Abstract
The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $Σ_{\mathrm{m}}$ of $m$-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $\left|a_{m+1}\right|$ and $\left|a_{2 m+1}\right|$ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. The results presented would generalize and improve on some r
Results & Lemmas (2)
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Lemma 1
Lemma 1. [2]. If with h(z) given by (1.2), then for each.
Lemma 1. [2]. If $h \in \mathcal{P}$ with h(z) given by (1.2), then $|h_k| \leq 2$ for each $k \in \mathbb{N}$ .
Theorem 1
Theorem 1. Let be given by (1.5). Then (2.3) and where and
Theorem 1. Let $f \in Q_{\Sigma_m}(\tau, \lambda, \gamma, \delta; \alpha)$ be given by (1.5). Then
$$|a_{m+1}| \le \frac{2\sqrt{2}|\tau|\alpha}{\sqrt{(\delta+1)|\tau\alpha(\delta+2)(m+1)\Phi_1(\lambda,\gamma,m)+2(1-\alpha)(\delta+1)\Phi_2(\lambda,\gamma,m)|}},$$
(2.3)
and
$$|a_{2m+1}| \le \frac{2|\tau|\alpha}{(\delta+1)(\delta+2)\Phi_1(\lambda,\gamma,m)} + \frac{2|\tau|^2\alpha^2(m+1)}{(\delta+1)^2\Phi_2(\lambda,\gamma,m)},\tag{2.4}$$
where
$$\Phi_1(\lambda, \gamma, m) = 1 + 2(\lambda + \gamma)m + \lambda\gamma\left((2m+1)^2 + 1\right),$$
and
$$\Phi_2(\lambda, \gamma, m) = \left(1 + (\lambda + \gamma)m + \lambda\gamma\left((m+1)^2 + 1\right)\right)^2.$$
Definitions (1)
Def 1
Definition 1. A function given by (1.5) is called in the class if it satisfies the conditions: and <span id="page-3-3"></span>where and the…
Definition 1. A function $f \in \Sigma_m$ given by (1.5) is called in the class $Q_{\Sigma_m}(\tau, \lambda, \gamma, \delta; \alpha)$ if it satisfies the conditions:
$$\left| \arg \left( 1 + \frac{1}{\tau} \left[ (1 - \lambda)(1 - \gamma) \frac{\mathcal{R}^{\delta} f(z)}{z} \right] + (\lambda(\gamma + 1) + \gamma) \left( \mathcal{R}^{\delta} f(z) \right)' + \lambda \gamma \left( z \left( \mathcal{R}^{\delta} f(z) \right)'' - 2 \right) - 1 \right] \right) \right| < \frac{\alpha \pi}{2}, \quad (2.1)$$
and
$$\left| \arg \left( 1 + \frac{1}{\tau} \left[ (1 - \lambda)(1 - \gamma) \frac{\mathcal{R}^{\delta} g(w)}{z} \right] + (\lambda(\gamma + 1) + \gamma) \left( \mathcal{R}^{\delta} g(w) \right)' + \lambda \gamma \left( w \left( \mathcal{R}^{\delta} g(w) \right)'' - 2 \right) - 1 \right] \right) \right| < \frac{\alpha \pi}{2}, \quad (2.2)$$
<span id="page-3-3"></span>where $z, w \in \mathbb{U}$ and the function $g = f^{-1}$ is given by (1.7).
Function classes studied:
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