The Fekete-Szegö functional and the Hankel determinant for a certain class of analytic functions involving the Hohlov operator

1.   Introduction and preliminaries

Geometric function theory is one of the most exciting areas of research in complex analysis, with applications in a wide range of mathematical fields including mathematical physics. Due to its many uses in analytical solutions to issues such as those in electrostatics, aerodynamics and fluid mechanics, researchers in the field of complex analysis have been investigating various families of analytic (or holomorphic) functions.

Analytic functions such as $\psi(z)$ can be expressed in the Taylor-Maclaurin series expansion about the origin $z = 0$ as follows:

which can be normalized in the following way:

where

and the series expansion in (1.1) is convergent in the open unit disk $\mathbb{U}$. Let $\mathcal{A}$ denote a class of functions $f(z)$ that are analytic (or holomorphic) in $\mathbb{U}$, have the form (1.1) and are normalized by the constraints $f'(0)-1 = f(0) = 0$.

The class of functions $\varphi$ that are holomorphic in $\mathbb{U}$ and have the form

with

is denoted by $\mathcal{P}$.

In the geometric function theory of complex analysis, studies of the concept of convolution are crucial. Various new and interesting subclasses of holomorphic and univalent functions have been introduced and investigated through the use of the Hadamard product (or convolution) in the direction of well-known ideas such as the integral mean, Hankel determinant, subordination, partial sums, superordination inequalities and so on. The Hadamard product (or convolution) of $f$ and $g$, represented by $f\ast g$, is defined by

for functions $f$ and $g$ in $\mathcal{A}$ given by the following series:

The Gauss hypergeometric function $_2{F}_{1}(u, v, w;z)$ is defined as follows:

where $(\delta)_j$ signifies the Pochhammer symbol (or the shifted factorial) defined in terms of the Gamma function $\Gamma$, as follows:

Hohlov (see ^[19,20]) proposed and investigated a linear operator denoted by $\mathcal{L}_{u, v}^{w}$ and defined by $\mathcal{L}_{u, v}^{w}f:\mathcal{A}\longrightarrow\mathcal{A}$, with

The above-specified three-parameter family of operators unifies several other linear operators that have been introduced and explored previously when the parameters are appropriately chosen. The works in ^{[7,9,11,12,40,41,42,43,51,53,59,72]} provide special examples of this operator. For more details, see ^{[17,47,70,71]}. It should be remarked in passing that much more general convolution operators, such as the Dziok-Srivastava operator (see ^[14,15]) and the Srivastava-Wright operator (see ^[62]), have also been investigated rather extensively in the vast literature in geometric function theory of complex analysis.

The $n$th coefficient of a function belonging to the class $\mathcal{S}$ is well-known to be bounded by $n$, and the coefficient bounds provide information about the geometric properties of the function. For example, the $n$th coefficient of functions in the family $\mathcal{S}$ yields the growth and distortion properties of the function, whereas the second coefficient of functions in the family $\mathcal{S}$ yields the growth and distortion properties of the function itself. Studying a functional composed of combinations of the coefficients of the original function is a common issue in the geometric function theory of complex analysis. In most cases, the extremal value of the functional is required across a parameter. Some of our findings are related to the Fekete-Szegö functional, which is a key functional of this kind.

The famous problem solved by Fekete and Szegö ^[16] is to determine the greatest value of the coefficient functional $\Omega_{\sigma}(f): = |a_3-\sigma a_2^2|$ over the class $\mathcal{S}$ for each $\sigma\in [0, 1]$, which was demonstrated by using the Loewner chain technique. For various subclasses of the class of $\mathcal{S}$ and associated subclasses of functions in $\mathcal{A}$, several scholars solved the Fekete-Szegö issue. For example, see ^{[8,13,23,25,28,29,30,40,44]}, and so on. We refer to ^[68] for a thorough study on the Fekete-Szegöproblem of the traditional univalent function class $\mathcal{S}$. Srivastava et al. claimed that the inequality was sharp in ^[68]. However, Peng (see ^[54]) has demonstrated that the extremal function presented there for the situation of $\varrho\in (2/3, 1)$ is not sharp. Cho et al. ^[10] discovered the Fekete-Szegö inequalities for close-to-convex functions with regard to a certain convex function, which improves the bound explored in ^[68]. Using the Hankel or Toeplitz determinants is another approach to look at the sharp bound for the nonlinear functional. We recall that Noonan and Thomas ^[49] introduced and investigated the $q$th Hankel determinant of $f$ for $q\geqq 1$ and $n\geqq 1$ as follows:

Several writers, notably Noor ^[50], have investigated the determinant $\mathcal{H}_q(j)$, with topics ranging from the rate of development of $H_q(j)$ (as $j\longrightarrow\infty$) to the determinant of exact limits for particular subclasses of analytic functions in the unit disk $\mathbb{U}$ with specified values of $j$ and $q$. When $q = 2,$ $j = 1$, and $b_1 = 1$, the Hankel determinant is $H_2(1) = |b_3-b_2^2|$. The Hankel determinant simplifies to $H_2(2) = |b_2b_4-b^2_3|$ when $j = q = 2$. Fekete and Szegö ^[12] consider the Hankel determinant $H_2(1)$ and refer to $H_2(2)$ as the second Hankel determinant. If $f$ is univalent in $\mathbb{U}$, then the sharp upper inequality $H_2(1) = |b_3-b_2^2|\leqq1$ is known (see^[16]). Janteng et al. ^[21] obtained sharp bounds for the functional $H_2(2)$ for the function $f$ in the subclass $\mathcal{RT}$ of $\mathcal{S}$, which was introduced by MacGregor ^[37] and which consists of functions whose derivative has a positive real part. They demonstrated that $H_2(2) = |b_2b_4-b_3^2|\leqq 4/9$ for each $f\in \mathcal{RT}$. They also discovered the sharp second Hankel determinant for the classical subclasses of $\mathcal{S}$, namely, the classes $S^*$ and $\mathcal{K}$ of starlike and convex functions, respectively (see ^[22]). These two classes have bounds of $|b_2b_4-b^2_3|\leqq1/8$ and $|b_2b_4-b^2_3|\leqq1$. The Hankel determinants for starlike and convex functions with respect to symmetric points were recently discovered by Reddy and Krishna ^[57]. For functions belonging to subclasses of the Ma-Minda type starlike and convex functions, Lee et al. ^[34] found bounds for the second Hankel determinant.

Mishra and Gochhayat ^[41] found the sharp bound to the nonlinear functional $|b_2b_4-b^2_3|$ for the subclass of analytic functions given by

and defined as follows:

using the Owa-Srivastava operator in ^[53]. Similar coefficient constraints are found for a variety of analytic function subclasses that are constructed by using other appropriate linear operators (see, for example, ^{[1,32,45,46,74,75]}).

In the case when $q = 3$ and $j = 1$, the Hankel determinant, represented by $H_3(1)$, is given by

Clearly, we have

Babalola (see ^[5]) recently obtained the sharp upper bound of $H_3(1)$ for functions in the classes $\mathcal{S}^*$, $\mathcal{K}$ and $\mathcal{RT}$ classes.

Krishna et al. ^[31] defined $\mathcal{RT}(\alpha)$ as $\Re\big(h'(z)\big) > \alpha$ and found the bound on $H_3(1)$. Ayinla and Opoola ^[4] introduced the class defined by using the Sălăgean derivative operator as follows:

and obtained inequalities for the Fekete-Szegö functional and the second Hankel determinant. Additionally, Bansal et al. ^[6] and Raza and Malik ^[56] found the bound for $H_3(1)$ for a subclass of univalent functions. Gochhayat et al. ^[17] recently introduced the class $\mathcal{R}_{a, b}^c$ and obtained the sharp bounds of $H_2(2)$ and $H_3(1)$ in terms of the Gauss hypergeometric function by utilizing the Hohlov operator. See also ^{[2,3,26,27,33,48,55,61,65,69,73,76]} for some of the recent works on the third Hankel determinant and ^[66] for some developments on the fourth Hankel determinant.

Here, in this paper, we introduce a subclass of the normalized univalent function class $\mathcal{S}$ by using the Hohlov operator, as inspired by some of the above-mentioned researches.

Definition 1.1. A function $f(z)$ of the form (1.1) that is holomorphic and univalent in $\mathbb{U}$ is said to belong to the class $\mathcal{J}_{\mu}^{\phi}(u, v, w)$ if it satisfies the following geometric criterion:

where $z\in \mathbb{U}$, $0\leqq \mu\leqq 1$ and $\phi\in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Remark 1.1. Choosing $u = 2,$ $v = w = 1$ and $\mu = 0$ gives we get the class

This class $\mathcal{J}^{\phi}$ was introduced and studied by Noshiro ^[52].

Remark 1.2. Choosing $u = 2,$ $v = w = 1$ and $\mu = \phi = 0$ gives we get the class

This class $\mathcal{J}$ was introduced and studied by MacGregor ^[37].

Remark 1.3. Choosing $u = 2,$ $v = w = 1,$ $\phi = 0$ and $\mu = 1$ gives we get the class

This class $\mathcal{J}_1$ was introduced and studied by Hengartner and Schober ^[18].

Remark 1.4. Choosing $u = 2,$ $v = w = 1$ and $\mu = 1$ gives we get the class

This class $\mathcal{J}_1^{\phi}$ was introduced and studied by Royster and Ziegl ^[58].

Remark 1.5. Choosing $u = 2,$ $v = w = 1$ and $\phi = 0$ gives we get the class

This class $\mathcal{J}_{\mu}$ was introduced and studied by Kanas and Lecko ^[24].

Remark 1.6. Choosing $u = 2$ and $v = w = 1$ gives we get the class

This class $\mathcal{J}_{\mu}^{\phi}$ was introduced and studied by Lecko ^[35].

In this article, we establish the coefficient estimates, Fekete-Szegö type inequality, and the bounds for the second and the third Hankel determinants for functions belonging to the class $\mathcal{J}_{\mu}^{\phi}(u, v, w)$.

Lemma 1.1. (see ^[12]) Let $\varphi(z)\in \mathcal{P}$. Then

Lemma 1.2. (see ^[38]) Let $\varphi(z)\in \mathcal{P}$. Then

for $\upsilon\in \mathbb{R}$.

Lemma 1.3. (see ^[36]) Let $\varphi(z)\in\mathcal{P}$. Then

for some complex numbers $x$ and $z$ such that $|x|\leqq 1$ and $|z|\leqq 1$.

Theorem 1.1. Let $f(z)\in \mathcal{J}_{\mu}^{\phi}(u, v, w)$. Then

and

where $\mu\in[0, 1]$ and $\phi\in(-\frac{\pi}{2}, \frac{\pi}{2})$.

Proof. Consider the function $\vartheta(z)$ given by

Then, by (1.5), we can have

Also, from (1.11) and (1.12), we get

As a result, the right-hand side of (1.13) is given by

which implies that

Furthermore, from the left-hand side of (1.13), we have

Now, upon comparing the coefficients of $z, z^2, z^3, z^4$ and $z^5$ in (1.14) and (1.15), we get

and

The desired estimate is obtained by first applying the triangle inequality to (1.16) to (1.20) and then using Lemma 1.1. The proof of Theorem 1.1 is thus completed.

Theorem 1.2. Let $f(z)\in \mathcal{J}_{\mu}^{\phi}(u, v, w),$ $0\leqq\mu\leqq 1$ and $-\frac{\pi}{2} < \phi < \frac{\pi}{2}$. Then

where

for any real number $\sigma$.

Proof. By applying (1.16) and (1.17), we have

where

By Lemma 1.2 and for $\upsilon\leqq 0$, we get

and, for $\upsilon\leqq 0$ in (1.21), we have

Also, by applying Lemma 1.2, and for $0\leqq \upsilon \leqq 2$, we obtain

and for $0\leqq \upsilon \leqq 2$ in (1.21), we get

Next, for $\upsilon\geqq2$ in Lemma 1.2, we have

which gives

and, for $\upsilon\geqq 2$ in (1.21), we get

This completes the proof of Theorem 1.2.

Theorem 1.3. Let $f(z)\in \mathcal{J}_{\mu}^{\phi}(u, v, w),$ $0\leqq\mu\leqq 1$ and $-\frac{\pi}{2} < \phi < \frac{\pi}{2}$. Then

Proof. By applying (1.16) and (1.17), we have

Thus, by applying Lemma 1.2, we find that

Theorem 1.4. Let $f(z)\in \mathcal{J}_{\mu}^{\phi}(u, v, w),$ $0\leqq\mu\leqq1$ and $-\frac{\pi}{2} < \phi < \frac{\pi}{2}$. Then

Proof. From the equations (1.16) to (1.18), we have

Applying Lemma 1.3, and after some simplification, we find that

Let $r_{1} = r$ and recall that $|r_{1}|\leqq 2$. We may assume without restriction that $r\in[0, 2]$. Then, by using the triangle inequality, we get

Now, putting $\lambda = |x|\leqq 1$, we have

which implies that

Now, maximizing the function $G_1(r, \lambda)$ in the closed interval $0\leqq \lambda\leqq 1$, we obtain

for $0\leqq r \leqq 1$. Thus, clearly, $G_{1}(\lambda, r)$ is an increasing function. Hence, it has the maximum point at $\lambda = 1$ and we have

After simplifying and differentiating with respect to $r$, we have

By equating $G'(r)$ to zero and doing some simple calculations, we have the critical points at

The maximum point occurs at $r_1 = \sqrt{\mu^2+3}$, so by using (1.30), we get

Hence, we have

Theorem 1.5. Let $f(z)\in \mathcal{J}_{\mu}^{\phi}(u, v, w),$ $0\leqq\mu\leqq1$ and $-\frac{\pi}{2} < \phi < \frac{\pi}{2}$. Then

Proof. Applying the equations (1.16) to (1.18), we have

Applying Lemma 1.3, we obtain

Let $r_{1} = r$, assuming that $|r| = |r_{1}|\leqq 2$, so that without restriction, $r\in [0, 2]$, and by applying triangle inequality with $|x| = \lambda \leqq 1$ and $|z|\leqq 1$, we find that

By differentiating with respect to $\lambda$, we have

for $0\leqq \lambda \leqq 1$. Since $G_{2}'(\lambda, r) > 0$ for $0\leqq \lambda \leqq 1$, it means that $G_{2}(\lambda, r)$ is an increasing function with its maximum point at $\lambda = 1$. Hence, from (1.32), we have

After some simple calculations and simplification, we get

By differentiating $G(r)$ with with respect to $r$ and equating it to zero, the critical point will be seen to occur at

Hence, we have

Also, we have

From (1.34), we get

Hence, we have

Theorem 1.6. Let $f(z)\in \mathcal{J}_{\mu}^{\phi}(u, v, w),$ $0\leqq\mu\leqq 1$ and $-\frac{\pi}{2} < \phi < \frac{\pi}{2}$. Then

where

Proof. Taking it from (1.4), we have

Applying Theorems 1.1 as well as 1.3 to 1.5, and by using the triangle inequality, we have

2.   Concluding remarks and observations

Our present investigation was motivated by a number of recent developments on the Fekete-Szegö functional, the Hankel determinants of the third and the fourth kinds, and the associated Taylor-Maclaurin coefficient estimates and coefficient inequalities. Here, in this paper, we have introduced and systematically studied a new subclass of normalized analytic and univalent functions in the open unit disk $\mathbb{U}$, which satisfies the following geometric criterion:

where $z\in \mathbb{U}$, $0\leqq \mu\leqq 1$ and $\phi\in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, and which is associated with the Hohlov operator $\mathcal{L}_{u, v}^{w}$. For functions in this class, we have investigated several coefficient bounds, as well as upper estimates for the Fekete-Szegö functional and the Hankel determinant.

It should be remarked that, in many recent investigations dealing with some of the topics of our presentation in this paper, the basic or quantum (or $q$-) calculus was extensively used (see ^[39,60,67]).

We conclude this paper by recalling a recently-published survey-cum-expository review article in which Srivastava ^[63] explored the mathematical applications of the $q$-calculus, the fractional $q$-calculus and the fractional $q$-derivative operators in geometric function theory of complex analysis, especially in the study of Fekete-Szegö functional. Srivastava ^[63] also exposed the not-yet-widely-understood fact that the so-called $(p, q)$-variation of the classical $q$-calculus is, in fact, a rather trivial and inconsequential variation of the classical $q$-calculus, the additional parameter $p$ being redundant or superfluous (see, for details, [63,p. 340]; see also [64,pp. 1511-1512]).

Conflicts of interest

The authors declare that they have no conflicts of interest.