On certain inclusion relations of functions with bounded rotations associated with Mittag-Leffler functions

1.   Introduction

The class of normalized analytic functions in the open unit disc $\Delta = \{z\in\mathbb{C}:|z| < 1\}$ denoted by $\Omega$ consists of the functions $f$ of the form

where $f^\prime(0)-1 = f(0) = 0$. Let $\ell(z)\in \Omega$ defined by

Then the Hadamard product, also known as the convolution of two function $f(z)$ and $\ell(z)$ denoted by $f\ast \ell$ is defined as

Moreover, $f(z)\prec \ell(z),$ if there exist a Schwartz function $\chi(z)$ in $A$, satisfying the conditions $\chi(0) = 0$ and $|\chi(z)| < 1,$ such that $f(z) = \ell(\chi(z))$. The symbol $\prec$ is used to denote subordination.

Let $S$ denote the subclass of $\Omega$ of univalent functions in $\Delta$. Let $P, C, S^*$ and $K$ represent the subclasses of $S$ known as the classes of Caratheodory functions, convex funtions, starlike functions, and close-to-convex functions, respectively.

The concept of bounded rotations was introduced by Brannan in ^[7]. A lot of quality work on the generalization of this concept has already been done. Working in the same manner, we have defined the following new classes.

Definition 1.1. Let

be analytic in $\Delta$ such that $\nu(0) = 1$. Then for $m\geq2$, $\nu(z)\in P_m(\hbar (z)),$ if and only if

where $\hbar (z)$ is a convex univalent function in $\Delta$ and $\nu_{i}(z)\prec \hbar (z)$ for $i = 1, 2$.

Particularly, for $m = 2$, we get the class $P(\hbar (z))$.

Definition 1.2. Let $f(z)$ and $\ell(z)$ be two analytic functions as defined in (1.1) and (1.2) such that $(f\ast \ell)^{\prime}(z)\neq0$. Let $\hbar (z)$ be a convex univalent function. Then for $m\geq2$, $f\in V_m[\hbar (z); \ell(z)]$ if and only if

Particularly, for $m = 2$, we will get the class $C[\hbar (z); \ell(z)]$. So, a function $f\in C[\hbar (z); \ell(z)]$ if and only if

Definition 1.3. Let $f(z)$ and $\ell(z)$ be the functions defined in (1.1) and (1.2), then $f(z)\in R_m[\hbar (z); \ell(z)]$ if and only if

Particularly, for $m = 2$, we get the class $S^{\Lambda }[\hbar (z); \ell(z)]$, i.e., $f\in S^{\Lambda }[\hbar (z); \ell(z)]$ if and only if

From (1.5) and (1.6) it can be easily noted that $f\in V_m[\hbar (z); \ell(z)]$ if and only if $zf^{\prime}(z)\in R_m[\hbar (z); \ell(z)]$. For $m = 2,$ this relation will hold for the classes $C[\hbar (z); \ell(z)]$ and $S^{\Lambda }[\hbar (z); \ell(z)]$.

Definition 1.4. Let $f(z)$ and $\ell(z)$ be analytic function as defined in (1.1) and (1.2) and $m\geq2$. Let $\hbar (z)$ be the convex univalent function. Then, $f\in T_m[\hbar (z); \ell(z)]$ if and only if there exists a function $\psi(z)\in S^{\Lambda }[\hbar (z); \ell(z)]$ such that

It is interesting to note that the particular cases of our newly defined classes will give us some well-known classes already discussed in the literature. Some of these special cases have been elaborated below.

Special Cases: Let $\ell(z)$ be the identity function defined as $\frac{z}{1-z}$ denoted by $I$ i.e., $f\ast \ell = f\ast I = f$. Then

(1) For $\hbar (z) = \frac{1+z}{1-z}$ we have $P_m(\hbar (z)) = P_m, R_m[\hbar (z); \ell(z)] = R_m$ introduced by Pinchuk ^[23] and the class $V_m[\hbar (z); \ell(z)] = V_m$ defined by Paatero ^[21]. For $m = 2$, we will get the well-known classes of convex functions $C$ and the starlike functions $S^{\Lambda }$.

(2) Taking $\hbar (z) = \frac{1+(1-2\delta)z}{1-z},$ we get the classes $P_m(\delta), R_m(\delta)$ and $V_m(\delta)$ presented in ^[22]. For $m = 2,$ we will get the classes $C(\delta)$ and $S^{\Lambda }(\delta).$

(3) Letting $\hbar (z) = \frac{1+Az}{1+Bz}$, with $-1\leq B < A \leq 1$ introduced by Janowski in ^[12], the classes $P_m[A, B], R_m[A, B]$ and $V_m[A, B]$ defined by Noor ^[16,17] can be obtained. Moreover, the classes $C[A, B]$ and $S^{\Lambda }[A, B]$ introduced by ^[12] can be derived by choosing $m = 2$.

A significant work has already been done by considering $\ell(z)$ to be different linear operators and $\hbar (z)$ to be any convex univalent function. For the details see (^{[4,9,18,19,24]}).

The importance of Mittag-Leffler functions have tremendously been increased in the last four decades due to its vast applications in the field of science and technology. A number of geometric properties of Mittag-Leffler function have been discussed by many researchers working in the field of Geometric function theory. For some recent and detailed study on the Geometric properties of Mittag-Leffler functions see (^[2,3,31]).

Special function theory has a vital role in both pure and applied mathematics. Mittag-Leffler functions have massive contribution in the theory of special functions, they are used to investigate certain generalization problems. For details see ^[11,26]

There are numerous applications of Mittag-Leffler functions in the analysis of the fractional generalization of the kinetic equation, fluid flow problems, electric networks, probability, and statistical distribution theory. The use of Mittag-Leffler functions in the fractional order integral equations and differential equations attracted many researchers. Due to its connection and applications in fractional calculus, the significance of Mittag-Leffler functions has been amplified. To get a look into the applications of Mittag-Leffler functions in the field of fractional calculus, (see ^{[5,27,28,29,30]}).

Here, in this article we will use the operator $H^{\gamma, \kappa} _{\lambda, \eta }: \Omega\rightarrow \Omega$, introduced by Attiya ^[1], defined as

where $\eta, \gamma\in C$, $\Re(\lambda) > max\{0, \Re(k)-1\}$ and $\Re(k) > 0$. Also, $\Re(\lambda) = 0$ when $\Re(k) = 1;\eta\neq0$. Here, $\mu^{\gamma, \kappa} _{\lambda, \eta }$ is the generalized Mittag-Leffler function, defined in ^[25]. The generalized Mittag-Leffler function has the following representation.

So, the operator defined in (1.8) can be rewritten as:

Attiya ^[1] presented the properties of the aforesaid operator as follows:

and

However, as essential as real-world phenomena are, discovering a solution for the commensurate scheme and acquiring fundamentals with reverence to design variables is challenging and time-consuming. Among the most pragmatically computed classes, we considered the new and novel class which is very useful for efficiently handling complex subordination problems. Here, we propose a suitably modified scheme in order to compute the Janowski type function of the form $\hbar (z) = \left(\frac{1+Az}{1+Bz}\right)^\beta,$ where $0 < \beta\leq1$ and $-1\leq B < A \leq1,$ which is known as the strongly Janowski type function. Moreover, for $\ell(z),$ we will use the function defined in (1.9). So, the classes defined in Definition 1.1–1.4 will give us the following novel classes.

Definition 1.5. A function $\nu(z)$ as defined in Eq (1.3) is said to be in the class $P_{(m, \beta)}[A, B]$ if and only if for $m\geq2$ there exist two analytic functions $\nu_1(z)$ and $\nu_2(z)$ in $\Delta$, such that

where $\nu_i(z)\prec \left(\frac{1+Az}{1+Bz}\right)^\beta$ for $i = 1, 2.$ For $m = 2$, we get the class of strongly Janowki type functions $P_{\beta}[A, B]$.

Moreover,

where $\eta, \gamma\in C$, $\Re(\lambda) > max\{0, \Re(k)-1\}$ and $\Re(k) > 0$. Also, $\Re(\lambda) = 0$ when $\Re(k) = 1;\eta\neq0$. It can easily be noted that there exists Alexander relation between the classes $V_{(m, \beta)}[A, B; \gamma, \eta]$ and $R_{(m, \beta)}[A, B; \gamma, \eta]$, i.e.,

Throughout this investigation, $-1\leq B < A\leq1$, $m\geq2$ and $0 < \beta\leq1$ unless otherwise stated.

2.   Preliminaries

Lemma 2.1. (^[13]) Let $\nu(z)$ as defined in (1.3) be in $P_{(m, \beta)}[A, B].$ Then $\nu(z)\in P_{m}(\varrho)$, where $0\leq\varrho = \left(\frac{1-A}{1-B}\right)^\beta < 1$.

Lemma 2.2. (^[8]) Let $\hbar (z)$ be convex univalent in $\Delta$ with $h(0) = 1$ and $\Re{(\zeta \hbar (z)+\alpha)} > 0 \quad (\zeta\in C).$ Let $p(z)$ be analytic in $\Delta$ with $p(0) = 1,$ which satisfy the following subordination relation

then

Lemma 2.3. (^[10]) Let $\hbar (z)\in P.$ Then for $|z| < r,$ $\frac{1-r}{1+r}\leq\Re(\hbar (z))\leq\ |\hbar (z)|\leq\frac{1+r}{1-r},$ and $|h^{\prime}(z)|\leq\frac{2r\Re \hbar (z)}{1-r^2}.$

3.   Main results

Theorem 3.1. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. Then for $\Re(\frac{\gamma}{\kappa}) > -\varrho$,

Proof. Let $f(z)\in R_{(m, \beta)}[A, B, \gamma+1, \eta]$. Set

then $\varphi(z)\in P_{(m, \beta)}[A, B].$ Now, Assume that

Plugging (1.10) in (3.2), we get

It follows that

After performing logarithmic differentiation and simple computation, we get

Now, for $m\geq2,$ consider

Combining (3.3) and (3.4) with the similar technique as used in Theorem 3.1 of ^[20], we get

where

for $i = 1, 2.$ Since $\varphi(z)\in P_{(m, \beta)}[A, B],$ therefore

for $i = 1, 2.$ By using Lemma 2.1 and the condition $\Re(\frac{\gamma}{\kappa}) > -\varrho$, we have

where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta.$ Hence, in view of Lemma 2.2, we have

for i = 1, 2. This implies $\psi(z)\in P_{(m, \beta)}[A, B]$, so

which is required to prove.

Theorem 3.2. If $\Re(\frac{\lambda}{\eta}) > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, then

Proof. Let $f(z)\in R_{(m, \beta)}[A, B, \gamma, \eta]$. Taking

we have $\varphi(z)\in P_{(m, \beta)}[A, B].$ Now, suppose that

Applying the relation (1.11) in the Eq (3.6), we have

arrives at

So by the logarithmic differentiation and simple computation we get,

Therefore, for $m\geq2,$ take

Combining Eqs (3.6) and (3.7) using the similar technique as in Theorem 3.1 of ^[20], we get

where

for $i = 1, 2.$ Since $\varphi(z)\in P_{(m, \beta)}[A, B],$ therefore

for $i = 1, 2.$ Applying Lemma 2.1 and the condition $\Re(\frac{\eta}{\lambda}) > -\varrho$, we get

where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta.$ Hence, by Lemma 2.2, we have

for i = 1, 2. This implies $\psi(z)\in P_{(m, \beta)}[A, B]$, so

which completes the proof.

Corollary 3.1. For $m = 2$, if $\Re(\frac{\gamma}{\kappa}) > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. Then

Moreover, if $\Re(\frac{\lambda}{\eta}) > -\varrho$, then

Theorem 3.3. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. Then for $\Re(\frac{\gamma}{\kappa}) > -\varrho$,

Proof. By means of theorem 3.1 and Alexander relation defined in (1.12), we get

Hence the result.

Analogously, we can prove the following theorem.

Theorem 3.4. If $\Re(\frac{\lambda}{\eta}) > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, then

Corollary 3.2. For $m = 2$, if $\Re(\frac{\gamma}{\kappa}) > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. Then

Moreover, if $\Re(\frac{\lambda}{\eta}) > -\varrho$, then

Theorem 3.5. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta,$ and $\Re(\frac{\gamma}{\kappa}) > -\varrho$. Then

Proof. Let $f(z)\in T_{(m, \beta)}[A, B, \gamma+1, \eta]$. Then there exist $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma+1, \eta]$ such that

Now consider

Since $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma+1, \eta]$ and $\Re(\frac{\gamma}{\kappa}) > -\varrho$, therefore by Corollary 3.3, $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$. So

By doing some simple calculations on (3.11), we get

Now applying the relation (1.10) on (3.10), we get

Differentiating both sides of (3.13), we have

By using (3.12) and with some simple computations, we get

with $\Re(q(z)+\frac{\gamma}{\kappa}) > 0$, since $q(z)\in P_\beta[A, B]$, so by Lemma 2.1, $\Re(q(z) > \varrho$ and $\Re(\frac{\gamma}{\kappa}) > -\varrho$. Now consider

Combining (3.14) and (3.15) with the similar technique as used in Theorem 3.1 of ^[20], we get

where

for $i = 1, 2$. Since $\varphi(z)\in P_{(m, \beta)}[A, B]$, therefore

Using the fact of Lemma 2.2, we can say that

So, $\phi(z)\in P_{(m, \beta)}[A, B].$ Hence we get the required result.

Theorem 3.6. If $\Re(\frac{\lambda}{\eta}) > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, then

Let $f(z)\in T_{(m, \beta)}[A, B, \gamma, \eta]$. Then there exist $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$ such that

Taking

As we know that, $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$ and $\Re(\frac{\eta}{\lambda}) > -\varrho$, therefore by Corollary 3.3, $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta+1]$. So

By doing some simple calculations on (3.19) with the help of (1.11), we get

Now, applying the relation (1.11) on (3.18), we get

Differentiating both sides of Eq (3.21), we have

some simple calculations along with using (3.20) give us

with $\Re(q(z)+\frac{\eta}{\lambda}) > 0$. Since $q(z)\in P_\beta[A, B]$, so applying Lemma 2.1, we have $\Re(q(z) > \varrho$ and $\Re(\frac{\eta}{\lambda}) > -\varrho$.

Assume that

Combining (3.22) and (3.23), along with using the similar technique as in Theorem 3.1 of ^[20], we get

where

for $i = 1, 2$. Since $\varphi(z)\in P_{(m, \beta)}[A, B]$, therefore

Applying the fact of Lemma 2.2, we have

So $\phi(z)\in P_{(m, \beta)}[A, B].$ Which gives us the required result.

Corollary 3.3. If $\varrho > -min\lbrace\Re\left(\frac{\gamma}{\kappa}\right), \Re\left(\frac{\lambda}{\eta}\right)\rbrace$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta,$ then we have the following inclusion relations:

(i) $R_{(m, \beta)}[A, B, \gamma+1, \eta]\subset R_{(m, \beta)}[A, B, \gamma, \eta]\subset R_{(m, \beta)}[A, B, \gamma, \eta+1].$

(ii)$V_{(m, \beta)}[A, B, \gamma+1, \eta]\subset V_{(m, \beta)}[A, B, \gamma, \eta]\subset V_{(m, \beta)}[A, B, \gamma, \eta+1].$

(iii)$T_{(m, \beta)}[A, B, \gamma+1, \eta]\subset T_{(m, \beta)}[A, B, \gamma, \eta]\subset T_{(m, \beta)}[A, B, \gamma, \eta+1].$

Now, we will discuss some radius results for our defined classes.

Theorem 3.7. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, and $\Re\left(\frac{\gamma}{\kappa}\right) > -\varrho$. Then

whenever

Proof. Let $f(z)\in R_{(m, \beta)}[A, B, \gamma, \eta]$. Then

In view of Lemma 2.1 $P_{(m, \beta)}[A, B]\subset P_m(\varrho),$ for $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, therefore $\psi(z)\in P_m(\varrho).$ So by the Definition of $P_m(\varrho)$ given in ^[22], there exist two functions $\psi_1(z), \psi_2(z) \in P(\varrho)$ such that

with $m\geq2$ and $\Re(\psi_i(z)) > \varrho, i = 1, 2.$ We can write

where $h_i(z)\in P$ and $\Re(h_i(z) > 0$, for $i = 1, 2$. Now, let

We have to check when $\phi(z)\in P_m(\varrho)$. Using relation (1.10) in (3.25), we get

So, by simple calculation and logarithmic differentiation, we get

Now, consider

where

To derive the condition for $\phi_i(z)$ to be in $P(\varrho)$, consider

In view of (3.27), we have

We have, $\Re(\frac{\gamma}{\kappa}+\varrho) > 0$ since $\Re(\frac{\gamma}{\kappa}) > -\varrho$. Since $h_i(z)\in P,$ hence by using Lemma 2.3 in inequality (3.30), we have

Since $1-r^2 > 0$, letting $T(r) = r^2(1-\varrho)-2r(2-\varrho)+(1-\varrho).$ It is easy to note that $T(0) > 0$ and $T(1) < 0$. Hence, there is a root of $T(r)$ between $0$ and $1$. Let $r_o$ be the root then by simple calculations, we get

Hence $\phi(z)\in P_m(\varrho)$ for $|z| < r_o$. Thus for this radius $r_o$ the function $f(z)$ belongs to the class $R_{(m, \beta)}[\varrho, \gamma+1, \eta],$ which is required to prove.

Theorem 3.8. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, and $\Re\left(\frac{\lambda}{\eta}\right) > -\varrho$. Then

whenever

Proof. Let $f(z)\in R_{(m, \beta)}[A, B, \gamma, \eta+1]$. Then

By applying of Lemma 2.1, we get $P_{(m, \beta)}[A, B]\subset P_m(\varrho),$ for $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, therefore $\psi(z)\in P_m(\varrho).$ Hence, the Definition of $P_m(\varrho)$ given in ^[22], there exist two functions $\psi_1(z), \psi_2(z) \in P(\varrho)$ such that

with $m\geq2$ and $\Re(\psi_i(z)) > \varrho, i = 1, 2.$ We can say that

where $h_i(z)\in P$ and $\Re(h_i(z) > 0$, for $i = 1, 2$. Now, assume

Here, We have to obtain the condition for which $\phi(z)\in P_m(\varrho)$. Using relation (1.11) in (3.51), we get

Thus, by simple calculation and logarithmic differentiation, we have

Now, consider

where

To derive the condition for $\phi_i(z)$ to be in $P(\varrho)$, consider

In view of (3.34), we have

Here, $\Re(\frac{\eta}{\lambda}+\varrho) > 0$ since $\Re(\frac{\eta}{\lambda}) > -\varrho$. We know that $h_i(z)\in P,$ therefore by using Lemma 2.3 in inequality (3.37), we have

Since $1-r^2 > 0$, letting $T(r) = r^2(1-\varrho)-2r(2-\varrho)+(1-\varrho).$ It can easily be seen that $T(0) > 0$ and $T(1) < 0$. Hence, there is a root of $T(r)$ between $0$ and $1$. Let $r_o$ be the root then by simple calculations, we get

Hence $\phi(z)\in P_m(\varrho)$ for $|z| < r_o$. Thus for this radius $r_o$ the function $f(z)$ belongs to the class $R_{(m, \beta)}[\varrho, \gamma, \eta],$ which is required to prove.

Corollary 3.4. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. Then, for $m = 2$, and $|z| < r_o = \frac{1-\varrho}{2-\varrho+\sqrt{3-2\varrho}},$

(i) If $\Re\left(\frac{\gamma}{\kappa}\right) > -\varrho,$ then $S^{\Lambda }_{\beta}[A, B, \gamma, \eta]\subset S^{\Lambda }_{\beta}[\varrho, \gamma+1, \eta].$

(ii) If$\Re\left(\frac{\lambda}{\eta}\right) > -\varrho,$ then $S^{\Lambda }_{\beta}[A, B, \gamma, \eta+1]\subset S^{\Lambda }_{\beta}[\varrho, \gamma, \eta].$

Theorem 3.9. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. Then for $|z| < r_o = \frac{1-\varrho}{2-\varrho+\sqrt{3-2\varrho}}$, we have

(1)$V_{(m, \beta)}[A, B, \gamma, \eta] \subset V_{(m, \beta)}[\varrho, \gamma+1, \eta]$, if $\Re\left(\frac{\gamma}{\kappa}\right) > -\varrho$.

(2)$V_{(m, \beta)}[A, B, \gamma, \eta+1] \subset V_{(m, \beta)}[\varrho, \gamma, \eta]$, if $\Re\left(\frac{\lambda}{\eta}\right) > -\varrho.$

Proof. The above results can easily be proved by using Theorem 3.10, Theorem 3.11 and the Alexander relation defined in (1.12).

Theorem 3.10. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, and $\Re\left(\frac{\gamma}{\kappa}\right) > -\varrho$. Then

whenever

Proof. Let $f\in T_{(m, \beta)}[A, B, \gamma, \eta]$, then there exist $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$ such that

Since by Lemma 2.1 we know that $P_{(m, \beta)}[A, B]\subset P_m(\varrho)$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, therefore $\varphi(z)\in P_m(\varrho)$. So by using the Definition of $P_m(\varrho)$ defined in ^[22], there exist two functions $\varphi_1(z)$ and $\varphi_2(z)$ such that

where $\varphi_i(z)\in P(\varrho), i = 1, 2.$ We can write

where $h_i(z)\in P$. Now, let

Since $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$, therefore

then by using relation (1.10) and doing some simple computation on Eq (3.42), we have

Now, using relation (1.10) in (3.39), we get

By some simple calculations along with differentiation of both sides of (3.44) and then applying (3.43) we get the following relation

Let us consider

where

$i = 1, 2.$ Since $q(z)\in P_{\beta}[A, B]\subset P(\varrho)$. Therefore, we can write

where $q_o(z)\in P$. We have to check when $\phi_i(z)\in P_m(\varrho).$ For this consider

Using (3.41) and (3.45), we have

where $h_i(z), q_o(z) \in P.$

Since $\Re(\frac{\gamma}{\kappa}) > -\varrho,$ so $\Re\left(\varrho+\frac{\gamma}{\kappa}\right) > 0$. Now by using the distortion results of Lemma 2.3, we have

Since $h_i(z)\in P,$ so $\Re(h_i(z)) > 0$ and $\Re(\frac{\gamma}{\kappa}+\varrho) > 0$ for $\Re(\frac{\gamma}{\kappa}) > -\varrho$. Hence, by using Lemma 2.3 in inequality (3.46), we have

Since $1-r^2 > 0$, taking $T(r) = r^2(1-\varrho)-2r(2-\varrho)+(1-\varrho).$ Let $r_o$ be the root then by simple calculations, we get

Hence $\phi(z)\in P_m(\varrho)$ for $|z| < r_o$. Thus for this radius $r_o$ the function $f(z)$ belongs to the class $T_{(m, \beta)}[\varrho, \gamma+1, \eta],$ which is required to prove.

Using the analogous approach used in Theorem 3.14, one can easily prove the following theorem.

Theorem 3.11. Let $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$, and $\Re\left(\frac{\eta}{\lambda}\right) > -\varrho$. Then

whenever

Integral Preserving Property: Here, we will discuss some integral preserving properties of our aforementioned classes. The generalized Libera integral operator $I_\sigma$ introduced and discussed in ^[6,14] is defined by:

where $f(z)\in A$ and $\sigma > -1.$

Theorem 3.12. Let $\sigma > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. If $f\in R_{(m, \beta)}[A, B, \gamma, \eta]$ then $I_\sigma(f)\in R_{(m, \beta)}[A, B, \gamma, \eta].$

Proof. Let $f\in R_{(m, \beta)}[A, B, \gamma, \eta]$, and set

where $\psi(z)$ is analytic and $\psi(0) = 1$. From definition of $H^{\gamma, \kappa} _{\lambda, \eta }(f)$ given by ^[1] and using Eq (3.47), we have

Then by using Eqs (3.48) and (3.49), we have

Logarithmic differentiation and simple computation results in

with $\Re(\psi(z)+\sigma) > 0$, since $\Re(\sigma) > -\varrho.$ Now, consider

Combining (3.50) and (3.51), we get

where $\phi_i(z) = \psi_i(z)+\frac{z\psi_i^{\prime}(z)}{\psi_i(z)+\sigma}$, $i = 1, 2.$ Since $\phi(z)\in P_{(m, \beta)}[A, B]$, therefore

which implies

Therefore, using Lemma 2.2 we get

or $\psi(z)\in P_{(m, \beta)}[A, B]$. Hence the result.

Corollary 3.5. Let $\sigma > -\varrho$. Then for $m = 2$, if $f\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$ then $I_\sigma(f)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta],$ where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$.

Theorem 3.13. Let $\sigma > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. If $f\in V_{(m, \beta)}[A, B, \gamma, \eta]$ then $I_\sigma(f)\in V_{(m, \beta)}[A, B, \gamma, \eta].$

Proof. Let $f\in V_{(m, \beta)}[A, B, \gamma, \eta]$. Then by using relation (1.12), we have

so by using Theorem 3.16, we can say that

equivalently

so again by using the relation (1.12), we get

Theorem 3.14. Let $\sigma > -\varrho$, where $\varrho = \left(\frac{1-A}{1-B}\right)^\beta$. If $f\in T_{(m, \beta)}[A, B, \gamma, \eta]$ then $I_\sigma(f)\in T_{(m, \beta)}[A, B, \gamma, \eta].$

Proof. Let $f\in T_{(m, \beta)}[A, B, \gamma, \eta]$. Then there exists $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$, such that

Consider

Since $\psi(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$, then by Corollary 3.17, $I_\sigma(\psi)(z)\in S^{\Lambda }_{\beta}[A, B, \gamma, \eta]$. Therefore

By using (3.47) and Definition of $H^{\gamma, \kappa} _{\lambda, \eta }$, we get

or we can write it as

Now using the relation (3.47) and the Definition of $H^{\gamma, \kappa} _{\lambda, \eta },$ in (3.53), we have

Differentiating both sides of (3.56), we have

then by simple computations and using (3.53)–(3.55), we get

with $\Re(\sigma) > -\varrho$, so $\Re(q(z)+\sigma) > 0$, since $q(z)\in P_{\beta}[A, B]\subset P(\varrho).$ Consider

Combining Eqs (3.57) and (3.58), we have

where $\varphi_i(z) = \phi_i(z)+\frac{z\phi_i^{\prime}(z)}{q(z)+\sigma}$, $i = 1, 2.$

Since $\varphi(z)\in P_{(m, \beta)}[A, B]$, thus we have

then

Since $\Re(q(z)+\sigma) > 0$, therefore using Lemma 2.2 we get

thus $\phi(z)\in P_{(m, \beta)}[A, B]$. Hence the result.

4.   Conclusions

Due to their vast applications, Mittag-Leffler functions have captured the interest of a number of researchers working in different fields of science. The present investigation may help researchers comprehend some stimulating consequences of the special functions. In the present article, we have used generalized Mittag-Leffler functions to define some novel classes related to bounded boundary and bounded radius rotations. Several inclusion relations and radius results for these classes have been discussed. Moreover, it has been proved that these classes are preserved under the generalized Libera integral operator. Finally, we can see that the projected solution procedure is highly efficient in solving inclusion problems describing the harmonic analysis. It is hoped that our investigation and discussion will be helpful in cultivating new ideas and applications in different fields of science, particularly in mathematics.

5.   List of Notations

$\Delta$ Open Unit Disc.

$\Omega$ Class of normalized analytic functions.

$\Re$ Real part of complex number.

$\Gamma$ Gamma function.

$\chi(z)$ Schwartz function.

Conflict of interest

The authors declare that they have no competing interests.

Acknowledgements

The authors would like to thank the Rector of COMSATS Univeristy Islamabad, Pakistan for providing excellent research oriented environment. The author Thabet Abdeljawad would like to thank Prince Sultan University for the support through TAS research Lab.