Coefficient functionals for a class of bounded turning functions related to modified sigmoid function

1.   Introduction

Let $\mathcal{A}$ represent the collections of analytic functions defined in open unit disc $\mathbb{D} = \left \{ z\in \mathbb{C} :\left \vert z\right \vert < 1\right \}$ whose normalization is of the form

Let $\mathcal{S}$ denote the subclass of $\mathcal{A}$ comprising of functions of the form $\left(1.1\right)$ which are also univalent in $\mathbb{D}$.

Let $\mathcal{P}$ represent the class of all functions $p$ that are analytic in $\mathbb{D}$ with $\Re (p(z)) > 0$ and has the series representation

Next we recall the definition of subordination. For two functions $h_{1}, h_{2}\in \mathcal{A}$, we say that $h_{1}$ is subordinate to $h_{2}$ and is symbolically written as $h_{1}\prec h_{2}$ if there exists an analytic function $w$ with the property $\left \vert w\left(z\right) \right \vert \leq \left \vert z\right \vert$ and $w\left(0\right) = 0$ such that $h_{1}\left(z\right) = h_{2}\left(w\left(z\right) \right)$ for $z\in \mathbb{D}$. Further, if $h_{2}\in \mathcal{S}$, then the condition becomes

Now we consider the following class $\mathcal{S}^{\ast }(\varphi)$ as follows:

where $\varphi$ is an analytic univalent function with positive real part in $\mathbb{D}$, $\varphi(\mathbb{U})$ is symmetric about the real axis and starlike with respect to $\varphi(0) = 1$ and $\varphi^\prime(0) > 0$. The class $\mathcal{S}^{\ast }(\varphi)$ was introduced by Ma and Minda ^[20]. In particular, if we take $\varphi(z) = {(1+z)}/{(1-z)}$, then the class $\mathcal{S}^{\ast }(\varphi)$ is the well-known class of starlike functions. If we vary the function $\varphi$ on the right hand side of $\left(1.3\right)$, then we obtain some several subclasses of $\mathcal{S}$ whose image domains have some interesting geometrical configurations as follows:

$(1)$ The class $\mathcal{S}^*(\varphi)$ with $\varphi(z) = 1+\sin z$ is introduced and studied by Cho et al. ^[6].

$(2)$ The class $\mathcal{S}^{\ast }(\varphi)$ with $\varphi (z) = 1+z-\frac{1 }{3}z^{3},$ which is a nephroid shaped domain, was introduced and investigated by Wani and Swaminathan ^[39].

$(3)$ The class $\mathcal{S}^*(\varphi)$ with $\varphi(z) = \sqrt{1+z},$ which is bounded by lemniscate of Bernoulli in right half plan, was developed by Sokól and Stankiewicz ^[30].

$(4)$ The class $\mathcal{S}^{\ast }(\varphi)$ with $\varphi (z) = 1+\frac{4}{ 3}z+\frac{2}{3}z^{2}$ was introduced by Sharma et al. ^[29].

$(5)$ The class $\mathcal{S}^*(\varphi)$ with $\varphi(z) = e^{z}$ was introduced and studied by Mendiratta et al. ^[21].

$(6)$ The class $\mathcal{S}^{\ast }(\varphi)$ with $\varphi (z) = z+\sqrt{ 1+z^{2}},$ which maps $\mathbb{D}$ to crescent shaped region, was introduced by Raina and Sokól ^[26].

Also we note that lately many subclasses of starlike functions are introduced see ^[7,9,12] by choosing some particular functions such as functions associated with Bell numbers, shell-like curve connected with Fibonacci numbers, functions connected with conic domains and rational functions instead of $\varphi$ in $\left(1.3\right)$.

Pommerenke ^[24,25] introduced the Hankel determinant $H_{q, n}\left(f\right)$ for function $f\in \mathcal{S}$ of the form (1.1), where the parameters $q, n\in \mathbb{N} = \left \{ 1, 2, 3, \cdots \right \}$ as follows:

The Hankel determinants for different orders are obtained for different values of $q$ and $n.$ When $q = 2$ and $n = 1,$ the determinant is

Note that $H_{2, 1}\left(f\right) = a_{3}-a_{2}^{2},$ is the classical Fekete-Szegö functional. For various subclasses of $\mathcal{A}$, the best possible value of the upper bound for $\left \vert H_{2, 1}\left(f\right) \right \vert$ was investigated by different authors (see ^[13,14,15] for details). Furthermore, when $q = 2$ and $n = 2,$ the second Hankel determinant is

The upper bound of $\left \vert H_{2, 2}\left(f\right) \right \vert$ has been studied by several authors in the last few decades. For instance, the readers may refer to the works of Hayman ^[11], the Noonan and Thomas ^[22], Ohran et al. ^[23] and Shi et al. ^[34]. Moreover, Babalola ^[3] studied the Hankel determinant $H_{3, 1}\left(f\right)$ for some subclasses of analytic functions. For some recent works on third order Hankel determinant we may refer the interested reader to such more recent works as (for example) ^[28,32,38]. The bound of the fourth Hankel determinant for a class of analytic functions with bounded turning associated with cardoid domain was approximated by Srivastava et al. in ^[37]. It should be remarked that a wide variety of applications of Hankel systems arise in linear filtering theory, discrete inverse scattering, and discretization of certain integral equations arising in mathematical physics ^[40].

Evaluating these Hankel determinants for various new subclasses has been an attracting area lately. One such field of interest is the Quantum Calculus ($q$-calculus), which is a generalization of classical calculus by replacing the limit by a parameter $q$. For the basics and preliminaries, the readers are advised to see the works and expositions in ^[31,35,37]. It is important to mention here the work on a $q$-differential operator by Srivastava et al. ^[33], in which they determined the upper bound of second Hankel determinant for a subclass of bi-univalent functions in $q$-analogue. Recently, the upper bound estimate for $q$-analogue of a subclass of starlike functions in connection with exponential function were evaluated in ^[36].

Recently, a class of starlike functions associated with Modified sigmoid function was defined by Goel and Kumar ^[10], i.e,

Motivated by all the works mentioned above and ^[4], in this article we introduce and investigate the class $\mathcal{R}_{\mathcal{S}_{G}},$ which is defined as follows:

We also establish some sharp results such as coefficient bounds, Fekete-Szeg ö inequality, second-order determinant, Zalcman conjecture and Krushkal inequality for functions belonging to the class $\mathcal{R}_{\mathcal{S} _{G}}$. Moreover, we estimate bounds of the third and forth-order Hankel determinants for this class $\mathcal{R}_{\mathcal{S}_{G}}$ and for the 2-fold and 3-fold symmetric functions.

2.   A set of lemmas

For the proofs of our main findings, we need the following lemmas.

Lemma 1. Let $p\in {\mathcal{P}}$ have the series expansion of the form $(1.2)$. Then, for $x$ and $\sigma$ with $\left \vert x\right \vert \leq 1, \left \vert \sigma \right \vert \leq 1,$ such that

We note that $\left(2.1\right)$ and $\left(2.2\right)$ are taken from ^[18].

Lemma 2. If $p\in {\mathcal{P}}$ and has the series of the form $(1.2)$, then

We note that the inequalities (2.3), (2.4) and (2.6) in the above can be found in ^[2,25] and (2.5) is given by ^[13].

Lemma 3. ^[2] If $p\in {\mathcal{P}}$ and has the series of the form $(1.2)$, then

where $J, K$ and $L$ are real numbers.

Lemma 4. ^[27] Let $m, n, l$ and $r$ satisfy the inequalities $0 < m < 1, \ 0 < r < 1$ and

If $p\in \mathcal{P}$ and has power series $\left(1.2\right)$, then

3.   Bounds of $\left \vert H_{3, 1}\left(f\right) \right \vert$ for the class $\mathcal{R}_{\mathcal{S}_{G}}$

Theorem 1. Let $f\in \mathcal{R}_{\mathcal{S}_{G}}$ and be of the form $(1.1)$. Then

The first four inequalities are sharp for the functions defined below respectively

Proof. Let $f\in \mathcal{R}_{\mathcal{S}_{G}}$. Then, (1.5) can be put in the form of Schwarz function $w\left(z\right)$ as

Also, if $p\in {\mathcal{P}}$, then it may be written in terms of the Schwarz function $w$ as

or equivalently,

Now

By a simplification and using the series expansion (3.9), we have

Comparing (3.10) and (3.11), we get

and

For $a_{2}$, putting (2.4) in (3.12), we have

For $a_{3}$, simplifying (3.13), we get

and applying (2.3), we have

For $a_{4}$, using (3.14), we obtain

By applying Lemma 3 to (3.18), we get

For $a_{5},$ applying Lemma 4 to (3.15), we get

For $a_{6},$ re-arranging (3.16) and applying the triangle inequality, we get

By applying $\left(2.3\right)$ and $\left(2.4\right)$ to the above, we get

For $a_{7}$, re-arranging (3.17) and applying the triangle inequality, we get

Also by using $\left(2.3\right)$ and $\left(2.4\right)$ to the above, we obtain

Next, we consider the Fekete-Szegö problem and the Hankel determinants for the class $\mathcal{R}_{\mathcal{S}_{G}}$.

Theorem 2. If $f$ of the form $(1.1)$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

The result is sharp for the function $f_{2}$ defined by $(3.7)$ for $|\zeta |\leq 8/3$ and the functiom $f_{1}$ defined by $(3.7)$ for $|\zeta |\geq 8/3$.

Proof. Using (3.12) and (3.13), we can write

By rearranging we have

Applying (2.5) we get

Then with simple calculations, we obtain

For the sharpness consider the function

which gives equality in $\left(3.19\right)$ when $\left \vert \zeta \right \vert \leq \frac{8}{3},$ namely

For the case $\left \vert \zeta \right \vert \geq \frac{8}{3}$ consider

which gives

If we put $\zeta = 1,$ then the above result becomes:

Corollary 1. If $f$ of the form $(1.1)$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

Theorem 3. If $f$ of the form $(1.1)$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

The result is sharp for the function $f_3$ defined by $(3.7)$.

Proof. From (3.12)–(3.14), we get

Using Lemma 3, we get the required result.

Theorem 4. If $f$ of the form $(1.1)$ belongs to $\mathcal{R}_{\mathcal{S }_{G}}$, then

The result is sharp for the function $f_2$ defined by $(3.7)$.

Proof. From (3.12)–(3.14), we have

Applying (2.1) and (2.2) to express $c_{2}$ and $c_{3}$ in terms of $c_{1} = c$, with $0\leq c\leq 2,$ we get

With the aid of the triangle inequality and replacing $|\sigma |\leq 1$, $|x| = b$, with $b\leq 1$, we obtain

It is a simple calculation to show that $\frac{\partial \phi (c, b)}{\partial b}\geq 0$ on $[0, 1]$, so that $\phi (c, b)\leq \phi (c, 1)$. Putting $b = 1$ gives

Also, $\phi ^{\prime }(c, 1) = 0$, has only root $c = 0\in \left[0, 2\right]$ and so $\phi ^{\prime \prime }(0, 1) < 0$. Thus, the maximum value at $c = 0$ is

Theorem 5. If $f\in \mathcal{A}$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

Proof. From (3.12)–(3.15), we have

Rearranging the above term, we get

Using (2.3) and (2.4), we get the required result.

Theorem 6. If $f\in \mathcal{A}$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

The result is sharp for function $f_4$ defined by $(3.7)$.

Proof. From (3.12)–(3.15), we have

By applying of Lemma 4, we get the desired result.

Theorem 7. If $f\in \mathcal{A}$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

Proof. From (3.13)–(3.15), we have

Now using (2.3)–(2.5), we get the required result.

We will now determine the bound of the third Hankel determinant ${H} _{3, 1}(f)$ for $f\in \mathcal{R}_{\mathcal{S}_{G}}$.

Theorem 8. If $f\in \mathcal{A}$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

Proof. We have the third Hankel determinant form as follows:

where $a_{1} = 1$. This yields

By using $\left(3.2\right)$–$\left(3.4\right)$ and (3.21)–(3.23), we obtain the desired result.

4.   Bounds of $|{H}_{4, 1}(f)|$ for the class $\mathcal{R}_{ \mathcal{S} _{G}}$

From (1.4), we can write ${H}_{4, 1}(f)$ as

where

In recent years, researchers start to find the fourth-order Hankel determinant for different subclasses of analytic functions. The trend of finding fourth-order Hankel determinant in geometric function theory started in 2018, when Arif et al. ^[1] studied and successfully obtained the upper bound for the class of bounded turning functions. Recently Khan et al. ^[16] obtained the third and fourth-order Hankel determinant for the class of bounded turning functions associated with sine function. Also, Zhang and Tang ^[41] studied the fouth-order Hankel determinat for class of starlike functions connected with sine function. Inspired from the above works, we discuss here the fourth-order Hankel determiant for the class $\mathcal{R}_{\mathcal{S}_{G}}.$

Theorem 9. If $f\in \mathcal{A}$ belongs to $\mathcal{R}_{\mathcal{S}_{G}}$, then

Proof. From (4.1), we have

where $\delta _{1},$ $\delta _{2}$ and $\delta _{3}$ are defined as in (4.2)–(4.4). Now by using the triangle inequality, we have

Since

by using (3.2), $\left(3.3\right), \left(3.5\right),$ (3.21), (3.24) and (3.25), we get

Since

by using (3.2), $\left(3.4\right), \left(3.5\right)$, (3.22), (3.25) and (3.26), we get

Also, since

by using (3.3)–$\left(3.5\right)$, (3.23), (3.24) and (3.26), we get

Now by using the values of (4.6)–(4.8) along with $\left(3.3\right)$–$\left(3.6\right)$ and (3.27) to (4.5), we get the desired estimate.

5.   Bounds of $|{H}_{4, 1}(f)|$ for the 2-fold and 3-fold symmetric functions

A function $f$ is said to be $m$-fold symmetric if the following condition holds true for $\varepsilon = \exp \left(\frac{2\pi i}{m}\right)$,

The set of all $m$-fold symmetric functions belonging to the familiar class $\mathcal{S}$ of univalent functions is denoted by $\mathcal{S}^{\left(m\right) }$, represented by the following series expansion

An analytic function $f$ of the form $\left(5.1\right)$ belongs to the class $\mathcal{R}_{\mathcal{S}_{G}}^{\left(m\right) }$ if and only if

where $p\left(z\right)$ belong to the class $\mathcal{P}^{\left(m\right) }$ which is defined as follows:

If a function $f$ belongs to $\mathcal{S}^{\left(2\right) }$, then its series representation is

and

Further, if a function $f$ belongs to $\mathcal{S}^{\left(3\right) },$ then its series representation is

and

Theorem 10. If $f\in \mathcal{R}_{\mathcal{S}_{G}}^{\left(2\right) },$ then

Proof. Let $f\in \mathcal{R}_{\mathcal{S}_{G}}^{\left(2\right) }.$ Then by the series $\left(5.1\right)$–$\left(5.3 \right)$ for $m = 2,$ we have

After comparing, we get

Then by substituting the above values to $\left(5.4\right),$ we get

and after rearranging, we get

Now by using the triangle inequality along with $\left(2.3\right)$ and $\left(2.4\right)$, we get the required result.

Theorem 11. If $f\in \mathcal{R}_{\mathcal{S}_{G}}^{\left(3\right) },$ then

Proof. Let $f\in \mathcal{R}_{\mathcal{S}_{G}}^{\left(3\right) }.$ Then by $\left(5.1\right)$–$\left(5.3\right)$ for $m = 3,$ we have

After comparing $\left(5.6\right)$ and $\left(5.7\right)$, we get

Then by substituting the above values to $\left(5.5\right)$, we get

Now by using the $\left(2.3\right)$ and $\left(2.4\right)$, we get the required result.

6.   Zalcman functional

One of the main conjectures in Geometric function theory, suggested by Lawrence Zalcman in 1960, is that the coefficients of class $\mathcal{S}$ satisfy the inequality,

Only the well-known Koebe function $k(z) = \frac{z}{(1-z)^{2}}$ and its rotations have equality in the above form. For the popular Fekete-Szego inequality, when $n = 2$, the equality holds. Many researchers have researched Zalcman functional in the literature ^[5,8,19].

Theorem 12. Let $f\in \mathcal{A}$ belong to $\mathcal{R}_{\mathcal{S}_{G}}$. Then

The result is sharp for the function $f_4$ defined by (3.7).

Proof. We use the Eqs (3.13) and (3.15) to get the Zalcman functional, and then we get

Using Lemma 4$,$ we can get the necessary result for the last expression.

7.   Krushkal inequality for the class $\mathcal{R}_{\mathcal{S}_{G}}$

In this section we will give a direct proof of the inequality

over the class $\mathcal{R}_{\mathcal{S}_{G}}$ for the choice of thinspace $n = 4,$ $p = 1,$ and for $n = 5,$ thinspace $p = 1.$ Krushkal introduced and proved this inequality for the whole class of univalent functions in ^[17].

Theorem 13. Let $f\in \mathcal{A}$ belong to $\mathcal{R}_{\mathcal{S}_{G}}$. Then

The result is sharp for the function $f_3$ defined by $(3.7)$.

Proof. From Eqs (3.12) and (3.14), we get

By applying (2.6) to the above, we get the required result.

Theorem 14. Let $f\in \mathcal{A}$ belong to $\mathcal{R}_{\mathcal{S}_{G}}$. Then

The result is sharp for the function $f_4$ defined by (3.7).

Proof. From Eqs (3.12) and (3.14), we get

By using Lemma 4$,$ we can get the necessary result for the last expression.

8.   Conclusions

In the present study, we have defined the class of bounded turning functions associated with modified sigmoid function. Also we have determined the sharp results for some coefficient functionals which play a very important role in the study of the geometric function theory. Furthermore, we have evaluated bounds of the third and fourth-order Hankel determinants for the 2-fold and 3-fold symmetric functions.

Recently, the usages of the quantum (or $q$-) calculus happens to provide another popular direction for researches in geometric function theory of complex analysis. This is evidenced by the recently-published survey-cum-expository review article by Srivastava ^[31]. Therefore the quantum (or $q$-) extensions of the results, which we have presented in this paper, are worthy of investigation.

Acknowledgements

The authors would like to express their gratitude to the editor and the reviewers for their valuable comments. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861).

Conflict of interest

The authors declare that they have no competing interests.