Some Results on a Class of Harmonic Univalent Functions Defined by Generalised Derivative Operator   

Some Results on a Class of Harmonic Univalent Functions Defined by Generalised Derivative Operator   

Entisar El-Yagubi^*, Samira Ziada

Mathematical Department, Gharyan University, Faculty of Science, Gharyan, Libya 

^*Corresponding Author

DOI: https://doi.org/10.51244/IJRSI.2024.1109033

Received: 28 August 2024; Accepted: 18 September 2024; Published: 03 October 2024

ABSTRACT

Harmonic function is one of an important branches of complex analysis. The first study of complex – valued, harmonic mappings defined on a domain ꓓ ⊂ C was given by Clunie and Sheil-Small [1]. Harmonic functions have been studied by different researchers such as Silverman [6]. In the present paper, a new class of harmonic univalent functions will be introduced. Various properties of functions belong to this class which include coefficient bounds, growth bounds, a closure property, extreme points, neighborhood and a convex combination will be obtained.

Keywords: Univalent functions, harmonic functions, derivative operator, distortion inequalities.

INTRODUCTION

Let \( U = \{ z \in \mathbb{C} : |z| < 1 \} \) be the open unit disc, and let \( S_H \) denote the class of all complex-valued, harmonic, sense-preserving, univalent functions \( f \) in \( U \), normalized by \( f(0) = 0 \) and \( f'(0) – 1 = 0 \). Each function \( f \in S_H \) is expressed as \[ f(z) = h(z) + \overline{g(z)}, \] where \( h \) and \( g \) belong to the linear space \( H(U) \) of all analytic functions on \( U \), taking the form \[ h(z) = z + \sum_{n=2}^{\infty} a_n z^n, \quad g(z) = \sum_{n=1}^{\infty} b_n z^n. \tag{1.1} \] Thus, for each \( f \in S_H \), we have \[ f(z) = z + \sum_{n=2}^{\infty} a_n z^n + \overline{\sum_{n=1}^{\infty} b_n z^n}, \quad z \in U. \tag{1.2} \] Clunie and Sheil-Small [1] proved that \( S_H \) is not compact, and the necessary and sufficient condition for \( f \) to be locally univalent and sense-preserving in any simply connected domain \( U \) is that \( |h'(z)| > |g'(z)| \).

Darus and Ibrahim [2] introduced the generalized derivative operator, denoted by \( D^{k}_{\delta, \beta, \lambda} f(z) \) for \( f \in A \), as follows:
\[
D^{k}_{\delta, \beta, \lambda} f(z) = z + \sum_{n=2}^{\infty} [\beta(n-1)(\lambda – \delta) + 1]^k a_n z^n, \tag{1.3}
\]
where \( \delta \geq 0 \), \( \beta > 0 \), \( \lambda > 0 \), \( \delta \neq \lambda \), and \( k \in \mathbb{N}_0 = \{0, 1, 2, \dots \} \).

In this paper, the operator \( D^{k}_{\delta, \beta, \lambda} f(z) \) will be introduced for \( f = h + \overline{g} \) where \( h \) and \( g \) are given by (1.1), and it will be expressed as
\[
D^{k}_{\delta, \beta, \lambda} f(z) = D^{k}_{\delta, \beta, \lambda} h(z) + (-1)^k \overline{D^{k}_{\delta, \beta, \lambda} g(z)}, \quad z \in U, \tag{1.4}
\]
where
\[
D^{k}_{\delta, \beta, \lambda} h(z) = z + \sum_{n=2}^{\infty} [\beta(n-1)(\lambda – \delta) + 1]^k a_n z^n,
\]
\[
D^{k}_{\delta, \beta, \lambda} g(z) = \sum_{n=1}^{\infty} [\beta(n-1)(\lambda – \delta) + 1]^k b_n z^n. \tag{1.5}
\]

For \( \delta \geq 0 \), \( \beta > 0 \), \( \lambda > 0 \), \( \delta \neq \lambda \), and \( k \in \mathbb{N}_0 \), we further denote by \( \overline{S_H} \), the subclass of \( S_H \) consisting of harmonic functions of the form
\[
f_k = h + \overline{g_k}, \tag{1.6}
\]
where
\[
h(z) = z – \sum_{n=2}^{\infty} |a_n| z^n,
\]
\[
g_k(z) = (-1)^k \sum_{n=1}^{\infty} |b_n| z^n, \quad z \in U, \quad |b_1| < 1.
\]

A class of harmonic univalent functions is introduced as follows:

Definition 1.1: The function \( f = h + \overline{g} \) defined by (1.2) is in the class \( S_H^k(\delta, \beta, \lambda, \alpha) \) if
\[
\Re\left(\frac{D^{k+1}_{\delta, \beta, \lambda} f(z)}{D^k_{\delta, \beta, \lambda} f(z)}\right) \geq \alpha, \tag{1.7}
\]
where \( 0 \leq \alpha < 1 \), \( \delta \geq 0 \), \( \beta > 0 \), \( \lambda > 0 \), \( \delta \neq \lambda \), and \( k \in \mathbb{N}_0 \).

Note that the class \( \overline{S_H}^0(0, 0, 0, \alpha) \equiv S_H(\alpha) \) is the class of sense-preserving harmonic univalent functions \( f \), which are starlike of order \( \alpha \) in \( U \), as studied by Jahangiri [3]. The class \( \overline{S_H}^k(0, 1, 1, \alpha) \) is the class of Salagean-type harmonic univalent functions introduced by Jahangiri et al. [4].

We further denote by \( \overline{S_H}^k(\delta, \beta, \lambda, \alpha) \), the subclass of \( S_H^k(\delta, \beta, \lambda, \alpha) \), where \( \overline{S_H}^k(\delta, \beta, \lambda, \alpha) = S_H^k(\delta, \beta, \lambda, \alpha) \cap \overline{S_H} \).

MAIN RESULTS

In the next theorem, a sufficient coefficient bound related to the class \( S_H^k (\delta, \beta, \lambda, \alpha) \) shall be obtained.

Theorem 2.1. Let \( f = h + \overline{g} \) be given by (1.2). Furthermore, let

\[
\sum_{n=2}^{\infty} \Omega^k [\Omega – \alpha] |a_n| + \sum_{n=1}^{\infty} \Omega^k [\Omega + \alpha] |b_n| \leq 1 – \alpha \tag{2.1}
\]

where

\[
\Omega^k = [\beta(n-1)(\lambda – \delta) + 1]^k, \quad \Omega = [\beta(n-1)(\lambda – \delta) + 1],
\]
\[
a_1 = 1, \quad 0 \leq \alpha < 1, \quad \delta \geq 0, \quad \beta > 0, \quad \lambda > 0, \quad \delta \neq \lambda, \quad k \in \mathbb{N}_0,
\]

then \( f \in S_H^k (\delta, \beta, \lambda, \alpha) \).

Proof. According to (1.7), we have

\[
\Re \left( \frac{D^{k+1}_{\delta, \beta, \lambda} f(z)}{D^k_{\delta, \beta, \lambda} f(z)} \right) \geq \alpha.
\]

This is equivalent to \( \Re \left( \frac{A(z)}{B(z)} \right) > \alpha \), where \( A(z) = D^{k+1}_{\delta, \beta, \lambda} f(z) \) and \( B(z) = D^k_{\delta, \beta, \lambda} f(z) \).

Using the fact that \( \Re(w) > \alpha \) if \( |1 – \alpha + w| \geq |1 + \alpha – w| \), it suffices to show that

\[
|A(z) + (1 – \alpha) B(z)| \geq |A(z) – (1 + \alpha) B(z)|.
\]

Substituting values of \( A(z) \) and \( B(z) \), and with simple calculations, we get

\[
|D^{k+1}_{\delta, \beta, \lambda} f(z) + (1 – \alpha) D^k_{\delta, \beta, \lambda} f(z)| \geq |D^{k+1}_{\delta, \beta, \lambda} f(z) – (1 + \alpha) D^k_{\delta, \beta, \lambda} f(z)|.
\]

\[
|z + \sum_{n=2}^{\infty} \Omega^{k+1} a_n z^n + (-1)^{k+1} \sum_{n=1}^{\infty} \Omega^{k+1} \overline{b_n} \overline{z^n} + (1 – \alpha) \left[ z + \sum_{n=2}^{\infty} \Omega^k a_n z^n + (-1)^k \sum_{n=1}^{\infty} \Omega^k \overline{b_n} \overline{z^n} \right]|
\]

\[
\geq |z + \sum_{n=2}^{\infty} \Omega^{k+1} a_n z^n + (-1)^{k+1} \sum_{n=1}^{\infty} \Omega^{k+1} \overline{b_n} \overline{z^n} – (1 + \alpha) \left[ z + \sum_{n=2}^{\infty} \Omega^k a_n z^n + (-1)^k \sum_{n=1}^{\infty} \Omega^k \overline{b_n} \overline{z^n} \right]|.
\]

\[
|(2 – \alpha)z + \sum_{n=2}^{\infty} \Omega^k [\Omega + (1 – \alpha)] a_n z^n – (-1)^k \sum_{n=1}^{\infty} \Omega^k [\Omega – (1 – \alpha)] \overline{b_n} \overline{z^n}|
\]

\[
– |-\alpha z + \sum_{n=2}^{\infty} \Omega^k [\Omega – (1 + \alpha)] a_n z^n + (-1)^k \sum_{n=1}^{\infty} \Omega^k [-\Omega – (1 + \alpha)] \overline{b_n} \overline{z^n}|
\]

\[
\geq 2(1 – \alpha) |z| – \sum_{n=2}^{\infty} \Omega^k [2\Omega – 2\alpha] |a_n| |z|^n – \sum_{n=1}^{\infty} \Omega^k [2\Omega + 2\alpha] |\overline{b_n}| |\overline{z}|^n
\]

\[
\geq 2(1 – \alpha) |z| \left\{ 1 – \sum_{n=2}^{\infty} \Omega^k \left[ \frac{\Omega – \alpha}{1 – \alpha} \right] |a_n| |z|^{n-1} – \sum_{n=1}^{\infty} \Omega^k \left[ \frac{\Omega + \alpha}{1 – \alpha} \right] |\overline{b_n}| |\overline{z}|^{n-1} \right\}
\]

\[
\geq 0
\]

by assumption. Hence, the proof is complete.

Theorem 2.2. Let \( f_k = h + \overline{g_k} \) be given by (1.6). Then \( f_k \in \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \) if and only if

\[
\sum_{n=2}^{\infty} \Omega^k [\Omega – \alpha] |a_n| + \sum_{n=1}^{\infty} \Omega^k [\Omega + \alpha] |b_n| \leq 1 – \alpha,
\tag{2.2}
\]

where

\[
\Omega^k = [\beta(n-1)(\lambda – \delta) + 1]^k, \quad \Omega = [\beta(n-1)(\lambda – \delta) + 1],
\]

\( a_1 = 1, \ 0 \leq \alpha < 1, \ \delta \geq 0, \ \beta > 0, \ \lambda > 0, \ \delta \neq \lambda, \ k \in \mathbb{N}_0 \).

Proof. Since \( \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \subseteq S_H^k (\delta, \beta, \lambda, \alpha) \), we only need to prove the “only if” part of the theorem. Note that a necessary and sufficient condition for \( f_k = h + \overline{g_k} \) given by (1.6) to be in \( \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \) is that

\[
\Re \left\{ \frac{D_{\delta, \beta, \lambda}^{k+1} f(z)}{D_{\delta, \beta, \lambda}^k f(z)} \right\} \geq \alpha,
\]

which is equivalent to

\[
\Re \left\{ \frac{D_{\delta, \beta, \lambda}^{k+1} f_k(z) – \alpha D_{\delta, \beta, \lambda}^k f_k(z)}{D_{\delta, \beta, \lambda}^k f_k(z)} \right\} \geq 0.
\]

This is further equivalent to:

\[
\Re \left\{ \frac{(1 – \alpha)z – \sum_{n=2}^{\infty} \Omega^k [\Omega – \alpha] a_n z^n – (-1)^k \sum_{n=1}^{\infty} \Omega^k [\Omega + \alpha] b_n \overline{z^n}}{z – \sum_{n=2}^{\infty} \Omega^k a_n z^n + (-1)^k \sum_{n=1}^{\infty} \Omega^k b_n \overline{z^n}} \right\} \geq 0.
\]

The above condition must hold for all values of \( z \), where \( |z| = r < 1 \). Choosing \( z \) on the positive real axis where \( 0 \leq z = r < 1 \), we have:

\[
\frac{(1 – \alpha) – \sum_{n=2}^{\infty} \Omega^k [\Omega – \alpha] a_n r^{n-1} – (-1)^k \sum_{n=1}^{\infty} \Omega^k [\Omega + \alpha] b_n \overline{r^{n-1}}}{1 – \sum_{n=2}^{\infty} \Omega^k a_n r^{n-1} + (-1)^k \sum_{n=1}^{\infty} \Omega^k b_n \overline{r^{n-1}}} \geq 0.
\tag{2.3}
\]

If the condition (2.2) does not hold, then the numerator in (2.3) is negative for \( r \) sufficiently close to 1. Thus there exists \( z_0 = r_0 \) in \( (0,1) \) for which the quotient in (2.3) is negative. This contradicts the required condition for \( f_k \in \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \), and so the proof is complete.

In this section, growth bounds for \( f_k \in \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \) are obtained.

Theorem 2.3. Let \( f_k \in \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \), then for \( |z| = r < 1 \), we have:

\[
|f_k(z)| \leq (1 + |b_1|)r + \frac{1}{\Omega^k} \left( \frac{1 – \alpha}{\Omega – \alpha} – \frac{\Omega^k (\Omega + \alpha)}{\Omega – \alpha} |b_1| \right) r^2,
\]

\[
|f_k(z)| \geq (1 – |b_1|)r – \frac{1}{\Omega^k} \left( \frac{1 – \alpha}{\Omega – \alpha} – \frac{\Omega^k (\Omega + \alpha)}{\Omega – \alpha} |b_1| \right) r^2,
\]

where \( \Omega^k = [\beta(\lambda – \delta) + 1]^k \), \( \Omega = [\beta(\lambda – \delta) + 1] \), and \( a_1 = 1, \ 0 \leq \alpha < 1, \ \delta \geq 0, \ \beta > 0, \ \lambda > 0, \ \delta \neq \lambda, \ k \in \mathbb{N}_0 \).

Proof. The first inequality will be proved. The argument for the second inequality is similar and will be omitted. Let \( f_k \in \overline{(S_H)}^k (\delta, \beta, \lambda, \alpha) \). Taking the absolute value of \( f_k \), we obtain

\[
|f_k(z)| \leq (1 + |b_1|)r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^n
\]

\[
\leq (1 + |b_1|)r + \sum_{n=2}^{\infty} (|a_n| + |b_n|) r^2
\]

\[
\leq (1 + |b_1|)r + \frac{(1 – \alpha)}{(\Omega^k (\Omega – \alpha))} \sum_{n=2}^{\infty} \left( \frac{(\Omega^k (\Omega – \alpha))}{(1 – \alpha)} (|a_n| + |b_n|) \right) r^2
\]

\[
\leq (1 + |b_1|)r + \frac{(1 – \alpha)}{(\Omega^k (\Omega – \alpha))} \left( 1 – \frac{(\Omega^k (\Omega + \alpha))}{(1 – \alpha)} |b_1| \right) r^2
\]

\[
\leq (1 + |b_1|)r + \frac{1}{\Omega^k} \left( \frac{(1 – \alpha)}{(\Omega – \alpha)} – \frac{(\Omega^k (\Omega + \alpha))}{(\Omega – \alpha)} |b_1| \right) r^2.
\]

Next, we prove the closure property related to the class \( \overline{(S_H)}^k (\delta, \beta, \lambda, \alpha) \).

Theorem 2.4. Let the functions \( f_{(k_i)}(z) \) defined by

\[
f_{(k_i)}(z) = z – \sum_{n=2}^{\infty} |a_{(n,i)}| z^n + (-1)^k \sum_{n=1}^{\infty} |b_{(n,i)}| \overline{(z^n)},
\]

be in the class \( \overline{(S_H)}^k (\delta, \beta, \lambda, \alpha) \), for every \( i = 1, 2, \ldots, m \). Then the convex combination of \( f_{(k_i)} \), denoted by \( \sum_{i=1}^m t_i f_{(k_i)}(z) \), are also in the class \( \overline{(S_H)}^k (\delta, \beta, \lambda, \alpha) \), where \( \sum_{i=1}^m t_i = 1 \) and \( 0 \leq t_i \leq 1 \).

Proof. According to the definition of the convex combination of \( f_{(k_i)} \), we can write

\[
\sum_{i=1}^m t_i f_{(k_i)}(z) = z – \sum_{n=2}^{\infty} \left( \sum_{i=1}^m t_i |a_{(n,i)}| \right) z^n + (-1)^k \sum_{n=1}^{\infty} \left( \sum_{i=1}^m t_i |b_{(n,i)}| \right) \overline{(z^n)}.
\]

Further, since \( f_{(k_i)}(z) \) are in \( \overline{(S_H)}^k (\delta, \beta, \lambda, \alpha) \) for every \( i = 1, 2, \ldots, m \), then by (2.2), we have

\[
\sum_{n=2}^{\infty} \frac{\Omega^k [\Omega – \alpha]}{1 – \alpha} \left( \sum_{i=1}^m t_i |a_{(n,i)}| \right) + \sum_{n=1}^{\infty} \frac{\Omega^k [\Omega + \alpha]}{1 – \alpha} \left( \sum_{i=1}^m t_i |b_{(n,i)}| \right)
\]

\[
= \sum_{i=1}^m t_i \left[ \sum_{n=2}^{\infty} \frac{\Omega^k [\Omega – \alpha]}{1 – \alpha} |a_{(n,i)}| + \sum_{n=1}^{\infty} \frac{\Omega^k [\Omega + \alpha]}{1 – \alpha} |b_{(n,i)}| \right]
\leq 1.
\]

Therefore, \( \sum_{i=1}^m t_i f_{(k_i)} \in \overline{(S_H)}^k (\delta, \beta, \lambda, \alpha). \

Next, we present the extreme points related to the class \( \overline{S_H}^k (\delta, \beta, \lambda, \alpha). \)

Theorem 2.5. A function \( f \in \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \) if and only if \( f \) can be expressed as

\[
f(z) = \sum_{n=1}^{\infty} \left( X_n h_n(z) + Y_n g_n(z) \right), \tag{2.4}
\]

where

\[
h_1(z) = z, \quad h_n(z) = z – \frac{(1 – \alpha)}{\Omega^k (\Omega – \alpha)} z^n, \quad n \geq 2,
\]

\[
g_n(z) = z + (-1)^k \frac{(1 – \alpha)}{\Omega^k (\Omega – \alpha)} \overline{(z^n)}, \quad n \geq 1,
\]

and

\[
\sum_{n=1}^{\infty} (X_n + Y_n) = 1, \quad X_n \geq 0, \quad Y_n \geq 0,
\]

\[
\Omega^k = [\beta(n – 1)(\lambda – \delta) + 1]^k, \quad \Omega = [\beta(n – 1)(\lambda – \delta) + 1], \quad 0 \leq \alpha < 1, \quad \delta \geq 0, \quad \beta > 0, \quad \lambda > 0, \quad \delta \neq \lambda, \quad k \in \mathbb{N}_0.
\]

In particular, the extreme points of \( \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \) are \( \{ h_n \} \) and \( \{ g_n \} \).

Proof. Note that for \( f \) of the form (2.4), we may write

\[
f(z) = \sum_{n=1}^{\infty} (X_n h_n + Y_n g_n) = \sum_{n=1}^{\infty} (X_n + Y_n) z – \sum_{n=2}^{\infty} \frac{(1 – \alpha)}{\Omega^k (\Omega – \alpha)} Y_n \overline{(z^n)}.
\]

Then

\[
\sum_{n=2}^{\infty} \frac{\Omega^k (\Omega – \alpha)}{(1 – \alpha)} \left( \frac{(1 – \alpha)}{\Omega^k (\Omega – \alpha)} X_n \right) + \sum_{n=1}^{\infty} \frac{\Omega^k (\Omega – \alpha)}{(1 – \alpha)} \left( \frac{(1 – \alpha)}{\Omega^k (\Omega – \alpha)} Y_n \right)
\]

\[
= \sum_{n=2}^{\infty} X_n + \sum_{n=1}^{\infty} Y_n
\]

\[
= \sum_{n=1}^{\infty} X_n – X_1 + \sum_{n=1}^{\infty} Y_n = 1 – X_1 \leq 1.
\]

So, \( f \in \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \). Conversely, suppose that \( f \in \overline{S_H}^k (\delta, \beta, \lambda, \alpha) \).

Set

\[
X_n = \frac{\Omega^k (\Omega – \alpha)}{1 – \alpha} |a_n|, \quad 0 \leq \alpha < 1, \quad 0 \leq X_n \leq 1, \quad n \geq 2,
\]

\[
Y_n = \frac{\Omega^k (\Omega – \alpha)}{1 – \alpha} |b_n|, \quad 0 \leq \alpha < 1, \quad 0 \leq Y_n \leq 1, \quad n \geq 1.
\]

We define

\[
X_1 = 1 – \sum_{n=2}^{\infty} X_n – \sum_{n=1}^{\infty} Y_n.
\]

Therefore, \( f \) can be written as

\[
f(z) = z – \sum_{n=2}^{\infty} |a_n| z^n + (-1)^k \overline{\left( \sum_{n=1}^{\infty} |b_n| z^n \right)}
\]

\[
= z – \sum_{n=2}^{\infty} \frac{(1 – \alpha) X_n}{\Omega^k (\Omega – \alpha)} z^n + (-1)^k \overline{\left( \sum_{n=1}^{\infty} \frac{(1 – \alpha) Y_n}{\Omega^k (\Omega – \alpha)} z^n \right)}
\]

\[
= z + \sum_{n=2}^{\infty} (h_n(z) – z) X_n + \sum_{n=1}^{\infty} (g_n(z) – z) Y_n
\]

\[
= \sum_{n=2}^{\infty} h_n(z) X_n + \sum_{n=1}^{\infty} g_n(z) Y_n + z \left( 1 – \sum_{n=2}^{\infty} X_n – \sum_{n=1}^{\infty} Y_n \right)
\]

\[
= \sum_{n=1}^{\infty} (h_n(z) X_n + g_n(z) Y_n)
\]

as required.

Following Avci and Zlotkiewicz [5], we refer to the \( \gamma \)-neighborhood of a function \( f \in S_H^* (\alpha) \) as defined by

\[
N_{\gamma}(f) = \{ F(z) = z + \sum_{n=2}^{\infty} A_n z^n + \sum_{n=1}^{\infty} \overline{B_n z^n} : \sum_{n=2}^{\infty} n (|a_n – A_n| + |b_n – B_n|) + |b_1 – B_1| \leq \gamma \}.
\]

In our case, we define the generalized \( \gamma \)-neighborhood of \( f \) to be the set

\[
N_{\gamma}^D(f) = \{ F(z) : \sum_{n=2}^{\infty} ( \beta(n – 1)(\lambda – \delta) + 1)^k \left[ (\Omega – \alpha)(|a_n – A_n| + (\Omega + \alpha)|b_n – B_n|) \right] + (1 + \alpha)|b_1 – B_1| \leq (1 – \alpha)\gamma \}. \tag{2.5}
\]

Now, we see the following theorem:

Theorem 2.6. Let \( f \in S_H^k(\delta, \beta, \lambda, \alpha) \) be given by (1.2). If \( f \) satisfies the conditions

\[
\sum_{n=2}^{\infty} n (\beta(n-1)(\lambda – \delta) + 1)^k \left( (\Omega – \alpha) |a_n| + (\Omega + \alpha) |b_n| \right) \leq (1 – \alpha) – (1 + \alpha) |b_1|, \tag{2.6}
\]

where

\[
\Omega^k = [\beta(n – 1)(\lambda – \delta) + 1]^k, \quad \Omega = [\beta(n – 1)(\lambda – \delta) + 1], \quad 0 \leq \alpha < 1, \quad \delta \geq 0, \quad \beta > 0, \quad \lambda > 0, \quad \delta \neq \lambda, \quad k \in \mathbb{N}_0,
\]

and

\[
\gamma \leq \frac{1}{2} \left( 1 – \frac{(1 + \alpha)}{(1 – \alpha)} |b_1| \right), \tag{2.7}
\]

then \( N_{\gamma}^D(f) \subset S_H^k(\delta, \beta, \lambda, \alpha) \).

Proof. Let \( f \) satisfy (2.6) and let \( F(z) \) be given by

\[
F(z) = z + \overline{B_1 z} + \sum_{n=2}^{\infty} \left( A_n z^n + \overline{B_n z^n} \right),
\]

which belongs to \( N_{\gamma}^D(f) \). In other words, it suffices to show that \( F \) satisfies the condition

\[
\sum_{n=2}^{\infty} \left[ \frac{(\Omega – \alpha)}{(1 – \alpha)} |A_n| + \frac{(\Omega + \alpha)}{(1 – \alpha)} |B_n| \right] (\beta(n – 1)(\lambda – \delta) + 1)^k + \frac{(1 + \alpha)}{(1 – \alpha)} |B_1| \leq 1.
\]

We observe that

\[
N_{\gamma}^D(f) = \sum_{n=2}^{\infty} \left[ \frac{(\Omega – \alpha)}{(1 – \alpha)} |A_n| + \frac{(\Omega + \alpha)}{(1 – \alpha)} |B_n| \right]
(\beta(n – 1)(\lambda – \delta) + 1)^k + \frac{(1 + \alpha)}{(1 – \alpha)} |B_1|
\]

\[
= \sum_{n=2}^{\infty} \left[ \frac{(\Omega – \alpha)}{(1 – \alpha)} |A_n – a_n + a_n| + \frac{(\Omega + \alpha)}{(1 – \alpha)} |B_n – b_n + b_n| \right]
(\beta(n – 1)(\lambda – \delta) + 1)^k + \frac{(1 + \alpha)}{(1 – \alpha)} |B_1 – b_1 + b_1|
\]

\[
= \sum_{n=2}^{\infty} \left[ \frac{(\Omega – \alpha)}{(1 – \alpha)} |A_n – a_n| + \frac{(\Omega + \alpha)}{(1 – \alpha)} |B_n – b_n| \right]
(\beta(n – 1)(\lambda – \delta) + 1)^k
\]

\[
+ \sum_{n=2}^{\infty} \left[ \frac{(\Omega – \alpha)}{(1 – \alpha)} |a_n| + \frac{(\Omega + \alpha)}{(1 – \alpha)} |b_n| \right]
(\beta(n – 1)(\lambda – \delta) + 1)^k + \frac{(1 + \alpha)}{(1 – \alpha)} |B_1 – b_1| + \frac{(1 + \alpha)}{(1 – \alpha)} |b_1|
\]

\[
= \sum_{n=2}^{\infty} \left[ \frac{(\Omega – \alpha)}{(1 – \alpha)} |A_n – a_n| + \frac{(\Omega + \alpha)}{(1 – \alpha)} |B_n – b_n| \right]
(\beta(n – 1)(\lambda – \delta) + 1)^k
\]

\[
+ \frac{(1 + \alpha)}{(1 – \alpha)} |B_1 – b_1| + \sum_{n=2}^{\infty} \left[ \frac{(\Omega – \alpha)}{(1 – \alpha)} |a_n| + \frac{(\Omega + \alpha)}{(1 – \alpha)} |b_n| \right]
(\beta(n – 1)(\lambda – \delta) + 1)^k + \frac{(1 + \alpha)}{(1 – \alpha)} |b_1|
\]

\[
= \gamma + \frac{(1 + \alpha)}{(1 – \alpha)} |b_1| + \frac{1}{2} \sum_{n=2}^{\infty} n \left[ \frac{(\Psi – \alpha)}{(1 – \alpha)} |a_n| + \frac{(\Psi + \alpha)}{(1 – \alpha)} |b_n| \right]
(\beta(n – 1)(\lambda – \delta) + 1)^k
\]

\[
\leq \gamma + \frac{(1 + \alpha)}{(1 – \alpha)} |b_1| + \frac{1}{2} \left( 1 – \frac{(1 + \alpha)}{(1 – \alpha)} |b_1| \right).
\]

Now, this last expression is never greater than one provided that

\[
\gamma \leq 1 – \frac{(1 + \alpha)}{(1 – \alpha)} |b_1| – \frac{1}{2} \left( 1 – \frac{(1 + \alpha)}{(1 – \alpha)} |b_1| \right) = \frac{1}{2} \left( 1 – \frac{(1 + \alpha)}{(1 – \alpha)} |b_1| \right).
\]

Remark 2.1. Other works related to harmonic univalent functions can be found in [[7]-[10]].

CONCLUSION

In this paper, we obtained some results concerning the coefficient bounds, growth bounds, a closure property, extreme points, neighborhood and a convex combination of harmonic univalent Function in the open unit disc, which are related to the differential operator. We suggest to introduce a new subclass of p-valent starlike functions with negative coefficients in the open unit disc which is defined by a generalised derivative operator.

REFERENCES

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