta , \eta )$ with $0\leqq \alpha <1$, then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{400 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,2} ) ^{2}}{ ( [3]_{q}! ) ^{2} ( 1+2\mu ) ^{2} ( 1+2\vartheta ) ^{2}b_{3}^{2}}. $$

(2.2)

Proof

Assume that $\mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta , \eta )$. Therefore

$$ \frac{(1-\delta +2\eta )\frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta }+(\delta -2\eta ) ( \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) ) ^{{\prime }}+\eta \zeta ( \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) ) ^{{{\prime \prime }}}-\alpha }{1-\alpha } \in \mathcal{P}, $$

and there is a function

$$ p(\zeta )=1+\omega _{1}\zeta +\omega _{2}\zeta ^{2}+\cdots \in \mathcal{P} $$

such that

$$\begin{aligned} &(1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F} (\zeta )}{\zeta }+(\delta -2\eta ) \bigl( \mathcal{H}_{\Upsilon }^{ \lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{\prime }}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F}(\zeta ) \bigr) ^{{{\prime \prime }}}-\alpha \\ &\quad = ( 1-\alpha ) p(\zeta ). \end{aligned}$$

In view of (2.1), this equation becomes

$$\begin{aligned} &(1+\mu \vartheta -\mu -\vartheta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F}(\zeta )}{\zeta }+( \mu +\vartheta -\mu \vartheta ) \bigl( \mathcal{H}_{\Upsilon }^{ \lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{\prime }}+\mu \vartheta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}(\zeta ) \bigr) ^{{{\prime \prime }}}-\alpha \\ &\quad = ( 1-\alpha ) p(\zeta ). \end{aligned}$$

(2.3)

Thus setting

$$ p(\zeta )=1+\sum_{j=1}^{\infty }\omega _{j}\zeta ^{j} $$

and

$$ \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )=\zeta +\sum _{j=2}^{\infty }\rho _{j}c_{j} \zeta ^{j} $$

in (2.3), we obtain

$$\begin{aligned} & ( 1-\alpha ) +\sum_{j=2}^{\infty } \bigl[ \mu \vartheta j^{2}+(\mu +\vartheta -2\mu \vartheta )j+(1+\mu \vartheta - \mu -\vartheta ) \bigr] \rho _{j}c_{j} \zeta ^{j-1} \\ &\quad = ( 1-\alpha ) \Biggl( 1+\sum_{j=1}^{ \infty } \omega _{j}\zeta ^{j} \Biggr) \end{aligned}$$

or, equivalently,

$$ ( 1-\alpha ) +\sum_{j=2}^{\infty } \bigl( 1+(j-1) \mu \bigr) \bigl( 1+(j-1)\vartheta \bigr) \rho _{j}c_{j} \zeta ^{j-1}= ( 1-\alpha ) \Biggl( 1+\sum _{j=1}^{\infty }\omega _{j} \zeta ^{j} \Biggr) . $$

Equating the corresponding coefficients, we get

$$\begin{aligned}& c_{2}= \frac{ ( 1-\alpha ) }{ ( 1+\mu ) ( 1+\vartheta ) \rho _{2}}\omega _{1}, \end{aligned}$$

(2.4)

$$\begin{aligned}& c_{3}= \frac{ ( 1-\alpha ) }{ ( 1+2\mu ) ( 1+2\vartheta ) \rho _{3}}\omega _{2}, \end{aligned}$$

(2.5)

$$\begin{aligned}& c_{4}= \frac{ ( 1-\alpha ) }{ ( 1+3\mu ) ( 1+3\vartheta ) \rho _{4}}\omega _{3}, \end{aligned}$$

(2.6)

and

$$\begin{aligned} c_{5}& = \frac{ ( 1-\alpha ) }{ ( 1+4\mu ) ( 1+4\vartheta ) \rho _{5}}\omega _{4}. \end{aligned}$$

(2.7)

Let us now consider

$$ \mathcal{Y}= ( 1+\mu ) ( 1+\vartheta ) ( 1+3\mu ) ( 1+3\vartheta ) \rho _{2}\rho _{4} $$

and

$$ \mathcal{T}= ( 1+2\mu ) ^{2} ( 1+2\vartheta ) ^{2} \rho _{3}^{2}. $$

Note that for $0\leqq \mu \leqq 1$ and $0\leqq \vartheta \leqq 1$,

$$ \mathcal{T}>0,\qquad \mathcal{Y}>0,\qquad \mathcal{T}-\mathcal{Y}\geqq 0, \quad \text{and}\quad \mathcal{T}-2\mathcal{Y}< 0. $$

Using (2.4) in (2.7) when paired with the values of $\mathcal{Y}$ and $\mathcal{T}$, the second Hankel determinant becomes

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}= \bigl\vert c_{2}c_{4}-c_{3}^{2} \bigr\vert = ( 1-\alpha ) ^{2} \biggl\vert \frac{\omega _{1}\omega _{3}}{\mathcal{Y}}- \frac{\omega _{2}^{2}}{\mathcal{T}} \biggr\vert = \frac{ ( 1-\alpha ) ^{2}}{\mathcal{YT}} \bigl\vert \mathcal{T} {\omega _{1}\omega _{3}}- \mathcal{Y} {\omega _{2}^{2}} \bigr\vert . $$

By Lemma 1 this becomes

$$\begin{aligned} {{ \bigl\vert H_{2}(2) \bigr\vert }}& = \frac{ ( 1-\alpha ) ^{2}}{4\mathcal{YT}} \bigl\vert ( \mathcal{T}- \mathcal{Y} ) \omega _{1}^{4}+2 ( \mathcal{T}- \mathcal{Y} ) \omega _{1}^{2}x \bigl( 4- \omega _{1}^{2} \bigr) -\mathcal{T}\omega _{1}^{2} \bigl( 4-\omega _{1}^{2} \bigr) x^{2} \\ & \quad{}-\mathcal{Y}x^{2} \bigl( 4-\omega _{1}^{2} \bigr) ^{2}+2\mathcal{T}\omega _{1} \bigl( 4- \omega _{1}^{2} \bigr) \bigl( 1- \vert x \vert ^{2} \bigr) \zeta \bigr\vert . \end{aligned}$$

Now, without any loss of generality, we normalize $\omega _{1}$ so that $\omega _{1}=\omega$ for $0\leqq \omega \leqq 2$. Then by the triangle inequality we get

$$\begin{aligned} {{ \bigl\vert H_{2}(2) \bigr\vert }}& \leqq \frac{ ( 1-\alpha ) ^{2}}{4\mathcal{YT}} \bigl\{ ( \mathcal{T}- \mathcal{Y} ) \omega ^{4}+2 ( \mathcal{T}-\mathcal{Y} ) \omega ^{2} \vert x \vert \bigl( 4-\omega ^{2} \bigr) + \mathcal{T}\omega ^{2} \bigl( 4-\omega ^{2} \bigr) \vert x \vert ^{2} \\ & \quad{}+\mathcal{Y} \vert x \vert ^{2} \bigl( 4-\omega ^{2} \bigr) ^{2}+2\mathcal{T}\omega \bigl( 4- \omega ^{2} \bigr) \bigl( 1- \vert x \vert ^{2} \bigr) \bigr\} \\ & =\frac{ ( 1-\alpha ) ^{2}}{4\mathcal{YT}}\phi \bigl( \vert x \vert \bigr) . \end{aligned}$$

Differentiating $\phi ( |x| )$ with respect to $|x|$, we have

$$ \phi ^{\prime } \bigl( \vert x \vert \bigr) =2 ( \mathcal{T}- \mathcal{Y} ) \omega ^{2} \bigl( 4-\omega ^{2} \bigr) +2 \vert x \vert \bigl( 4- \omega ^{2} \bigr) ( 2-\omega ) \bigl( 2 \mathcal{Y}- \omega ( \mathcal{T}-\mathcal{Y} ) \bigr) . $$

We can see that for $0\leqq \mu \leqq 1$, $0\leqq \vartheta \leqq 1$, and $0\leqq \omega \leqq 2$, $\phi ^{\prime } ( |x| ) \geqq 0$. Thus $\phi ( |x| ) \leqq \phi ( 1 )$, and hence

$$\begin{aligned} {{ \bigl\vert H_{2}(2) \bigr\vert }}& \leqq \frac{ ( 1-\alpha ) ^{2}}{4\mathcal{YT}} \bigl\{ ( \mathcal{T}- \mathcal{Y} ) \omega ^{4}+2 ( \mathcal{T}-\mathcal{Y} ) \omega ^{2} \bigl( 4- \omega ^{2} \bigr) +\mathcal{T}\omega ^{2} \bigl( 4-\omega ^{2} \bigr) + \mathcal{Y} \bigl( 4-\omega ^{2} \bigr) ^{2} \bigr\} \\ & =\frac{ ( 1-\alpha ) ^{2}}{4\mathcal{YT}}\mathcal{G}( \omega ). \end{aligned}$$

Solving the equation

$$ \mathcal{G}^{\prime }(\omega )=0, $$

we have

$$ \omega =0,\qquad \omega =\sqrt{ \frac{3\mathcal{T}-4\mathcal{Y}}{\mathcal{T-Y}}},\quad {\text{and}}\quad \omega =-\sqrt{ \frac{3\mathcal{T}-4\mathcal{Y}}{\mathcal{T}-\mathcal{Y}}}. $$

Since $3\mathcal{T}-4\mathcal{L}<0$ for $0\leqq \mu$, $\vartheta \leqq 1$, $\mathcal{G}(\omega )$ has only one critical point at $\omega =0$, and also

$$ \mathcal{G}^{\prime \prime }(\omega )\vert _{\omega =0}=8 ( 3\mathcal{T}-4 \mathcal{Y} ) < 0. $$

Therefore $\mathcal{G}(\omega )$ obtains its maximum value at $\omega =0$, i.e., $\mathcal{G}(\omega )\leqq \mathcal{G}(0)$ for all $\omega \in {}[ 0,2]$. Hence

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{ ( 1-\alpha ) ^{2}}{4\mathcal{YT}}16 \mathcal{Y}= \frac{4 ( 1-\alpha ) ^{2}}{\mathcal{T}}= \frac{400 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,2} ) ^{2}}{ ( [3]_{q}! ) ^{2} ( 1+2\mu ) ^{2} ( 1+2\vartheta ) ^{2}b_{3}^{2}}. $$

The proof of Theorem 1 is now complete. □

We will obtain different known results from Theorem 1 for specific choices of δ and η. Taking $\delta =\eta =0$ (i.e., $\mu =\vartheta =0$) in Theorem 1, we obtain the following:

Corollary 1

If $\mathcal{F}\in \mathcal{A}$ satisfies the inequality

$$ {\Re} \biggl( \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) }{\zeta } \biggr) >\alpha $$

with $0\leqq \alpha <1$, then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{400 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,2} ) ^{2}}{ ( [3]_{q}! ) ^{2}b_{3}^{2}}. $$

If $\eta =\alpha =0$, then $\mu =0$ and $\vartheta =\delta =1$, and we obtain the following:

Corollary 2

If $\mathcal{F}\in \mathcal{A}$ satisfies

$$ {\Re} \bigl(\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) \bigr)^{{ \prime }}>0, $$

then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{400 ( [\lambda +1]_{q,2} ) ^{2}}{9 ( [3]_{q}! ) ^{2}b_{3}^{2}}. $$

If $\delta =1+2\eta$ with $\eta >0$, $\alpha =0$, and $\mu =1$, then $\vartheta =\eta >0$, and we have the following:

Corollary 3

If $\mathcal{F}\in \mathcal{A}$ satisfies

$$ {\Re} \bigl( \bigl( \mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}( \zeta ) \bigr) ^{{\prime }}+\eta \zeta \bigl( \mathcal{H}_{ \Upsilon }^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{{\prime \prime }}} \bigr) >0 $$

for $\eta \geqq 0$, then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{400 ( [\lambda +1]_{q,2} ) ^{2}}{9 ( [3]_{q}! ) ^{2} ( 1+2\eta ) ^{2}b_{3}^{2}}. $$

If $\eta =\alpha =0$, then $\mu =0$ and $\vartheta =\delta >0$, and we get the following:

Corollary 4

If $\mathcal{F}\in \mathcal{A}$ satisfies

$$ {\Re} \biggl( (1-\delta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta }+\delta \bigl( \mathcal{H}_{\Upsilon }^{\lambda ,q} \mathcal{F}(\zeta ) \bigr) ^{{\prime }} \biggr) >0, $$

then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{400 ( [\lambda +1]_{q,2} ) ^{2}}{ ( [3]_{q}! ) ^{2} ( 1+2\delta ) ^{2}b_{3}^{2}}. $$

Putting $b_{j}= \frac {(-1)^{j-1}\Gamma (\upsilon +1)}{4^{j-1}(j-1)!\Gamma (j+\upsilon )}$, $\upsilon >0$, in Theorem 1, we obtain the following:

Example 1

Let $0\leqq \mu \leqq 1$ and $0\leqq \vartheta \leqq 1$ satisfy (2.1). If $\mathcal{F}\in \mathcal{P}_{\alpha ,\upsilon }^{\lambda ,q}(\delta , \eta )$ with $0\leqq \alpha <1$, then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{400 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,2} ) ^{2}}{ ( [3]_{q}! ) ^{2} ( 1+2\mu ) ^{2} ( 1+2\vartheta ) ^{2}\varkappa _{3}^{2}}. $$

Putting $b_{j}= ( \frac {n+1}{n+j} ) ^{r}$, $r>0$, $n\geqq 0$, in Theorem 1, we obtain the following:

Example 2

Let $0\leqq \mu \leqq 1$ and $0\leqq \vartheta \leqq 1$ satisfy (2.1). If $\mathcal{F}\in \mathcal{R}_{\alpha ,n,r}^{\lambda ,q}(\delta ,\eta )$ with $0\leqq \alpha <1$, then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{400 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,2} ) ^{2} ( n+3 ) ^{2r}}{ ( [3]_{q}! ) ^{2} ( 1+2\mu ) ^{2} ( 1+2\vartheta ) ^{2} ( n+1 ) ^{2r}}. $$

Putting $b_{j}=\frac {\rho ^{j-1}}{(j-1)!}e^{-\rho }$, $\rho >0$, in Theorem 1, we obtain the following:

Example 3

Let $0\leqq \mu \leqq 1$ and $0\leqq \vartheta \leqq 1$ satisfy (2.1). If $\mathcal{F}\in \mathcal{D}_{\alpha ,\rho }^{\lambda ,q}(\delta , \eta )$ with $0\leqq \alpha <1$, then

$$ {{ \bigl\vert H_{2}(2) \bigr\vert }}\leqq \frac{1600 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,2} ) ^{2}}{ ( [3]_{q}! ) ^{2} ( 1+2\mu ) ^{2} ( 1+2\vartheta ) ^{2}\rho ^{4}e^{-2\rho }}. $$

Theorem 2

Let the paramaters μ $(0\leqq \mu \leqq 1)$ and ϑ $(0\leqq \vartheta \leqq 1)$ satisfy condition (2.1). If $\mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta , \eta )$ with $0\leqq \alpha <1$, then

$$ \bigl\vert c_{3}-c_{2}^{2} \bigr\vert \leqq \frac{20 ( 1-\alpha ) [\lambda +1]_{q,2}}{ ( 1+2\mu ) ( 1+2\vartheta ) [3]_{q}!b_{3}}. $$

(2.8)

Proof

From (2.4)–(2.7) we can see that

$$ \bigl\vert c_{3}-c_{2}^{2} \bigr\vert = \biggl\vert \frac{ ( 1-\alpha ) }{ ( 1+2\mu ) ( 1+2\vartheta ) \rho _{3}} \omega _{2}- \frac{ ( 1-\alpha ) ^{2}}{ ( 1+\mu ) ^{2} ( 1+\vartheta ) ^{2}\rho _{2}^{2}}\omega _{1}^{2} \biggr\vert . $$

(2.9)

Let $\mathcal{S}= ( 1+2\mu ) ( 1+2\vartheta ) \rho _{3}$ and $\mathcal{R}= ( 1+\mu ) ^{2} ( 1+\vartheta ) ^{2} \rho _{2}^{2}$. Note that for $0\leqq \mu \leqq 1$ and $0\leqq \vartheta \leqq 1$,

$$ \mathcal{S}>0,\qquad \mathcal{R}>0,\qquad \mathcal{R}-\mathcal{S}\geqq 0, \quad\text{and}\quad \mathcal{R}-2\mathcal{S}>0. $$

Using $\mathcal{S}$ and $\mathcal{R}$, equation (2.9) becomes

$$ \bigl\vert c_{3}-c_{2}^{2} \bigr\vert = \frac{ ( 1-\alpha ) }{\mathcal{S}} \biggl\vert \omega _{2}- \frac{ ( 1-\alpha ) \mathcal{S} }{\mathcal{R}}\omega _{1}^{2} \biggr\vert . $$

Taking $\upsilon = \frac{ ( 1-\alpha ) \mathcal{S}}{\mathcal{R}}$ in Lemma 2, we obtain

$$\begin{aligned} \biggl\vert \omega _{2}- \frac{ ( 1-\alpha ) \mathcal{S}}{\mathcal{R} }\omega _{1}^{2} \biggr\vert &\leqq 2\max \biggl\{ 1, \biggl\vert \frac{2 ( 1-\alpha ) \mathcal{S}}{\mathcal{R}}-1 \biggr\vert \biggr\} \\ &=2\max \biggl\{ 1, \biggl\vert \frac{2 ( 1-\alpha ) \mathcal{S}- \mathcal{R}}{\mathcal{R}} \biggr\vert \biggr\} . \end{aligned}$$

Since $\mathcal{R}-\mathcal{S}\geqq 0$ and $0\leqq \alpha <1$, we get

$$ -\mathcal{R}< 2 ( 1-\alpha ) \mathcal{S}-\mathcal{R}\leqq 2 \mathcal{S }-\mathcal{R}\leqq 0, $$

and so we have

$$ \biggl\vert \frac{2 ( 1-\alpha ) \mathcal{S}-\mathcal{R}}{\mathcal{R }} \biggr\vert \leqq 1. $$

Then

$$ \biggl\vert \omega _{2}- \frac{ ( 1-\alpha ) \mathcal{S}}{\mathcal{R} }\omega _{1}^{2} \biggr\vert \leqq 2\max \biggl\{ 1, \biggl\vert \frac{2 ( 1-\alpha ) \mathcal{S}}{\mathcal{R}}-1 \biggr\vert \biggr\} =2. $$

Hence

$$ \bigl\vert c_{3}-c_{2}^{2} \bigr\vert = \frac{ ( 1-\alpha )}{ \mathcal{S}} \biggl\vert \omega _{2}- \frac{ ( 1-\alpha ) \mathcal{S} }{\mathcal{R}}\omega _{1}^{2} \biggr\vert \leqq \frac{2 ( 1-\alpha ) }{\mathcal{S}}= \frac{20 ( 1-\alpha ) [\lambda +1]_{q,2}}{ ( 1+2\mu ) ( 1+2\vartheta ) [3]_{q}!b_{3}}. $$

 □

Theorem 3

Let the parameters μ $(0\leqq \mu \leqq 1)$ and ϑ $(0\leqq \vartheta \leqq 1)$ satisfy condition (2.1). If $\mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta , \eta )$ with $0\leqq \alpha <1$, then

$$ \vert c_{4}-c_{2}c_{3} \vert \leqq \frac{84 ( 1-\alpha ) [\lambda +1]_{q,3}}{ ( 1+3\mu ) ( 1+3\vartheta ) [4]_{q}!b_{4}}. $$

(2.10)

Proof

From (2.4)–(2.7) we get

$$ \vert c_{4}-c_{2}c_{3} \vert = \biggl\vert \frac{ ( 1-\alpha ) }{ ( 1+3\mu ) ( 1+3\vartheta ) \rho _{4}} \omega _{3}- \frac{ ( 1-\alpha ) ^{2}}{ ( 1+\mu ) ( 1+\vartheta ) ( 1+2\mu ) ( 1+2\vartheta ) \rho _{2}\rho _{3}} \omega _{1}\omega _{2} \biggr\vert . $$

(2.11)

Now let

$$ \mathcal{M}= ( 1+3\mu ) ( 1+3\vartheta ) \rho _{4} $$

and

$$ \mathcal{N}= ( 1+\mu ) ( 1+\vartheta ) ( 1+2\mu ) ( 1+2\vartheta ) \rho _{2}\rho _{3}. $$

Note that for $0\leqq \mu$, $\vartheta \leqq 1$,

$$ \mathcal{M}>0,\qquad \mathcal{N}>0,\qquad \mathcal{N}-\mathcal{M}\geqq 0. $$

(2.12)

Using $\mathcal{M}$ and $\mathcal{N}$, equation (2.11) becomes

$$ \vert c_{4}-c_{2}c_{3} \vert = \frac{ ( 1-\alpha ) }{\mathcal{M}} \biggl\vert \omega _{3}- \frac{ ( 1-\alpha ) \mathcal{M}}{\mathcal{N}}\omega _{1}\omega _{2} \biggr\vert . $$

Now letting $2\mathcal{C}= \frac{ ( 1-\alpha ) \mathcal{M}}{\mathcal{N}}$ and $\mathcal{D}=0$ in Lemma 3, we have

$$\begin{aligned} \biggl\vert \omega _{3}- \frac{ ( 1-\alpha ) \mathcal{M}}{\mathcal{N} }\omega _{1}\omega _{2} \biggr\vert &\leqq 2 \\ &=2\max \biggl\{ 1, \biggl\vert \frac{2 ( 1-\alpha ) \mathcal{M}- \mathcal{N}}{\mathcal{N}} \biggr\vert \biggr\} , \end{aligned}$$

provided that

$$ 0\leqq \mathcal{C}\leqq 1\quad \text{and}\quad \mathcal{C}(2\mathcal{C}-1) \leqq \mathcal{D}\leqq \mathcal{C}. $$

Since $\mathcal{N}-\mathcal{M}\geqq 0$ and $0\leqq \alpha <1$, we get

$$ 0< \mathcal{C}= \frac{ ( 1-\alpha ) \mathcal{M}}{2\mathcal{N}}\leqq \frac{1}{2}< 1, $$

and since $\mathcal{D}=0$, we have

$$ \mathcal{C}(2\mathcal{C}-1)\leqq \mathcal{D}\leqq \mathcal{C}. $$

Then

$$\begin{aligned} \vert c_{4}-c_{2}c_{3} \vert &= \frac{ ( 1-\alpha ) }{ \mathcal{M}} \biggl\vert \omega _{3}- \frac{ ( 1-\alpha ) \mathcal{M} }{\mathcal{N}}\omega _{1}\omega _{2} \biggr\vert \\ &\leqq \frac{2 ( 1-\alpha ) }{\mathcal{M}} \\ &= \frac{84 ( 1-\alpha ) [\lambda +1]_{q,3}}{ ( 1+3\mu ) ( 1+3\vartheta ) [4]_{q}!b_{4}}. \end{aligned}$$

This completes the proof of Theorem 3. □

Theorem 4

Let the parameters μ $(0\leqq \mu \leqq 1)$ and ϑ $(0\leqq \vartheta \leqq 1)$ satisfy condition (2.1). If $\mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta , \eta )$ with $0\leqq \alpha <1$, then

$$\begin{aligned} \bigl\vert H_{3}(1) \bigr\vert &\leqq \frac{8000 ( [\lambda +1]_{q,2} ) ^{3} (1-\alpha ) ^{3}}{ ( 1+2\mu ) ^{3} ( 1+2\vartheta ) ^{3} ( [3]_{q}! ) ^{3}b_{3}^{3}}+ \frac{7056 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,3} ) ^{2} }{ ( 1+3\mu ) ^{2} (1+3\vartheta ) ^{2} ( [4]_{q}! ) ^{2}b_{4}^{2}} \\ &\quad{}+ \frac{8640 ( 1-\alpha ) ^{2}[\lambda +1]_{q,2}[\lambda +1]_{q,4}}{ ([3]_{q}! ) ( [5]_{q}! ) ( 1+2\mu ) ( 1+2\vartheta ) ( 1+4\mu ) ( 1+4\vartheta ) b_{3}b_{5}}. \end{aligned}$$

Proof

In view of (2.4)–(2.7), we can see that for

$$\begin{aligned}& \mathcal{F}\in \mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta , \eta ) \quad (0\leqq \alpha < 1), \\& c_{3}= \frac{ ( 1-\alpha ) }{ ( 1+2\mu ) ( 1+2\vartheta ) \rho _{3}} \omega _{2}, \\& c_{4}= \frac{ ( 1-\alpha ) }{ ( 1+3\mu ) ( 1+3\vartheta ) \rho _{4}} \omega _{3}, \\& c_{5}= \frac{ ( 1-\alpha )}{ ( 1+4\mu ) ( 1+4\vartheta ) \rho _{5}} \omega _{4}, \\& \bigl\vert c_{2}c_{4}-c_{3}^{2} \bigr\vert \leqq \frac{4 ( 1-\alpha ) ^{2}}{ ( 1+2\mu ) ^{2} ( 1+2\vartheta ) ^{2}\rho _{3}^{2}}, \\& \bigl\vert c_{3}-c_{2}^{2} \bigr\vert \leqq \frac{2 ( 1-\alpha ) }{ ( 1+2\mu ) ( 1+2\vartheta ) \rho _{3}}, \end{aligned}$$

and

$$\begin{aligned} \vert c_{4}-c_{2}c_{3} \vert \leqq \frac{2 ( 1-\alpha ) }{ ( 1+3\mu ) ( 1+3\vartheta ) \rho _{4}}. \end{aligned}$$

Substituting all these values into (1.16), we obtain

$$\begin{aligned} \bigl\vert H_{3}(1) \bigr\vert &\leqq \vert c_{3} \vert \bigl\vert c_{2}c_{4}-c_{3}^{2} \bigr\vert + \vert c_{4} \vert \vert c_{2}c_{3}-c_{4} \vert + \vert c_{5} \vert \bigl\vert c_{3}-c_{2}^{2} \bigr\vert \\ &= \frac{8 ( 1-\alpha ) ^{3}}{ ( 1+2\mu ) ^{3} ( 1+2\vartheta ) ^{3}\rho _{3}^{3}}+ \frac{4 ( 1-\alpha ) ^{2} }{ ( 1+3\mu ) ^{2} ( 1+3\vartheta ) ^{2}\rho _{4}^{2}} \\ &\quad{}+ \frac{4 ( 1-\alpha ) ^{2}}{ ( 1+2\mu ) ( 1+2\vartheta ) ( 1+4\mu ) ( 1+4\vartheta ) \rho _{3}\rho _{5}}, \\ &= \frac{8000 ([\lambda +1]_{q,2} ) ^{3} ( 1-\alpha ) ^{3}}{ ( 1+2\mu ) ^{3} ( 1+2\vartheta ) ^{3} ( [3]_{q}! ) ^{3}b_{3}^{3}}+ \frac{7056 ( 1-\alpha ) ^{2} ( [\lambda +1]_{q,3} ) ^{2}}{ ( 1+3\mu ) ^{2} ( 1+3\vartheta ) ^{2} ( [4]_{q}! ) ^{2}b_{4}^{2}} \\ &\quad{}+ \frac{ 8640 ( 1-\alpha ) ^{2}[\lambda +1]_{q,2}[\lambda +1]_{q,4}}{ ( [3]_{q}! ) ( [5]_{q}! ) ( 1+2\mu ) ( 1+2\vartheta ) ( 1+4\mu ) ( 1+4\vartheta ) b_{3}b_{5}}. \end{aligned}$$

 □

3 Concluding remarks and observations

In this study, we have used the q-derivative operator and the error function $\operatorname{erf}(\zeta )$ to obtain a new operator $\mathcal{H}_{\Upsilon }^{\lambda ,q}$ with a view to introducing some new subclasses of the class of analytic functions in the open unit disk Λ. We have established upper bounds for the second Hankel determinant, the Fekete–Szegö functional, and the third Hankel determinant for normalized analytic functions $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon }^{\lambda ,q}(\delta , \eta )$ together with other features and results. In addition, we have deduced a few examples of our main points (see Theorem 1).