Nuts for Physical Health and Fitness: A Review

1.   Introduction

One of the most fascinating areas of Complex analysis is the study of geometric characterization of univalent functions in the open unit disc $U$. Because of the challenging problem in studying the class $S_{(1-1)}$ of univalent functions in $U$ as a whole, several subclasses of it emerged. The most studied of these are the classes $C_{CV}$, $S_{ST}$, $C_{CV}(\beta)$ and $S_{ST}(\beta)\, \, (0\leq\beta < 1)$ of convex functions, starlike functions and, convex and starlike functions of order $\beta$, respectively. Since the image domains of $U$ plays a significant role in their geometric characterization, various subclasses of $S_{(1-1)}$ have been receiving attention in different directions and perspectives (see ^{[4,6,20,23,28,32,33,36,37,38]}). For this reason, Ma and Minda ^[17] gave a unified treatment of both $S_{ST}$ and $C_{CV}$. They considered the class $\Psi$ of analytic univalent functions $\psi(\varsigma)$ with $\rm{Re} \psi(\varsigma) > 0$ and for which $\psi(U)$ is symmetric with respect to the real axis and starlike with respect to $\psi(0)$ such that $\psi^{\prime}(0) > 0$. They initiated the following classes of functions that generalized and unified many renowned subclasses of $S_{(1-1)}$:

and

where $\mathcal{A}$ is the class of analytic functions $f(\varsigma)$ of the form

If

then $C_{CV}(\psi) = UCV$ is the Goodman class of uniformly convex functions ^[8,30], which was later modified and examined by Kanas and Wisniowska ^[13,14]. Similarly, $ST_{hpl}(s) = S^{\ast}\left(\frac{1}{(1-\varsigma)^{s}}\right)$, $CV_{hpl}(s) = C\left(\frac{1}{(1-\varsigma)^{s}}\right) (0 < s\leq 1)$, are made-known by Kanas and Ebadian ^[15,16], respectively. These consist of functions $f\in\mathcal{A}$ such that $\varsigma f^{\prime}(\varsigma)/f(\varsigma)$ and $(\varsigma f^{\prime}(\varsigma))^{\prime}/f^{\prime}(\varsigma)$ lie in the domain bounded by the right branch of a hyperbola

More special families of Ma and Minda classes can be found in ^{[3,9,10,24,25,26,31,34,39]}.

Recently, Kanas et al ^[18] introduced novel subclasses $ST_{\mathcal{L}}(s)$ and $CV_{\mathcal{L}}(s)$ of $S_{ST}$ and $C_{CV}$, respectively. Geometrically, they consist of functions $f(\varsigma)\in\mathcal{A}$ such that $\varsigma f^{\prime}(\varsigma)/f(\varsigma)$ and $(\varsigma f^{\prime}(\varsigma))^{\prime}/f^{\prime}(\varsigma)$ lie in the region bounded by the limaçon defined as

as shown in Figure 1 for different values of $s$. $s = 0.35, 0.5, 0.6, 0.71, 0.75$ and 1 corresponds to blue, red, green, gray, yellow and black. Some novel properties of these classes were derived in ^[18].

Motivated by this present work and other aforementioned articles, the goal in this paper is to continue with the investigation of some interesting properties of the classes $ST_{\mathcal{L}}(s)$ and $CV_{\mathcal{L}}(s)$. To this end, the sharp bounds of the Hankel determinant, subordination conditions as well as some radius results for these novel classes are investigated.

2.   Materials and method

To put our investigations in a clear perspective, some preliminaries and definitions are presented as follows:

Denoted by $\mathcal{W}$ is the class of analytic functions

such that $w(0) = 0$ and $\vert w(\varsigma)\vert < 1$. These functions are known as Schwarz functions. If $f(\varsigma)$ and $g(\varsigma)$ are analytic functions in ${U}$, then $f(\varsigma)$ is subordinate to $g(\varsigma)$ (written as $f(\varsigma)\prec g(\varsigma)$) if there exists a Schwarz function $w(\varsigma)\in\mathcal{W}$ such that $f(\varsigma) = g(w(\varsigma)), \, \, \varsigma\in {U}.$

Janowski ^[12] introduced the class ${P(\mathbb{A}_{\Game}, \mathbb{B}_{\Game})}, \, \, -1{\leq \mathbb{B}_{\Game} < \mathbb{A}_{\Game}\leq }1$ of functions $p(\varsigma)$ satisfying the subordination condition

or equivalently, satisfying the inequality

As a special cases, $P(1, -1)\equiv P$ and $P(1-2\beta, -1)\equiv P(\beta)\, (0\leq\beta < 1)$ are the classes of functions of positive real part and that whose real part is greater than $\beta$, respectively (see ^[7]).

Definition 2.1. Noonan and Thomas ^[22] defined for $q\geq1, n\geq1$, the $q$th Hankel determinant of $f(\varsigma)\in S_{1-1}$ of the form (1.1) as follows$\colon$

This determinant has been studied by many researchers. In particular Babalola ^[2] obtained the sharp bounds of $\mathcal{H}_{3}(1)$ for the classes $S_{ST}$ and $C_{CV}$. By this definition, $\mathcal{H}_{3}(1)$ is given as:

and the by triangle inequality,

Clearly, one can see that $\mathcal{H}_{2}(1) = \vert \delta_{3}-\delta_{2}^{2}\vert$ is a particular instance of the well-known Fekete Szeg$\ddot{ \rm{o}}$ functional $\vert \delta_{3}-\mu \delta_{2}^{2}\vert$, where $\mu$ is a real number.

Definition 2.2. ^[18] Let $p(\varsigma) = 1+\sum\limits_{n = 1}^{\infty}c_{n}\varsigma^{n}$. Then $p\in P(\mathbb{L}_{s})$ if and only if

or equivalently, if $p(\varsigma)$ satisfies the inequality

Demonstrated in ^[18], was the inclusion relation

It is worthy of note that the function $\mathbb{L}_{s}(\varsigma) = (1+s\varsigma)^{2}$ is the analytic characterization of $\mathbb{L}_{s}(U)$ given by (1.2). Also, $\mathbb{L}_{s}(\varsigma)$ is starlike and convex univalent in $U$ for $0 < s\leq\frac{1}{2}$. Furthermore, $\mathbb{L}_{s}(\varsigma)\in P(\beta)$, where $\beta = (1-s)^{2}, \, \, \, 0 < s\leq\frac{1}{2}$.

Definition 2.3. Let $f\in\mathcal{A}$. Then $f\in ST_{\mathcal{L}}(s)$ if and only if

Also, $f\in CV_{\mathcal{L}}(s)$ if and only if

Moreover, the integral representation of functions $f\in ST_{\mathcal{L}}(s)$ is given as

while that of $g\in CV_{\mathcal{L}}(s)$ is given as

Furthermore, the extremal functions for each of the classes are given by

and for $K_{s, n}(\varsigma)\in CV_{\mathcal{L}}(s)$,

3.   A set of lemmas

Lemma 3.1. ^[1] If $w\in\mathcal{W}$ is of the form (2.1), then for a real number $\sigma$,

When $\sigma < -1$ or $\sigma > 1$, equality holds if and only if $w(\varsigma) = \varsigma$ or one of its rotations. If $-1 < \sigma < 1$, then equality holds if and only if $w(\varsigma) = \varsigma^{2}$ or one of its rotations. Equality holds for $\sigma = -1$ if and only if $w(\varsigma) = \frac{\varsigma(x+\varsigma)}{1+x\varsigma}\, \, (0\leq x\leq 1)$ or one of its rotations while for $\sigma = 1$, equality holds if and only if $w(\varsigma) = -\frac{\varsigma(x+\varsigma)}{1+x\varsigma}\, \, (0\leq x\leq 1)$ or one of its rotations.

Also, the sharp upper bound above can be improved as follows when $-1\leq\sigma\leq 1$ :

and

Lemma 3.2. ^[16] If $w\in\mathcal{W}$ is of the form (2.1), then for some complex numbers $\xi$ and $\eta$ such that $\vert \xi\vert\leq 1$ and $\vert \eta\vert\leq 1$,

and

Lemma 3.3. [19,Theorem 3.4h,p. 132] Let $q(\varsigma)$ be univalent in $U$ and let $\theta$ and $\varphi$ be analytic in a domain $D$ containing $q(U)$ with $\varphi(\omega)\neq 0$, when $\omega\in q(U)$. Set $Q(\varsigma) = \varsigma q^{\prime}(\varsigma)\cdot\varphi(q(\varsigma)), \, \, h(\varsigma) = \theta(q(\varsigma))+Q(\varsigma)$, and suppose that either

(i) $h(\varsigma)$ is convex, or $Q(\varsigma)$ is starlike,

(ii)

If $p(\varsigma)$ is analytic in $U$ with $p(0) = q(0), \, \, p(U)\subset D$ and

then $p(\varsigma)\prec q(\varsigma)$, and $q(\varsigma)$ is the best dominant in the sense that $p\prec t\Rightarrow q\prec t$ for all $t$.

Lemma 3.4. ^[29] Let $h(\varsigma) = 1+\sum\limits_{n = 1}^{\infty}c_{n}\varsigma^{n}, \, G(\varsigma) = 1+\sum\limits_{n = 1}^{\infty}d_{n}\varsigma^{n}$ and $h(\varsigma)\prec G(\varsigma)$. If $G(\varsigma)$ is univalent in $U$ and $G(U)$ is convex, then $\vert c_{n}\vert\leq\vert d_{1}\vert$, for all $n \geq 1.$

Lemma 3.5. ^[11] Let $w\in\mathcal{W}$. If $\vert w(\varsigma)\vert$ attains its maximum value on the circle $\vert\varsigma\vert = r$ at a point $\varsigma_{0}\in U$, then we have $\varsigma_{0}w^{\prime}(\varsigma_{0}) = kw(\varsigma_{0})$, where $k\geq 1$.

Throughout this work $f(\varsigma)$ is taken to be of the form (1.1) while $w(\varsigma)$ is of the form (2.1). In the next sections, the main results are presented.

4.   Coefficient results

In this section, we assume $0 < s\leq\frac{1}{2}$. First, we establish a few auxiliary results whose applications will be needed hereafter.

Lemma 4.1. Let $f\in ST_{\mathcal{L}}(s)$. Then

Proof. From the definition of $f\in ST_{\mathcal{L}}(s)$, we have

where

Comparing the coefficients of $\varsigma^{n}$ in (4.2), it follows that

It is obvious from Lemma 3.4 and the fact that $\mathbb{L}_{s}(\varsigma)$ is convex for $0 < s\leq\frac{1}{2}$ that

Using this result in (4.4), we obtain

We need to show (4.1) by Mathematical induction. For this reason, assume (4.1) is true and proceed to prove

From (4.6),

and finally,

Therefore,

Hence, by Mathematical induction, we have the desired result.

In view of Theorem 4.1 and the definition of functions in $CV_{\mathcal{L}}(s)$, we are led to the following result.

Lemma 4.2. Let $f\in CV_{\mathcal{L}}(s)$. Then

Lemma 4.3. Let $f\in ST_{\mathcal{L}}(s)$. Then

The bound $\frac{19s^{4}}{12}$ is sharp for the function

Proof. For $f\in ST_{\mathcal{L}}(s)$,

where $w\in\mathcal{W}$. Comparing coefficients of $\varsigma, \varsigma^{2}$ and $\varsigma^{3}$ in (4.8), we arrive at

By Lemma 3.1, we obtain

Let $x = w_{1}, \, \xi = y$ with $0\leq x\leq 1$ and $\vert y\vert\leq 1$. Then the triangle inequality gives

where

and

This means that $\mathcal{F}(x, \vert y\vert)$ is increasing on the interval $[0, 1]$. So,

where

which implies that $\mathcal{F}(x)$ is an increasing function of $x$ on $[0, 1]$. Consequently,

Lemma 4.4. Let $f\in CV_{\mathcal{L}}(s)$. Then

The bound $\frac{s^{2}}{9}$ is sharp for the function

Proof. From the definition of $f\in CV_{\mathcal{L}}(s)$ and (4.9), it is easy to see that

The rest of the proof follows as in Theorem 4.3.

Lemma 4.5. Let $f\in ST_{\mathcal{L}}(s)$. Then

The bound $\frac{2s}{3}$ is best possible for the function

Proof. From (4.9), a computation gives

Employing Lemma 3.2, we write the expression for $w_{3}$, and applying the triangle inequality together with Lemma 3.1, we obtain

where we have taken $w_{1} = x, \, \xi = y$ with $0\leq x\leq 1$ and $\vert y\vert\leq 1$. Let $\mathcal{H}(x, \vert y\vert)$ represents the right side of (4.12). Then

Thus,

where

It is clear that $\mathcal{H}(x)$ attains its maximum value at $x = \frac{7s}{2}$. Thus, $\mathcal{H}(x)\leq\mathcal{H}\left(\frac{7s}{2}\right) = \frac{2s}{3}$. Consequently,

Lemma 4.6. Let $f\in CV_{\mathcal{L}}(s)$. Then

This bound cannot be improved since the function

attains the equality.

Proof. Using the definition of $f\in CV_{\mathcal{L}}(s)$ and (4.9), we find

Let $w_{1} = x\, (0\leq x < 1)$ and $\xi = y$ with $\vert y\vert\leq 1$. Then applying Lemma 3.2 and following the procedure of proof as in Theorem 4.5, we arrive at the desired result.

5.   Fekete Szeg$\ddot{ \rm{o}}$ inequality for the classes $ST_{\mathcal{L}}(s)$ and $CV_{\mathcal{L}}(s)$

Lemma 5.1. Let $f\in ST_{\mathcal{L}}(s)$. Then for a real number $\mu$,

It is asserted also that

and

These inequalities are sharp for the functions

where $\vert \lambda\vert = 1$ and

Proof. From (4.9), we have

Then using Lemma 3.1 with $\sigma = \frac{s(8\mu-5)}{2}$, we obtain the required result. For $\mu = 1$ in Theorem 5.1, we deduce the following sharp result.

Corollary 5.1. Let $f\in ST_{\mathcal{L}}(s)$. Then

Lemma 5.2. Let $f\in CV_{\mathcal{L}}(s)$. Then for a real number $\mu$,

It is asserted also that

and

These inequalities are sharp for the functions

where $\vert \lambda\vert = 1$ and

Proof. Using the definition of $CV_{\mathcal{L}}(s)$ and (4.9), we get

Then using Lemma 3.1 with $\sigma = \frac{s(6\mu-5)}{2}$, we obtain the desired result. For $\mu = 1$ in Theorem 5.2, we deduce the following sharp result.

Corollary 5.2. Let $f\in CV_{\mathcal{L}}(s)$. Then

Theorem 5.3. Let $f\in ST_{\mathcal{L}}(s)$. Then

Proof. The proof is immediate from (2.4), Lemma 4.1, Lemma 4.3, Lemma 4.5 and Corollary 5.1.

Theorem 5.4. Let $f\in CV_{\mathcal{L}}(s)$. Then

Proof. The proof is straightforward from (2.4), Lemma 4.2, Lemma 4.4, Lemma 4.6 and Corollary 5.2.

6.   $ST_{\mathcal{L}}(s)$ and $CV_{\mathcal{L}}(s)$ radii

Theorem 6.1. The $CV_{\mathcal{L}}(s)$-radius for the class $S_{ST}(\beta)$ (where $\beta = (1-s)^{2}$) is given by

This radius is sharp for the functions given by

Proof. Let $f\in S_{ST}(\beta)$. Then

This means that there exists $w\in\mathcal{W}$ such that

Let

Then by Schwarz lemma,

It follows from logarithmic differentiation of (6.3) that

It is known from ^[27] that for $p\in P(\beta)$,

Using this result together with (6.4), we write

We need to show that

However, it holds if

Let $\mathcal{T}(r) = 3(1-2\beta)r^{2}+2(2+\beta)r-1$. Then $\mathcal{T}(0) = -1 < 0$ and $T(1) = 6-4\beta > 0$ such that $\mathcal{T}(0)\mathcal{T}(1) < 0.$ Thus, there exists $R_{1}\in [0, 1]$ such that

Therefore, $3(1-2\beta)r^{2}+2(2+\beta)r-1\leq 0$ for all $r < R_{1}$ and $R_{1}$ is the smallest roots of (6.5) given by (6.1).

For sharpness, we consider the functions $f_{0}(\varsigma)$ defined by (6.2). At the point $z = R_{1}$, we have

Theorem 6.2. Let $f\in ST_{\mathcal{L}}(s).$ Then $f\in C_{CV}$ for all $z$ in the disc $\vert\varsigma\vert < R_{2}$, where $R_{2}$ is the positive roots of the equation

Proof. For $f\in ST_{\mathcal{L}}(s)$,

where $w\in\mathcal{W}$. Therefore

where we have used the extension of Schwarz lemma (see ^[21]). Thus,

if $(1-sr)^{3}(1-r^{2})-2r > 0$. Let $\mathcal{T}(r) = (1-sr)^{3}(1-r^{2})-2r$. Then $\mathcal{T}(0) = 1 > 0$ and $\mathcal{T}(1) = -2 < 0$ with $\mathcal{T}(0)\mathcal{T}(1) < 0.$ Therefore, there exists $R_{2}\in[0, 1]$ such that $(1-sR_{2})^{3}(1-R_{2}^{2})-2R_{2} = 0$. Hence, $(1-sr)^{3}(1-r^{2})-2r > 0$ holds for all $r < R_{2}$, where $R_{2}$ is the smallest positive roots of (6.6).

Theorem 6.3. Let $p\in P$. Then $p\in P(\mathbb{L}_{s})$ for all $z$ in the disc

Proof. Let $p\in P$. Then

We want to prove that

Now,

if (6.7) is satisfied. To show that the radius cannot be improved, we consider the function

Then for $\varsigma = \frac{2s-s^{2}}{2+2s-s^{2}}$,

which shows that equality is attained for (6.7).

Corollary 6.1. The $ST_{\mathcal{L}}(s)$-radius and $CV_{\mathcal{L}}(s)$-radius for the classes of starlike and convex functions are given by (6.7).

7.   Some properties of the function $\mathbb{L}_{s}(\varsigma)$

Theorem 7.1. If $p(\varsigma)$ is analytic in $U$ with $p(0) = 1$ and satisfies the condition

or

then $p(\varsigma)\prec(1+s\varsigma)^2$ for $-1 < s < 0$.

Proof. Let $p(\varsigma)$ be defined by

Clearly, $w(\varsigma)$ is analytic in $U$ with $w(0) = 0$. To prove our result, it is required to show that $\vert w(\varsigma)\vert < 1$ for all $\varsigma\in U.$ From (7.3), a simple calculation gives

Suppose there exists a point $\varsigma_{0}\in U$ such that

Then by Lemma 3.5, $w(\varsigma_{0}) = e^{i\theta}$ and $\varsigma_{0}w^{\prime}(\varsigma_{0}) = kw(\varsigma_{0})$. Thus,

This contradicts (7.1). Therefore, there exists no $\varsigma_{0}\in U$ such that $\vert w({\varsigma_{0}})\vert = 1$. Thus $\vert w(\varsigma)\vert < 1$ in $U$, so that $p(\varsigma)\prec(1+s\varsigma)^{2}$ for $-1 < s < 0$.

Similarly,

which contradicts the assumption (7.2). Hence, the proof is completed.

Following the discussion demonstrated by Sharma et al in ^[35] for Theorem 3, we present the following results.

Theorem 7.2. Let $-1 < \mathbb{B}_{\Game} < \mathbb{A}_{\Game}\leq 1$, $0 < s\leq\frac{1}{\sqrt{2}}$ and $p(\varsigma) = \frac{1+\mathbb{A}_{\Game}\varsigma}{1+\mathbb{B}_{\Game}\varsigma}$. Then $p\in P(\mathbb{L}_{s})$ if and only if

or, equivalently, if and only if

Proof. The proof follows the techniques presented in [35,Theorem 3] For $\mathbb{B}_{\Game} = 0, \mathbb{A}_{\Game} = 0$ and $\mathbb{B}_{\Game} = -\mathbb{A}_{\Game}$, we give the following consequences of Theorem 7.2.

Corollary 7.1.

(i) $p(\varsigma) = 1+\mathbb{A}_{\Game}\varsigma\in P(\mathbb{L}_{s}) \Longleftrightarrow 0 < \mathbb{A}_{\Game}\leq 2s-s^{2}$.

(ii) $p(\varsigma) = 1/(1+\mathbb{B}_{\Game}\varsigma)\in P(\mathbb{L}_{s}) \Longleftrightarrow(s^{2}-2s)/(1+2s-s^{2})\leq \mathbb{B}_{\Game} < 0$.

(iii) $p(\varsigma) = (1+\mathbb{A}_{\Game}\varsigma)/(1-\mathbb{A}_{\Game}\varsigma)\in P(\mathbb{L}_{s}) \Longleftrightarrow0 < \mathbb{A}_{\Game}\leq (2s-s^{2})/(2+2s-s^{2})$.

Corollary 7.2. Let $-1 < \mathbb{B}_{\Game} < \mathbb{A}_{\Game}\leq 1$ and consider

Then $f\in ST_{\mathcal{L}}(s)$ and $f\in CV_{\mathcal{L}}(s)$, respectively if and only if conditions (7.4) or (7.5) is satisfied

Applying Corollary 7.1 along with the integral representation for the classes $ST_{\mathcal{L}}(s)$ and $CV_{\mathcal{L}}(s)$, respectively, we present the following examples.

Example 7.3.

(i) For $0 < \mathbb{A}_{\Game}\leq 2s-s^{2}$,

(ii) For $\frac{s^{2}-2s}{1+2s-s^{2}}\leq \mathbb{B}_{\Game} < 0$,

(iii) For $0 < \mathbb{A}_{\Game}\leq\frac{2s-s^{2}}{2+2s-s^{2}}$,

8.   Sufficient conditions and related results

Theorem 8.1. Let $-1\leq \mathbb{B}_{\Game} < \mathbb{A}_{\Game}\leq 1, \, \, 0 < s\leq\frac{1}{2}$ and $p(\varsigma)$ be analytic in $U$ with $p(0) = 1$ such that

If

then $p\in P(\mathbb{L}_{s})$.

Proof. Let $q(\varsigma) = (1+s\varsigma)^{2}, \, \, \, 0 < s\leq\frac{1}{2}$. Then $q(\varsigma)$ is convex univalent in $U$. Consider the functions $\phi(\omega) = \rho$ and $\theta(\omega) = 1$. These functions are both analytic in a domain containing $q(U)$ with $\phi(\omega)\neq0$. A computation shows that

Further,

and

Using Lemma 3.3, the subordination condition

implies $p(\varsigma)\prec q(\varsigma)$. To complete the proof, it suffices to prove that the circular disc 2.2 is contained in the region bounded by the curve $h(e^{i\theta})\, (\theta\in[0, 2\pi))$. To this end, we must show that

Now,

Thus,

if (8.2) is satisfied.

Theorem 8.2. Let $0 < s\leq\frac{1}{2}$ and $p(\varsigma)$ be analytic in $U$ with $p(0) = 1$ such that

If

then $p\in P(\mathbb{L}_{s})$.

Proof. Following the same arguments as in the proof of Theorem 8.1, we arrive at where to show that

To achieve this, it is enough to show that the domain bounded by the limaçon is inside the region bounded by the curve $h(e^{i\theta})\, \, (\theta\in[0, 2\pi))$. As a result, we need to find $\rho$ for which

Now,

if

Theorem 8.3. Let $-1\leq \mathbb{B}_{\Game} < \mathbb{A}_{\Game}\leq 1, \, \, 0 < s\leq\frac{1}{\sqrt{2}}$ and $p(\varsigma)$ be analytic in $U$ with $p(0) = 1$ such that

If

then

Proof. Let $q(\varsigma) = \frac{1+\mathbb{A}_{\Game}\varsigma}{1+\mathbb{B}_{\Game}\varsigma}$. We have that $q(\varsigma)$ is convex univalent in $U$. Therefore, following the method of proof in Theorem 8.1, we arrive at where to show that

For this, we need to establish that the region bounded by the limaçon lies inside the domain bounded by the curve $h(e^{i\theta})\, \, (\theta\in[0, 2\pi)$. A simple observation of (2.5) suggests it suffices to show

Now,

But

provided (8.6) holds. This completes the proof.

Theorem 8.4. Let $0 < s\leq\frac{1}{\sqrt{2}}$ and $p(\varsigma)$ be analytic in $U$ with $p(0) = 1$ such that

If

then

Proof. Let $q(\varsigma) = (1+s\varsigma)^{2}$. Then $q(\varsigma)$ is convex univalent in $U$. The function $\phi(\omega) = \rho/\omega$ and $\theta(\omega) = 1$ are analytic in the domain containing $q(U)$. Set

and

Then

From Lemma 3.3, the differential subordination

implies $p(\varsigma)\prec q(\varsigma).$ To finalize the proof, we need to prove

which is equivalent to showing

It is easy to see that the transformation $h(\varsigma) = 1+\rho\frac{\varsigma q^{\prime}(\varsigma)}{q(\varsigma)}$ maps $U$ onto the disc $\mathcal{D}(a, r)$, where

Therefore (8.9) holds if and only if

which implies

We choose to omit the proof of the next theorem since it follows the same argument as in Theorem 8.3.

Theorem 8.5. Let $-1\leq \mathbb{B}_{\Game} < \mathbb{A}_{\Game}\leq 1$, $0 < s\leq\frac{1}{\sqrt{2}}$ and $p(\varsigma)$ be analytic in $U$ with $p(0) = 1$ such that

If

then

Remark 8.1.

(ⅰ) If we put $p(\varsigma) = \varsigma f^{\prime}(\varsigma)/f(\varsigma)$ and $p(\varsigma) = (\varsigma f^{\prime}(\varsigma))^{\prime}/f^{\prime}(\varsigma)$ in Theorem 8.1-8.5, we obtain the conditions on $\rho$ for which the respective subordination conditions (8.1), (8.3) and (8.7) imply $f\in ST_{\mathcal{L}}(s)$ and $f\in CV_{\mathcal{L}}(s)$.

(ⅱ) We note that our condition $0 < s\leq\frac{1}{2}$ cannot be relaxed in Theorem 8.1 and Theorem 8.2. Otherwise, starlikeness of $Q$ will not be achieved. As such, the proof of the theorems will be extremely difficult to obtain via Lemma 3.3.

9.   Conclusion

The Ma and Minda classes of functions are the comprehensive generalization of the classes $S_{ST}$ and $C_{CV}$. These classes are vital in (GFT) because of their importance in science and technology. To this end, continuous studies of their subfamily, which are related to Limaçon domain were investigated. Coefficients bounds, Fekete Szeg$\ddot{ \rm{o}}$ inequality as well as the upper bounds of the third Hankel determinants for these subclasses were derived. Finally, the techniques of differential subordination were also used to obtain some restrictions for which analytic functions belonged to these families. In addition, to have more new theorems under present examinations, new generalization and applications can be explored with some positive and novel outcomes in various fields of science, especially, in applied mathematics. These new surveys will be presented in future research work being processed by authors of the present paper.

Acknowledgments

The authors would like to thank the Rector of COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environment. The fourth author is supported by UKM grant: GUP-2019-032.

Conflict of interest

The authors declare that they have no conflicts of interest.