The Fekete-Szegö type inequalities for certain subclasses analytic functions associated with petal shaped region

1.   Introduction

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

which are analytic in the open unit disk $\mathbb{D} = \left\{z:\mid z\mid < 1\right\}$ and normalized by $f(0) = 0$ and $f^{\prime}(0) = 1$. Recall that, $\mathcal{S}\subset \mathcal{A}$ is the univalent function in $\mathbb{D} = \left\{z:\mid z\mid < 1\right\}$ and has the star-like and convex function classes as its sub-classes which their geometric conditions satisfy

respectively. The two well-known sub-classes have been used to define different subclasses of analytical functions in different direction with different perspective and their results are too voluminous in literature.

For two functions $f$ and $g$, $f$ is said to be subordinate to $g$, written as $f\prec g$, if there exists a Schwartz function $w(z)$ such that

where $w(0) = 0$ and $\mid w(z) \mid < 1$ for $z\in\mathbb{D}$.

Goodman ^[2] proposed the concept of conic domain to generalize convex function which generated the first parabolic region as an image domain of analytic function. Besides, he also introduced and studied the class $\mathcal{UCV}$ of uniformly convex functions which satisfy

Lately, Ma and Minda ^[8] and Rønning^[15] independently studied the underneath characterization

Rønning ^[15] also defined a class $\mathcal{ST}$ as below

Further, we say that $f$ of the form (1.1) is in $\mathcal{USD}$, if

The above characterization given in (1.4), resulted in the first parabolic region of the form

which was later generalized by Kanas and Wisniowska ^[3,4] to

We note that $\Omega_{k}$ represents the right half plane for $k = 0$, hyperbolic region for $0 < k < 1$, parabolic region for $k = 1$ and elliptic region for $k > 1$.

From then on, the generalized conic region (1.8) has been studied by many researchers (see ^[10,12] and also references cited therein). Moreover, the conic domain $\Omega$ was generalized to domain $\Omega[A, B]$, $-1\leq B < A\leq1$, by Noor and Malik ^[13] via

which is called petal shaped region (also see ^[11]).

A function $p(z) \in \mathcal{UP}[A, B]$, if and only if

where "$\prec$" denotes subordination, and $\tilde{p} = 1 + \dfrac{2}{\pi^2} \left(log \dfrac{1+\sqrt{z}}{1-\sqrt{z}} \right)^2.$

Fixing $A = 1$ and $B = -1$ in (1.9), the usual classes of functions studied by Goodman ^[1] and Kanas ^[3,4] can be obtained.

Furthermore, the classes $\mathcal{UCV}[A, B]$ and $\mathcal{ST} [A, B]$ are uniformly Janowski convex and starlike defined below:

A function $f \in \mathcal{A}$ is said to be in the class $\mathcal{UCV}[A, B]$ $-1 \leq B < A \leq 1,$ if and only if

or equivalently

A function $f \in \mathcal{A}$ is said to be in the class $\mathcal{ST} [A, B]$, $-1 \leq B < A \leq 1,$ if and only if

or equivalently

A function $f \in \mathcal{A}$ is said to be in the class $\mathcal{USD} [A, B]$, $-1 \leq B < A \leq 1,$ if and only if

or equivalently

It can easily be seen that $f(z) \in \mathcal{UCV}[A, B] \Leftrightarrow zf'(z)\in \mathcal{ST} [A, B]$. Setting $A = 1$ and $B = -1$ in (1.10) and (1.11), we obtained the classes of functions investigated by Goodman ^[2] and Rønning ^[15].

The relevant connection to Fekete-Szegö problem is a way of maximizing the non-linear functional $\left| a_3 - \mu a_2^2\right|$ for various subclasses of univalent functions. To know much more of history, we refer the readers to ^[5,7,16]. Inspired by earlier work in ^[14], in this paper we study the coefficient inequalities for certain subclasses of analytical functions related to petal type region. The first few coefficient bounds and the relevant connection to Fekete-Szegö inequalities were obtained for the classes of functions defined. Also note that, the results obtained here have not been in literature and the varying of parameters involved can give rise to new or known results. For the purpose of the main results, the following lemmas and definitions are needed.

Lemma 1.1. ^[1] Let $\mathcal{P}$ be the class of all analytic functions $h(z)$ of the following form

satisfying $\Re[h(z)] > 0$ and $h(0) = 1$. Then $\mid c_{n}\mid\leq2(n\in\mathbb{N})$ and the result is best possible for $h(z) = \frac{1 +\rho z}{1-\rho z}, |\rho| = 1.$

The next lemmas give us a majorant for the coefficients of the functions of the class $\mathcal{P},$ and more details may be found in ^[9]:

Lemma 1.2. ^[9] If $h(z)\in\mathcal{P}$ is given by (1.13), then, for any complex $\mu$,

and the result is sharp for the functions

Lemma 1.3. ^[9] (Lemma 1 and Remark, pp. 162-163) Let $h(z)\in\mathcal{P}$ be given by (1.13). Then

When $\mu < 0$ or $\mu > 1$, the equality holds if and only if $h_{1} = \dfrac{1+z}{1-z}$ or one of its rotations. If $0 < \mu < 1$, then equality holds if and only if $h_{2} = \dfrac{1+z^2}{1-z^2}$ or one of its rotations. If $\mu = 0$, the equality holds if and only if

or one of its rotations. If $\mu = 1$, then the sharp result holds for the following function

or one of its rotations. Although the above upper bound is sharp, when $0 < \mu < 1$, it can be improved as follows:

and

Definition 1.4. For $0\leq\lambda\leq1,$ and $-1\leq B < A\leq1$, a function $f\in \mathcal{A}$ is said to be in the class $\mathcal{N}[\lambda, A, B]$, if and only if

or equivalently $(f'(z))^{\lambda}\left(\frac{zf'(z)}{f(z)}\right)^{1-\lambda} = \mathcal{F}(z)\in \mathcal{UP}[A, B].$

Remark 1.5. We note that by fixing $\lambda = 0,$ then $\mathcal{N}[\lambda, A, B]\equiv \mathcal{ST}[A, B]$ and $\lambda = 1,$ then $\mathcal{N}[\lambda, A, B]\equiv \mathcal{USD}[A, B]$

Definition 1.6. For $0\leq\lambda\leq1,$ and $-1\leq B < A\leq1$, a function $f\in \mathcal{A}$ is said to be in the class $\mathcal{M}[\lambda, A, B]$, if and only if

or equivalently $(f'(z))^{\lambda}\left(1+\frac{zf''(z)}{f'(z)}\right)^{1-\lambda} = \mathcal{G}(z)\in \mathcal{UP}[A, B].$

Remark 1.7. We note that by taking $\lambda = 0,$ then $\mathcal{M}[\lambda, A, B]\equiv \mathcal{UCV}[A, B]$ and $\lambda = 1,$ then $\mathcal{M}[\lambda, A, B]\equiv \mathcal{USD}[A, B]$

Definition 1.8. For $0\leq\lambda\leq1,$ and $-1\leq B < A\leq1$, a function $f\in \mathcal{A}$ is said to be in the class $\mathcal{W}[\lambda, A, B]$, if and only if

or equivalently $\left(\dfrac{zf'(z)}{f(z)}\right)^{\lambda} \left(1+\dfrac{zf''(z)}{f'(z)}\right)^{1-\lambda} = \mathcal{Q}(z)\in \mathcal{UP}[A, B].$

Remark 1.9. Assuming $\lambda = 0,$ we note that $\mathcal{W}[\lambda, A, B]\equiv \mathcal{UCV}[A, B]$ and $\lambda = 1,$ then $\mathcal{W}[\lambda, A, B]\equiv \mathcal{ST}[A, B]$

2.   Coefficient bounds and Fekete-Szegö inequality

In this section, we let $-1\leq B < A\leq1$, $0\leq\lambda\leq1,$ unless otherwise stated. To prove our main results we recall the following:

Let $h \in \mathcal{P}$ of the form (1.13). Consider

where $w(z)$ is such that $w(0) = 0$ and $|w(z)| < 1$. Then, it follows easily that

Now, if ${\tilde p(z)} = 1+R_{1}z+R_{2}z^{2}+\cdots$, where $R_{1} = \frac{8}{ \pi^{2}}, \; R_{2} = \frac{16}{3 \pi^{2}}$ and $R_{3} = \frac{184}{ 45\pi^{2}}$ (see ^[6]), then we have

Hence, from (2.1) and (2.2) we get

If $p\in\mathcal{ UP}[A, B]$, from the relation (2.3), we may derive

Thereby, it implies that

Theorem 2.1. Let $f\in \mathcal{N}[\lambda, A, B]$ and $f$ be of the form (1.1). Then, we have

for

Furthermore, for a complex number $\mu$, we get

where

Proof. Assume that $f\in \mathcal{N}[\lambda, A, B]$. Then, it follows from the relation (1.9) and Definition 1.4 that

where $w(z)$ is such that $w (0) = 0$ and $\mid w(z) \mid < 1$. From (2.4) we get

If $f(z)$ is given by (1.1), then we have

From the comparison of coefficients of $z$ and $z^{2}$ in (2.8) and (2.9), we obtain

and

By using (2.10) in (2.11), we get

where

In view of Lemma 1.1, from (2.10) we get

and by applying Lemma 1.2 to (2.12), we get

so that we get the desired result in (2.5) and (2.6).

Now, for $\mu\in \mathbb{C}$, we note that

where

By Lemma 1.2, we get

which yields the desired result in (2.7).

Theorem 2.2. Let $f\in \mathcal{M}[\lambda, A, B]$ be of the form (1.1). Then

for

Further, for a complex number $\mu$, we have

for

Proof. Suppose that $f\in \mathcal{M}[\lambda, A, B]$ and of the form (1.1). Then, it follows from the relation (1.9) and Definition 1.6 that

where $w(z)$ is such that $w (0) = 0$ and $\mid w(z) \mid < 1$. From (2.4) we assert that

Since $f$ is given by (1.1), we know

By the comparison of coefficients of $z$ and $z^{2}$ in (2.16) and (2.17), we get

of which by Lemma 1.1 gives

and

where

By Lemma 1.2, we can deduce that

which yields the desired result in (2.14).

Now, for a complex number $\mu,$

where

Hence, by means of Lemma 1.2 we get the desired result (2.15).

Theorem 2.3. If the function $f\in \mathcal{W}[\lambda, A, B]$ is of the form (1.1), then

for

Moreover, for a complex number $\mu$, we obtain

for

Proof. Let $f\in \mathcal{W}[\lambda, A, B]$. Then, applying the relation (1.9) and Definition 1.8 we have

where $w(z)$ is such that $w(0) = 0$ and $|w(z)| < 1$. From (2.4) we get

In view of (1.1), we obtain

By the comparison of coefficients of $z$ and $z^{2}$ in (2.24) and (2.25), we get

and

Then, by (2.26),

Following the procedure as in the above theorems, we can get the desired results given by (2.20) and (2.21).

Now, for a complex number $\mu$,

where

Therefore, in light of Lemmas 1.1 and 1.2 we get the desired result (2.22).

If we choose real $\mu$, then by Lemma 1.3 we derive the next results for Fekete-Szegö problem for these classes above.

Theorem 2.4. If the function $f\in\mathcal{N}[\lambda, A, B]$ be of the form (1.1), and $\mu\in\mathbb{R}$ then

where

and

Moreover, we set

Then, each of the following results holds:

(A) For $\mu\in[\Upsilon_{1}, \Upsilon_{3}]$,

(B) For $\mu\in[\Upsilon_{3}, \Upsilon_{2}]$,

where

Proof. If $f\in\mathcal{N}[\lambda, A, B]$ is given by (1.1), based on the proof of Theorem 2.1 we see

where

For $\hbar\in\mathbb{R}$, we know that $\hbar\ge1$ is equivalent to $\mu\ge\Upsilon_2$, and $\hbar\le0$ is equivalent to $\mu\le\Upsilon_1$. Therefore, taking the modulus on both sides of the above equality, with the aid of the inequality in Lemma 1.3 we obtain the first estimates of Theorem 2.4.

For the proof of the second part, note that $0 < \hbar\le1/2$ is equivalent to $\Upsilon_1 < \mu\le\Upsilon_3$. By using the relations (2.28) and (2.10), and then by applying the inequality (1.14) of Lemma 1.3, we get

such that the required inequality (A) holds.

Similarly, we can easily check that $1/2\le \hbar < 1$ is equivalent to $\Upsilon_3\le\mu < \Upsilon_2$. From the relations (2.28) and (2.10), and the inequality (1.15) of Lemma 1.3 we obtain

which is exactly the inequality (B).

To show that the bounds are sharp, we define the functions $K_{p_n}$ $(n = 2, 3, \ldots)$ with $K_{p_n} (0) = 0 = [K_{p_n}]'(0)-1$, by

where $p(z)$ is as given in (1.9). Also, define the functions $F_{\eta}$ and $G_{\eta}$, $0\leq\eta\leq 1$, respectively, with $F_{\eta}(0) = 0 = F_{\eta}^{\prime}(0)-1$ and $G_{\eta}(0) = 0 = G_{\eta}^{\prime}(0)-1$ by

and

Clearly,

Hence, $K_{{p}_n}, F_\eta, \; G_\eta\in\mathcal{N}[\lambda, A, B]$. Also, we write

If $\mu < \Upsilon_1$ or $\mu > \Upsilon_2$, then the equality holds if and only if $f$ is $K_{{p}_2}(z)$ or one of its rotations. When $\Upsilon_1 < \mu < \Upsilon_2$, then the equality holds if and only if $f$ is $K_{{p}_3}(z)$ satisfying

or one of its rotations. If $\mu = \Upsilon_1$ then the equality holds if and only if $f$ is $F_{\eta}$ or one of its rotations. If $\mu = \Upsilon_2$ then the equality holds if and only if $f$ is $G_\eta$ or one of its rotations.

As it is similar to the above result in Theorem 2.4, we state the following results without proof.

Theorem 2.5. If the function $f\in\mathcal{M}[\lambda, A, B]$ is of the form (1.1) and $\mu\in\mathbb{R}$, then

where

and

Further, we set

Then, each of the following results holds:

(A) For $\mu\in[\Gamma_{1}, \Gamma_{3}]$,

(B) For $\mu\in[\Gamma_{3}, \Gamma_{2}]$,

where

Theorem 2.6. Let $\mu\in\mathbb{R}$. If $f\in\mathcal{A}$ is assumed to be in $\mathcal{W}[\lambda, A, B]$, then

where

and

Besides, we let

Then, each of the following results holds:

(A) For $\mu\in[\aleph_{1}, \aleph_{3}]$,

(B) For $\mu\in[\aleph_{3}, \aleph_{2}]$,

where

3.   The logarithmic coefficients

In this section we determine the coefficient bounds and Fekete-Szegö problems associated with logarithmic function $\mathcal{H}(z)$ defined by

where the coefficient $d_{n}$ of $\mathcal{H}(z)$ is called the logarithmic coefficient of $f\in\mathcal{A}$ defined in $\mathbb{D}$. Expanding (3.1) by series expansion of $\log(1+z)$ and equating the various coefficients, we assert that

Theorem 3.1. Let $f\in \mathcal{N}[\lambda, A, B]$ with the logarithmic coefficients in (3.1), and $\nu\in\mathbb{C}$. Then

for

and

for

Proof. From (2.10) and (2.12), and (3.2) and (3.3) we get

Furthermore, we have

where

In view of Lemmas 1.1 and 1.2, we get the desired results such that Theorem 3.1 holds true.

Theorem 3.2. Let $f\in \mathcal{M}[\lambda, A, B]$ with the logarithmic coefficients in (3.1), and $\nu\in\mathbb{C}$. Then

for

and

for

Proof. According to (2.18), (2.19), (3.2) and (3.3) we get

Further, we have

where

Applying Lemmas 1.2 and 1.1, we obtain the desire estimates and complete the proof of Theorem 3.2.

Theorem 3.3. Let $f\in \mathcal{W}[\lambda, A, B]$ with the logarithmic coefficients in (3.1), and $\nu\in\mathbb{C}$. Then

for

and

for

Proof. From (2.26) and (2.27), and (3.2) and (3.3) we derive

Therefore, we obtain

where

By applying Lemmas 1.1 and 1.2, we get the desired results, which prove Theorem 3.3.

If we consider real $\nu$, then by Lemma 1.3 we provide the several results for Fekete-Szegö problem with respect to the logarithmic coefficients.

Theorem 3.4. Let $\nu\in\mathbb{R}$. If $f\in\mathcal{N}[\lambda, A, B]$ with the logarithmic coefficients is of the form (3.1), then

where

and

Moreover, we put

Then, each of the following results holds:

(i) For $\nu\in[\widetilde{\Upsilon}_{1}, \widetilde{\Upsilon}_{3}]$,

(ii) For $\nu\in[\widetilde{\Upsilon}_{3}, \widetilde{\Upsilon}_{2}]$,

where

Theorem 3.5. Let $\nu\in\mathbb{R}$. If $f\in\mathcal{M}[\lambda, A, B]$ with the logarithmic coefficients is of the form (3.1), then

where

and

Moreover, we put

Then, each of the following results holds:

(i) For $\nu\in[\widetilde{\Gamma}_{1}, \widetilde{\Gamma}_{3}]$,

(ii) For $\nu\in[\widetilde{\Gamma}_{3}, \widetilde{\Gamma}_{2}]$,

where

Theorem 3.6. Let $\nu\in\mathbb{R}$. If $f\in\mathcal{W}[\lambda, A, B]$ with the logarithmic coefficients is of the form (3.1), then

where

and

Moreover, we let

Then, each of the following results holds:

(i) For $\mu\in[\widetilde{\aleph}_{1}, \widetilde{\aleph}_{3}]$,

(ii) For $\nu\in[\widetilde{\aleph}_{3}, \widetilde{\aleph}_{2}]$,

where

Concluding Remark: By fixing $A = 1\; and \; B = -1$ or $A = 1\; and \; B = 1-2\alpha$, one can deduce some interesting results.

Acknowledgements

The authors are profoundly grateful to all the reviewers for their careful readings and insightful suggestions such that the original manuscript for this article is greatly improved.

This work is supported by Natural Science Foundation of Ningxia (Grant No. 2020AAC03066) and Natural Science Foundation of China (Grant Nos. 42064004; 11762017).

Conflict of interest

The authors declare no conflict of interest.