Sharp coefficient problems of functions with bounded turning subordinated to the domain of cosine hyperbolic function

1.   Introduction

Complex analysis is one of the major disciplines nowadays due to its numerous applications in mathematical science and other fields. Geometric function theory is an intriguing topic of complex analysis that involves the geometrical characteristics of analytic functions. This area is crucial to applied mathematics, particularly in fields such as engineering, electronics, nonlinear integrable system theory, fluid dynamics, modern mathematical physics, and partial differential equation theory. The key problem that led to the rapid emergence of geometric function theory is the Bieberbach conjecture. It is about the coefficient bounds for functions belonging to the class $\mathcal{S}$ of univalent functions. This conjecture states that if $g\in \mathcal{S}$ with the Taylor-Maclaurin series expansion of the form

where $\mathbb{D}: = \left \{ z\in \mathbb{C}:\left \vert z\right \vert < 1\right \},$ then $\left \vert d_{k}\right \vert \leq k$ for all $k\geq 2.$ Let $\mathcal{A}$ be the class of analytic functions with the series representation given in $\left(1.1\right).$ The set $\mathcal{S}$ is a subclass of $\mathcal{A}$ which was first taken into account by Koebe in 1907. The above famous conjecture was proposed by Bieberbach ^[1] in 1916. He proved this for $k = 2$, and subsequent researchers, including L öwner ^[2], Garabedian and Schiffer ^[3], Pederson and Schiffer ^[4], and Pederson ^[5], confirmed it for $k = 3,$ $4,$ $5,$ and $6$, respectively. However, settling the conjecture for $k\geq 7$ remained elusive until 1985 when de Branges ^[6] used hypergeometric functions to prove it for every $k\geq 2$. Lawrence Zalcman proposed the inequality $\left \vert d_{k}^{2}-d_{2k-1}\right \vert \leq \left(k-1\right) ^{2}$ with $k\geq 2$ for $g\in \mathcal{S}$ in the late 1960's as a way of establishing the Bieberbach conjecture. Due to this, a number of articles ^[7,8,9] have been published on the Zalcman hypothesis and its generalized form $\left \vert \lambda d_{k}^{2}-d_{2k-1}\right \vert \leq \lambda k^{2}-2k+1$ with $\lambda \geq 0$ for different subclasses of the class $\mathcal{S}$. This conjecture has remained unsolved for a long time. Krushkal ^[10] established this hypothesis for $k\leq 6$. In an attempt to solve the Zalcman conjecture, Krushkal ^[11] investigated the inequality $\left \vert d_{k}^{l}-d_{2}^{l\left(k-1\right) }\right \vert \leq 2^{l\left(k-1\right) }-k^{l}$ with $k, l\geq 2$ for $g\in \mathcal{S}$. A broader Zalcman hypothesis for $g\in \mathcal{S}$ was proposed by Ma ^[12] later, in 1999, and is given by

He only proved it for a subclass of $\mathcal{S}$. The challenge is still open for the class $\mathcal{S}$.

Now, we recall the definition of subordination, which actually provides a relationship between analytic functions. We write $g_{1}\prec g_{2}$ to illustrate that $g_{1}$ is subordinate to $g_{2}.$ It is explained that for two given functions $g_{1}, g_{2}\in \mathcal{A}$, a Schwarz function $w$ exists such that $g_{1}(z) = g_{2}\left(w(z)\right)$ for $z\in \mathbb{D}.$ Once $g_{2}$ is univalent in $\mathbb{D},$ then this relation is equivalent to saying that

if and only if

The three classic subclasses of univalent functions are $\mathcal{C},$ $\mathcal{S}^{\ast }$ and $\mathcal{K},$ of which their functions are known respectively as convex functions, starlike functions and close-to-convex functions. These classes are defined by

and

for some $h\in \mathcal{S}^{\ast }$. Taking $h\left(z\right) = z,$ the class $\mathcal{K}$ reduces to the class $\mathcal{BT}$ of bounded turning functions. Further, replacing $\frac{1+z}{1-z}$ by some other special functions, various interesting subfamilies of the class $\mathcal{S}$ were studied; interested readers may refer to ^{[13,14,15,16,17,18]}.

The determinant $D_{\lambda, n}\left(g\right)$, where $n, \lambda \in \mathbb{N} = \{1, 2, \ldots \}$, is known as the Hankel determinant and was presented by Pommerenke ^[19,20]. It is formed by the coefficients of the function $g\in \mathcal{S}$ and is defined by

Hankel matrices are used in both pure mathematics and technological applications, including the theory of Markov processes, the theory of non-stationary signals in the Hamburger moment problem, and many other topics, see for example ^{[21,22,23,24]}. There are relatively few publications on the bounds of the Hankel determinant for functions in the general class $\mathcal{S}$. The best estimate for $g\in \mathcal{S}$ was determined by Hayman in ^[25], which asserted that $\left \vert D_{2, n}\left(g\right) \right \vert \leq \left \vert \eta \right \vert$, where $\eta$ is a constant. Additionally, for $g\in \mathcal{S}$, it was shown in ^[26] that the second-order Hankel determinant $\left \vert D_{2, 2}\left(g\right) \right \vert \leq \eta$ for $0\leq \eta \leq 11/3$. The two determinants $D_{2, 1}\left(g\right)$ and $D_{2, 2}\left(g\right)$ have been extensively studied in the literature for various subfamilies of univalent functions. The works ^{[27,28,29,30,31]}, in which the sharp bounds of the second-order Hankel determinant for some subclasses of $\mathcal{S}$ are determined, are particularly noteworthy.

In comparison to the second-order Hankel determinant, the sharp bound of the third-order Hankel determinant $D_{3, 1}\left(g\right)$ for certain analytic univalent functions is much harder to find. The investigation on $D_{3, 1}\left(g\right)$ for $\mathcal{S}$ was initiated by Babalola ^[32] in 2010. The exact bounds of this determinant were proved recently for the classes $\mathcal{C},$ $\mathcal{S}^{\ast },$ and $\mathcal{ BT}$ in ^[33], ^[34], and ^[35], respectively. These bounds are given by

By employing similar techniques, Khalil Ullah et al. ^[36] and Lecko et al. ^[37] derived the sharp bounds for $\left \vert D_{3, 1}\left(g\right) \right \vert$ when considering functions belonging to the families $\mathcal{S}_{\tanh }^{\ast }$ and $\mathcal{S}^{\ast }\left(1/2\right)$, respectively. Additionally, the works ^{[38,39,40,41,42,43]} proved the sharp bounds for the same third-order Hankel determinant in various novel subfamilies of analytic univalent functions.

Let us consider the two function classes defined respectively by

and

These classes were introduced and studied by Mundalia et al. ^[44] and Ghaffar et al. ^[45], respectively. In this paper, we improved the bound of the third-order Hankel determinant $\left \vert D_{3, 1}\left(g\right) \right \vert$, which was determined by Ghaffar et al. and published recently in AIMS Mathematics ^[45]. Furthermore, we obtain the sharp estimates of the Fekete-Szegö, Krushkal, and Zalcman functionals with logarithmic coefficients as entries.

2.   A set of Lemmas

In the theory of univalent functions, the Carathéodory functions are well studied. They are analytic in $\mathbb{D}$ with positive real part and take series representations of the form

We denote by $\mathcal{P}$ the set of these functions.

To prove the main theorems, we need the following lemmas.

Lemma 2.1 ^[46] Let $p\in \mathcal{P}$ be the form of (2.1) with $\tau_{1}\geq0$. Then

for some $\beta, \eta, \kappa \in \overline{\mathbb{D}}: = \left \{z\in\mathbb{ C}:\left \vert z\right \vert \leq1\right \}$.

Lemma 2.2 ^[47] If $p\in \mathcal{P}$ is of the form (2.1) and $\vartheta \in \mathbb{C}$, we have

for all $1\leq k\leq n-1$.

Lemma 2.3 ^[48] Let $\mu$, $\lambda$, $\zeta$, and $\varsigma$ satisfy $\lambda \in \left(0, 1\right)$, $\zeta \in \left(0, 1\right)$ and

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

Lemma 2.4 ^[49] Suppose that $p\in \mathcal{P}$ is provided by (2.1). If $R\in \left[0, 1\right]$ and $R\left(2R-1\right) \leq S\leq R$, then we have

3.   Coefficient results for the class $\mathcal{BT}_{\cosh }$

Theorem 3.1 If $g\in \mathcal{BT}_{\cosh }$ is of the form (1.1), then

This inequality is sharp.

Proof. Let $g\in \mathcal{BT}_{\cosh }.$ Then $g$ can be easily expressed by using the Schwarz function as

If $p\in \mathcal{P},$ then we can write it in the form of

It follows that

From (1.1), we have

Using the series expansion of (3.1) with simple calculation, we get

Comparing the coefficients in (3.2) and (3.3), we obtain

Employing (3.4), (3.6) and (3.7), we may write

where

These constants satisfy $\lambda \in \left(0, 1\right)$, $\zeta \in \left(0, 1\right)$, and

Hence, by Lemma 2.3 and (3.8), we deduce that

This required result is sharp and determined by

where we choose the branch of the square root function so that

□

Theorem 3.2 If $g\in \mathcal{BT}_{\cosh }$ is of the form of (1.1), then

This inequality is sharp.

Proof. From (3.5) and (3.7), we obtain

where

These constants satisfy $\lambda \in \left(0, 1\right)$, $\zeta \in \left(0, 1\right)$, and

Hence, by Lemma 2.3 and (3.11), we deduce that

This required outcome is sharp for the function $g_{\mathfrak{4}}$ given in (3.9). □

Theorem 3.3 If $g\in \mathcal{BT}_{\cosh }$ is of the form of (1.1), then

This inequality is sharp.

Proof. Using (3.4) and (3.6), we have

Let $R = \frac{11}{24}$ and $S = \frac{8}{45}$. It is clear that

All the conditions of Lemma 2.4 are satisfied, and thus we have

This result is the best possible and equality is attained by

where we choose the branch of the square root function so that

□

Theorem 3.4 If $g\in \mathcal{BT}_{\cosh }$ is of the form of (1.1), then

This inequality is sharp.

Proof. From (3.4) and (3.7), we obtain

where

These constants satisfy $\lambda \in \left(0, 1\right)$, $\zeta \in \left(0, 1\right)$, and

Hence, by Lemma 2.3 and (3.14), we deduce that

This required inequality is sharp for the function $g_{\mathfrak{4}}$ given in (3.9). □

Theorem 3.5 If $g\in \mathcal{BT}_{\cosh }$ has the form of (1.1), then

This inequality is sharp.

Proof. From the definition, we know

Let $g\in \mathcal{BT}_{\cosh }$ and $g_{\theta}(z) = e^{-i\theta}g\left(e^{i \theta}z\right)$ with $\theta \in \mathbb{R}$. It is noted that $g_{\theta}^{\prime}(z) = g^{\prime}\left(e^{i\theta}z\right)$ and thus $g_{\theta}\in \mathcal{BT}_{\cosh }$ for all $\theta \in \mathbb{R}$. Since $\left \vert D_{3, 1}\left(g_{\theta}\right)\right \vert = \left \vert D_{3, 1}\left(g\right)\right \vert$, we may choose the coefficient $d_{2}$ of $g$ to be a non-negative real number when estimating the functional $\left \vert D_{3, 1}\left(g\right)\right \vert$. Then, since $d_{2} = \frac{1}{8}\tau_{1}$ and $\tau_{1}$ is a coefficient of a function in $\mathcal{P}$, it follows that $\tau_{1}$ is real and $\tau_{1} = \tau\in[0, 2]$. Putting the estimations of $d_{i}$'s from (3.4), (3.5), (3.6), and (3.7) into $D_{3, 1}\left(g\right)$ with $\tau _{1} = \tau$, we have

Let $r = 4-\tau ^{2}.$ Then, by (2.2), (2.3), and (2.4), we obtain

It is seen that we can write $D_{3, 1}\left(g\right)$ in the form of

where $\beta, \eta, \kappa \in \overline{\mathbb{D}}$, and

By using $\left \vert \beta \right \vert = \varkappa, \left \vert \eta \right \vert = y$ and utilizing the fact $\left \vert \kappa \right \vert \leq 1,$ we obtain

where

with

Now, we have to maximize $M$ in the closed cuboid $\Upsilon : = \left[0, 2 \right] \times \left[0, 1\right] \times \left[0, 1\right].$

In light of $\left(\tau, \varkappa \right)\in \left[0, 2\right]\times \left[0, 1 \right]$, we observe that

Taking $g_{i}\left(\tau, \varkappa \right) = m_{i}\left(\tau, \varkappa \right)$ for $i = 1, 2, 4$ and

it is not hard to see that $M\left(\tau, \varkappa, y\right)\leq G\left(\tau, \varkappa, y\right)$ in the cuboid $\Upsilon$. In the following, we aim to find the maximum value of $G$ in $\Upsilon$.

By partially differentiating $G$ with respect to $y$, we have

In view of $g_{2}\left(\tau, \varkappa \right)\geq0$ and

on $[0, 2]\times[0, 1]$, we have $\frac{\partial G}{\partial y}\geq0$ for all $y\in[0, 1]$. It follows that

where

where

Then the problem reduces to finding the maximum value of $Q$ on $[0, 2]\times[0, 1]$. By noting that $q_{4}(\tau)\leq0$ for all $\tau \in[0, 2]$, we obtain that

Setting $\tau = 0,$ we obtain

for all $\varkappa \in[0, 1]$. Setting $\tau = 2,$ we get

It is left to consider the case of $\tau \in(0, 2)$. For the system of equations

with $\left(\tau, \varkappa\right)\in\left(0, 2\right) \times \left(0, 1\right),$ a numerical computation indicates that all the real approximate solutions are listed as $(1.6125, -1.0547)$, $(158.7578, -0.6573)$, $(2.0982, -0.4927)$, $(2.1274, 0.3361)$, $(1.0709, 0.9834)$, $(0, 0)$, and $(2, 2.1500)$. Thus, the only critical point of $W$ that lies in $(0, 2)\times(0, 1)$ is about $(1.0709, 0.9834)$. For this point, we have $W(1.0709, 0.9834)\approx1.9621\times 10^{8}$.

Thus, from above cases, we conclude that

on $\left[0, 2\right] \times \left[0, 1\right] \times \left[0, 1\right]$. From $\left(3.16\right)$ we get that

If $g\in \mathcal{BT}_{\cosh },$ then the sharp bound for this Hankel determinant is determined by

with an extremal function $g_{\mathfrak{3}}$ given in (3.12). □

4.   Logarithmic coefficient problems

The logarithmic coefficients $\xi _{k}$ of $g\in \mathcal{S}$ are given by

These coefficients contribute significantly in many estimations to the theory of univalent functions. In 1985, de Branges ^[6] completed the proof of the Milin conjecture ^[50], which asserted that for all positive integers $k\geq 1,$

and equality holds if and only if $g$ takes the form $z/\left(1-e^{i\varphi }z\right) ^{2}$ for some $\varphi \in \mathbb{R}.$ This inequality leads to the famous Bieberbach–Robertson–Milin conjectures. In 2005, Kayumov ^[51] was able to solve the Brennans conjecture for conformal mappings by considering the logarithmic coefficients. For some recent works on the study of logarithmic coefficients, see, for example, ^{[52,53,54,55,56]}.

If $g\in \mathcal{S}$ is in the form of (1.1), then its logarithmic coefficients are given by

Plugging (3.4), (3.5), (3.6), and (3.7) into (4.1), (4.2), (4.3), and (4.4), we get

Define

It is observed that $D_{2, 1}\left(G_{g}/2\right)$ resembles the well-known functional $D_{2, 1}(g) = d_{1}d_{3}-d_{2}^{2}$ over the class $\mathcal{S}$ or its subclasses.

Theorem 4.1 If $g\in \mathcal{BT}_{\cosh }$ is of the form $\left(1.1\right),$ then

This inequality is sharp.

Proof. By employing (4.5) and (4.6), we may write

An application of Lemma 2.2 leads to

The bound $\frac{1}{12}$ is achieved by the function $g_{\mathfrak{2}}$ given as

The bound $\frac{\left \vert 9 \vartheta +5\right \vert}{576}$ for $\vartheta \in \mathbb{C}$ is attained by the function $g_{\mathfrak{1}}$ given as

Here, we choose the branch of the square root function so that

and

□

Substituting $\vartheta = 1,$ we deduce the corollary stated below.

Corollary 4.1 If the function $g\in \mathcal{BT} _{\cosh }$ has the form of (1.1), then

This bound is achieved by the function $g_{\mathfrak{2}}$ given in (4.11).

Theorem 4.2 If $g\in \mathcal{BT}_{\cosh }$ has the form of (1.1), then

This inequality is sharp.

Proof. Using (4.5), (4.6), and (4.7), we have

Let $R = \frac{7}{12}$ and $S = \frac{1969}{5760}.$ It is clear that

By Lemma 2.4, we obtain

The equality is attained by the function $g_{\mathfrak{3}}$ given in (3.12). □

Theorem 4.3 Let $g\in \mathcal{BT}_{\cosh }$ have the series expansion (1.1). Then

The inequality is sharp.

Proof. From (4.5), (4.7), and (4.8), we have

where

These constants satisfy $\lambda \in \left(0, 1\right)$, $\zeta \in \left(0, 1\right)$, and

Hence, by Lemma 2.3 and (4.15), we deduce that

This equality is achieved by the function $g_{\mathfrak{4}}$ given in (3.9). □

Theorem 4.4 Let $g\in \mathcal{BT}_{\cosh }$ be in the form of (1.1). Then

This inequality is sharp.

Proof. From (4.6) and (4.8), we obtain

where

These constants satisfy $\lambda \in \left(0, 1\right)$, $\zeta \in \left(0, 1\right)$, and

Hence, by Lemma 2.3 and (4.16), we deduce that

The required inequality is sharp and determined by the function $g_{\mathfrak{4}}$ given in (3.9). □

Theorem 4.5 If $g\in \mathcal{BT}_{\cosh }$ has the form of (1.1), then

This inequality is sharp.

Proof. Using (4.5) and (4.7), we have

Let $R = \frac{13}{24}$ and $S = \frac{553}{1920}$. It is clear that

By Lemma 2.4, it follows that

This result is the best possible and the extremal function is $g_{\mathfrak{3}}$ as given in (3.12). □

Theorem 4.6 If $g\in \mathcal{BT}_{\cosh }$ is of the form (1.1), then

This inequality is sharp.

Proof. From (4.5) and (4.8), we obtain

where

These constants satisfy $\lambda \in \left(0, 1\right)$, $\zeta \in \left(0, 1\right)$, and

Hence, by Lemma 2.3 and (4.17), we deduce that

This required inequality is sharp and determined by using (4.1), (4.4) and (3.9). □

Theorem 4.7 If $g\in \mathcal{BT}_{\cosh }$ has the form of (1.1), then

This inequality is sharp.

Proof. Suppose that $g\in \mathcal{BT}_{\cosh }$ and $g_{\theta}(z) = e^{-i\theta}g \left(e^{i\theta}z\right)$ with $\theta \in \mathbb{R}$. Since $\left \vert D_{2, 2}\left(G_{g_{\theta}}/2\right)\right \vert = \left \vert D_{2, 2}\left(G_{g}/2\right)\right \vert$ for all $\theta \in \mathbb{R}$, we still assume that $\tau_{1} = \tau \in[0, 2]$. Putting (4.6), (4.7), and (4.8) into (4.10) with $\tau _{1} = \tau,$ we obtain

Let $r = 4-\tau ^{2}.$ Then, by (2.2), (2.3), and (2.4), we obtain

It is observed that we can write $D_{2, 2}\left(G_{g}/2\right)$ in the form of

where $\beta, \eta, \kappa \in \overline{\mathbb{D}},$ and

Now, by using $\left \vert \beta \right \vert = \varkappa, \left \vert \eta \right \vert = y$ and utilizing the fact $\left \vert \kappa \right \vert \leq 1,$ we get

where

with

Now, we have to maximize $\Lambda$ in the closed cuboid $\Upsilon$.

In view of

for all $\left(\tau, \varkappa \right)\in[0, 2]\times[0, 1]$, by setting $u_{i}\left(\tau, \varkappa \right) = t_{i}\left(\tau, \varkappa \right)(i = 1, 2, 4)$ and

it is not hard to see that $\Lambda \left(\tau, \varkappa, y\right)\leq \Theta \left(\tau, \varkappa, y\right)$ on $\Upsilon$. In the following, we aim to discuss the maximum value of $\Theta$ on $\Upsilon$.

By partially differentiating $\Theta$ with respect to $y$, we get

Because $u_{2}\left(\tau, \varkappa \right)\geq0$ and

on $[0, 2]\times[0, 1]$, we have $\frac{\partial \Theta}{\partial y}\geq0$ for all $y\in[0, 1]$. Hence, we obtain

where

where

Taking the fact of $v_{4}(\tau)\leq0$ for all $\tau \in[0, 2]$, we obtain

Now, we need to find the maximum value of $K$ on $[0, 2]\times[0, 1]$. For $\tau = 0$, we have

for all $\varkappa \in[0, 1]$. If $\tau = 2$, it is calculated that $K(2, \varkappa)\equiv170944$ with $\varkappa \in[0, 1]$. Hence, it is left to discuss the case of $(\tau, \varkappa)\in(0, 2)\times(0, 1)$.

For the system of equations

with $\left(\tau, \varkappa \right)\in(0, 2)\times(0, 1)$, a numerical computation indicates that the approximate solutions are $(0, 0)$, $(1.7758, -1.2237)$, $(2.0405, -0.4933)$, $(52.7638, 0.2085)$, $(-212.4757, 0.2665)$, $(-2.0293, 0.3246)$, $(-1.5447, 1.0839)$, $(5.1393, 2.5129)$, $(2, 2.1720)$, and $(-2, -1.1906)$. It is found that there are no critical points of $K$ that lie in $(0, 2)\times(0, 1)$.

From the above cases we conclude that

on $\left[0, 2\right] \times \left[0, 1\right] \times \left[0, 1\right]$. Using (4.18), we have

If $g\in \mathcal{BT}_{\cosh },$ then the sharp bound for this second-order Hankel determinant is determined by using (4.2), (4.3), (4.4), and (3.12). □

Use of AI tools declaration

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

Acknowledgments

The authors would like to extend their sincere-appreciation to the Researchers Supporting Project number (RSPD2024R802) King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

Prof. Dr. Nak Eun Cho is the Guest Editor of special issue “Geometric Function Theory and Special Functions” for AIMS Mathematics. Prof. Dr. Nak Eun Cho was not involved in the editorial review and the decision to publish this article.

The authors declare that they have no conflicts of interest.