Problems concerning sharp coefficient functionals of bounded turning functions

1.   Introduction

By $\mathcal{A}$, we denote an analytic (regular) function's family and $g$ defined in the following region:

with $g\left(0\right) = 0 = g^{^{\prime }}\left(0\right) -1.$ Thus, every function $g$ of a family $\mathcal{A}$ is of the form: $\ \ $

Moreover, let $\mathcal{S}$ indicates a subfamily of $\mathcal{A}$, whose members are univalent in $\mathbf{D}$. Let $h_{1}, h_{2}\in \mathcal{A}$; we state that the function $h_{1}$ is subordinate to $h_{2}$ (written as $h_{1}\prec h_{2}$) if there exists a regular function $u$ that satisfies

such that $h_{1}\left(\varepsilon \right) = h_{2}\left(u\left(\varepsilon \right) \right)$ for $\varepsilon \in \ \mathbf{D}$. Moreover, if $h_{2}\in \mathcal{S}$, then the above conditions imply the follwoing:

In 1992, Ma and Minda ^[1] utilized the idea of subordination and initiated the family ${\Lambda }^{\ast }(\Omega)$ as follows:

where the image of $\Omega$ under $\mathbf{D}$ is a star-shaped functions satisfying that $\Omega (0) = 1$ and $\Omega ^{\prime }(0) > 0$. Also, they investigated various beautiful geometric results like growth, distortion and covering results. If we pick $\Omega (\varepsilon) = { (1+}${$\varepsilon$}${)}/{(1-} \varepsilon )$ categorically, then the family ${\Lambda }^{\ast }(\Omega)$ reduces to the family of functions whose image domain is star-shaped. For the numerous choices of $\Omega (\varepsilon)$ on the right hand side of $\left(1.5\right)$, we get various subfamilies of $\mathcal{S}$ whose image domains have some beautiful geometrical interpretations. Among them some are recorded as follows:

$1)$ If we take $\Omega ( \varepsilon ) = 1+\sin \varepsilon \mathbf{, }$ then we obtain the family ${\Lambda }_{\sin }^{\ast } = {\Lambda }^{\ast }\left(1+\sin \varepsilon \right),$ which is described by the functions bounded by the eight shaped region, and which was established and studied by Cho et al. ^[2].

$2)$ By considering a function $\Omega (\varepsilon) = 1+\varepsilon -\frac{1 }{3}\varepsilon ^{3},$ we get the recently investigated family ${\Lambda } _{nep}^{\ast } = {\Lambda }^{\ast }\left(1+\varepsilon -\frac{1}{3} \varepsilon ^{3}\right),$ introduced by Wani and Swaminathan ^[3]. The image of the function $\Omega (\varepsilon) = 1+\varepsilon -\frac{1}{3} \varepsilon ^{3}$ under an open unit disc is bounded by a nephroid shaped region.

$3)$ The family ${\Lambda }_{L}^{\ast } = {\Lambda }^{\ast }\left(\sqrt{ 1+\varepsilon }\right),$ with $\Omega (\varepsilon) = \sqrt{1+\varepsilon },$ was established by Sokól et al. ^[4].

$4)$ The family ${\Lambda }_{car}^{\ast } = {\Lambda }^{\ast }\left(1+\frac{4 }{3}\varepsilon +\frac{2}{3}\varepsilon ^{2}\right)$ was recently investigated by Sharma et al. ^[5], and the image of $\Omega (\varepsilon) = 1+\frac{4}{3}\varepsilon +\frac{2}{3}\varepsilon ^{2}$ is cardioid shape under an open unit disc.

$5)$ By choosing $\Omega (\varepsilon) = e^{\varepsilon }$, we get the family ${\Lambda }_{\exp }^{\ast } = {\Lambda }^{\ast }\left(e^{\varepsilon }\right),$ which was established in ^[6]. On the other side, if we pick $\Omega (\varepsilon) = \varepsilon +\sqrt{1+\varepsilon ^{2}},$ we get the family ${\Lambda }_{cre}^{\ast } = {\Lambda }^{\ast }\left(\varepsilon + \sqrt{1+\varepsilon ^{2}}\right),$ which maps $\mathbf{D}$ to a crescent shaped region and was given by Raina ^[7].

For more particular subfamilies of the family of starlike functions, see the articles ^[8,9,10].

Pommerenke ^[11,12] was the first to initiate the idea of a Hankel determinant $H_{q, n}\left(g\right)$ for a function $g\in \mathcal{S}$ of the form (1.2), where the parameters $q, n\in \mathbb{N} = \left \{ 1, 2, 3, \cdots \right \}$ are as follows:

For particular values, e.g., $q = 2$ and $n = 1,$ we get the Hankel determinant

And for $q = 2$ and $n = 2,$ in $\left(1.6\right)$ we get the second order Hankel determinant

For the third order Hankel determinant we take $q = 3$ and $n = 1,$ which yields the following form

The functional

is known as the Fekete-Szego functional. For numerous subfamilies of a regular function's family $\mathcal{A}$, the best possible value of the upper bound for $\left \vert H_{2, 1}\left(g\right) \right \vert$ has been evaluated by various authors (see ^[13,14,15]). Moreover, the second Hankel determinant and the extreme value has been studied and investigated by several authors from many different directions and perspectives. For instance, the readers may refer to see ^{[16,17,18,19,20,21]}. Furthermore, Babalola ^[22] described the Hankel determinant $H_{3, 1}\left(g\right)$ for several subfamilies of regular functions. For some recent works on the third-order Hankel determinant, we refer the reader to ^{[23,24,25,26,27]} and the references therein.

Recently, Allah et al. ^[28] defined the family of starlike functions based on the trigonometric hyperbolic tangent function as follows:

Motivated by the work mentioned above, we now introduce the subfamily of analytic functions:

In this paper, we evaluate first three initial sharp coefficient bounds, sharp Fekete-Szegö functional, the sharp second Hankel determinant, sharp third Hankel determinant and Krushkal inequality for functions belonging to this family. Further, the sharp initial four logarithmic coefficient bounds and the second Hankel determinant are investigated.

2.   A collection of lemmas

We next indicate by ${\mathcal{P}}$ the family of all holomorphic functions $p$ satisfying that $Re(p(\varepsilon)) > 0$, $\varepsilon \in \mathbf{D}$, and that also has series representation:

Lemma 2.1. ^[29] Suppose that $p\in {\mathcal{P}}$. Then, for $x$ and $\delta$ with $\left \vert x\right \vert \leq 1 and \left \vert \delta \right \vert \leq 1,$ it follows that

Lemma 2.2. If $p\in {\mathcal{P}}$, then the following estimations hold

and for $\eta \in \mathbb{C}$, we have

For the inequalities (2.4)–(2.6) see ^[30] and (2.7) is given in ^[31].

Lemma 2.3. ^[32] If $p\in {\mathcal{P}}$ and it has the form (2.1), then

where $\alpha _{1}, \alpha _{2}$ and $\alpha _{3}$ are real numbers.

Lemma 2.4. ^[33] Let $m_{1}, n_{1}, l_{1}$ and $r_{1}$ satisfy that $m_{1}, r_{1}\in \left(0, 1\right)$ and

If $h\in \mathcal{P}$ and it is of the form $\left(2.1\right)$, then

3.   Main results

Theorem 3.1. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then

Equalities of these inequalities are obtained for functions as follows:

respectively.

Proof. Let $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ then by the definitions of subordinations there exists a Schwarz function $u\left(\varepsilon \right)$ with the properties given in (1.3)$,$ such that

Let $p\in \mathcal{P};$ then, it can be written in terms of Schwarz functions as

Or

Now, from $\left(3.7\right),$ we have

And

Comparing $\left(3.9\right)$ and $\left(3.10\right),$ we get

Applying $\left(2.4\right)$ to $\left(3.11\right)$, we get

From $\left(3.12\right)$, and by using $\left(2.6\right)$, we have

Clearly, $H\left(p_{1}\right)$ is a decreasing function with the maximum attained at $p_{1} = 0;$ hence,

Applying Lemma 2.3 to $\left(3.13\right)$, we get

□

Theorem 3.2. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then

Equalities of this inequality can be obtained for the function $g_{2}$ defined in $\left(3.5\right)$ for $\left \vert \lambda \right \vert \leq \frac{4}{ 3}$ and for the function $g_{1}$ defined by $\left(3.4\right)$ for $\left \vert \lambda \right \vert \geq \frac{4}{3}.$

Proof. From $\left(3.11\right)$ and $\left(3.12\right)$, we get

Applying $\left(2.7\right)$ to the above equation we get the required results. □

Corollary 3.3. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then

The equality of this inequality can be obtained for the function $g_{2}$ defined in $\left(3.5\right)$.

Theorem 3.4. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then

The equality of this inequality can be obtained for the function $g_{3}$ defined in $\left(3.6\right)$.

Proof. From (3.11)–(3.13), we get

Applications of Lemma 2.3 lead us to the required results. □

Theorem 3.5. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then

The equality of this inequality can be obtained for the function $g_{2}$ defined in $\left(3.5\right)$.

Proof. From (3.11)–(3.13), we get

Applying $\left(2.2\right)$ and $\left(2.3\right)$ to write $p_{2}$ and $p_{3}$ in terms of $p_{1} = p\in \left[0, 2\right]$, we get

Implementing the triangle inequality and using $\left \vert \delta \right \vert \leq 1$ and $\left \vert x\right \vert = y\leq 1,$ we have

Now, differentiating partially with respect to $y$, we get

Obviously, $\frac{\partial R\left(p, y\right) }{\partial y} > 0$ is an increasing function, so it has a maximum at $y = 1$; thus

Now, differentiating with respect to $p$, we get

Clearly, $G^{^{\prime }}\left(p, 1\right) = 0$ only has the root $p = 0\in \left[0, 2\right]$. Hence, $G^{^{\prime \prime }}\left(p, 1\right) < 0$ at $p = 0$, so the maximum value is attained, that is

□

Theorem 3.6. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then,

The equality of this inequality can be obtained for the function given by $\left(3.6\right)$.

Proof. We know that

Setting the values of (3.11)–(3.14), and putting $p_{1} = p$, we get

Now, supposing that $p_{1} = p$ and $t = \left(4-p_{1}^{2}\right)$ in (2.2), (2.3) and (3.18), we get

by putting the above values of $p_{2}, \; p_{3}$ and $p_{4}$ in $\left(3.18\right)$, we get

where $t = \left(4-p^{2}\right)$; then, we have

where

and

Now, let $\left \vert \delta \right \vert = y$, $\left \vert x\right \vert = x\; and\; \left \vert \rho \right \vert \leq 1.$ Then we have

where

Then,

Now, we need to attain the maxima of $H\left(p, x, y\right)$ in the interior of the closed cuboid $\Delta :\left[0, 2\right] \times \left[0, 1\right] \times \left[0, 1\right]$. In order to do this, we have to maximize $H\left(p, x, y\right)$ on all six internal faces and at the 12 edges of the cuboid $\Delta.$

1) First, we will check for the maximum of the function $H$ in the interior of $\Delta.$ Let $\left(p, x, y\right) \in \left(0, 2\right) \times \left(0, 1\right) \times \left(0, 1\right)$. Then differentiating $\left(3.22\right)$ partially with rspect to $y$, we get

by setting $\frac{\partial H\left(p, x, y\right) }{\partial y} = 0,$ we get

which is possible only if

and

Let $g(x) = \frac{4\left(15-x\right) }{27-x}$; it follows that

This shows that $g(x)$ is a decreasing function. Hence, $p^{2} > \frac{28}{13}$; a simple calculation show that the above inequality $\left(3.23 \right)$ does not hold true for the given values of $x\in \left(0, 1\right)$, so the function $L$ is no critical point in the interior of the cuboid.

2) We will check the maximum value on the six faces. First on $p = 0,$ we have

Now,

Hence, $H(0, x, y)$ is no optimal point in $\left(0, 1\right) \times \left(0, 1\right)$.

At $p = 2$,

At $x = 0$,

and

by taking $\frac{\partial m_{2}(p, y)}{\partial y} = 0$, we get that $y = \frac{5p^{3}}{ \left(54p^{2}-120\right) } = :y_{0}.$ For the provided range of $y,$ $p > p_{0} = 1.490711984999.$

Further,

Taking $\frac{\partial m_{2}}{\partial p} = 0$, we get

By putting $y = \frac{5p^{3}}{\left(54p^{2}-120\right) }$, we get

When solving for $p\in (0, 2)$, the solution is $p = 1.2751$; upon checking we conclude that there is no optimal solution for $H(p, 0, y) = m_{2}(p, y)\ $in $\left(0, 2\right) \times \left(0, 1\right)$.

At $x = 1$, we have

and

Now, for the critical point put $\frac{\partial m_{3}}{\partial p} = 0$; we obtain the solution to be $p = 1.1117$, at which $m_{3}$ yields the maximum value, which is

At $y = 0$, we obtain

From the computation it is clear that the system of equations has no solution for $\left(0, 2\right) \times \left(0, 1\right)$.

At $y = 1$,

Computation indicates that the solution for the system of equations associated with $\frac{ \partial m_{5}}{\partial x} = 0$ and $\frac{\partial m_{5}}{\partial p} = 0$ in the region $\left(0, 2\right) \times \left(0, 1\right) \ $does not exist.

3) Now, we will check the maximum of $H(p, x, y)$ at the 12 edges of cuboid.

By putting $x = 0$ and $y = 0$, we have

For the critical points put $\frac{\partial m_{6}}{\partial p} = 0$; its critical point is $p = 1.5059$, at which the maximum value of $m_{6}(p)$ is

By putting $x = 0$ and $y = 1$,

For critical point

As $\frac{\partial m_{7}}{\partial p} < 0$, for $p\in \left[0, 2\right], \ \frac{\partial m_{7}}{\partial p}$ is a decreasing function that achieves its maximum value at $p = 0$, which is

For $x = 0$ and $p = 0,$ we have

As $m_{8}(y)$ is an increasing function, its maximum occurs at $y = 1$, that is

Now, the equation

is free from $y$. So,

Then, $m_{9}(p) = -\frac{7}{2}p^{6}-144p^{4}+372p^{2}+1280$ has its maximum value at $p = 1.1117,$ which corresponds to

For $p = 0$ and $x = 1$,

For $p = 2$, all of the terms of $H(p, x, y)$ are free from $p, x$ and $y$. So,

At $p = 0$ and $y = 0$,

To find the critical point at which $m_{11}(x)$ gives the maximum value, put $\frac{\partial m_{11}}{\partial x} = 0$ which gives $x_{0} = 1.5$ at which the maximum value of $m_{11}(x)$ is given by

For $p = 0$ and $y = 1$,

As $\frac{\partial m_{12}}{\partial x} = -576x^{3}+3840x^{2}-4032x < 0,$ for $x\in \left(0, 1\right)$ which shows that it is decreasing function then its maximum occurs at $x = 0$, that is

Here from all of the calculations, we conclude that

For $\triangle :\left[0, 2\right] \times \left[0, 1\right] \times \left[0, 1\right]$, it follows that

□

4.   Krushkal inequality

In this section for the particular choice of $n = 4$ and $p = 1,$ we will give a direct proof of the inequality

over the family $\mathcal{R}_{\tanh }.$ For the whole family of univalent functions Krushkal ^[34] introduced and proved this inequality.

Theorem 4.1. Let $g\in \mathcal{A}$ belong to $\mathcal{R}_{\tanh }$. Then,

The equality associated with this inequality can be obtained for the function defined by (3.4).

Proof. From Eqs (3.11) and (3.13), we get

By applying Lemma 2.3 to the above equation, we get the required result. □

5.   Logarithmic coefficients for the family $\mathcal{R}_{\tanh }$

The logarithmic coefficients of $g\in \mathcal{S}$ denoted by $\gamma _{n} = \gamma _{n}\left(g\right),$ are defined by the following series expansion:

For the functions $g$ given by $\left(1.2\right)$, the logarithmic coefficients are as follows

Theorem 5.1. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then

The equality associated with these inequalities can be obtained for the function

Proof. Now from $\left(5.1\right)$ to $\left(5.5\right)$ and $\left(3.11\right)$ to $\left(3.14\right)$, we get

Applying $\left(2.4\right)$ to $\left(5.7\right)$, we get

From $\left(5.8\right)$ and by using $\left(2.5\right)$, we get

Applying Lemma 2.3 to $\left(5.9\right)$, we get

Also, applying Lemma 2.4 to $\left(5.10\right)$, we get

Proof of sharpness. Since

it follows that these inequalities are obtained for the functions $g_{n}\left(\varepsilon \right)$ for $n = 1, 2, 3, 4$ as defined in $\left(5.6 \right)$. □

Theorem 5.2. If $g\left(\varepsilon \right) \in \mathcal{R}_{\tanh }$ and it has the form given by (1.2), then

The equality in this inequality can be obtained for the function $g_{2}$ in $\left(5.6\right)$.

Proof. From (5.7)–(5.9), we have

Applying $\left(2.2\right)$ and $\left(2.3\right)$ to write $p_{2}$ and $p_{3}$ in terms of $p_{1} = p\in \left[0, 2\right]$, we get

By the triangle inequality, and by using $\left \vert \delta \right \vert \leq 1$ and $\left \vert x\right \vert = y\leq 1,$ we get

Now, differentiating Eq $\left(5.11\right)$ partially with respect to $y,$ we have

It is easy to observe that $\frac{\partial G\left(p, y\right) }{\partial y} \geq 0$ in the interval $\left[0, 1\right]$, so the maximum is attained at $y = 1$; thus

Now, differentiating with respect to $p,$ we get

Clearly, $G^{^{\prime }}\left(p, 1\right) = 0,$ has three roots namely $0,$ $\pm 4$, and the only root that lies in the interval $\left[0, 2\right]$ is $0,$ so

Thus, $G^{^{\prime \prime }}\left(0, 1\right) \leq 0,$ so the function has its maximum at $p = 0,$ that is

□

6.   Conclusions

Recently, the investigations of the Hankel determinant have attracted the attention of many researchers due to their applications in many diverse areas of mathematics and other sciences. In this paper, we have defined a new subfamily of analytic functions connected with the hyperbolic tangent function with bounded boundary rotation. We have also investigated the upper bound of the third Hankel determinant for this newly defined family of functions. On the other hand, we have obtained the Krushkal inequality and investigated the first four initial sharp bounds of the logarithmic coefficients and the sharp second Hankel determinant of the logarithmic coefficients for this defined family of functions.

Here, we want to remark on the fact that one can extend the suggested results investigated in this article to some other subclasses of analytic functions, and also that those interested scholars can use the $D_{q}$ derivative operator and generalize the work presented here.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgment

1) This work was supported by a National Research Foundation of Korea (NRF) grant funded by the South Korea government (MSIT) (No. NRF-2022R1A2C2004874).

2) This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry and Energy (MOTIE) of the Republic of Korea (No. 20214000000280).

Conflicts of Interest

The authors declare that they have no competing interest.