Abstract
In the present paper, we define a new general subclass of bi-univalent functions involving a differential operator in the open unit disk U. For this purpose, we use the Faber polynomial expansions. Several connections to some of the earlier known results are also pointed out.
Results & Lemmas (2)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 3.1 · coeff
Theorem 3.1. For,, and, let the function be given by (1.1). Then and
Theorem 3.1. For $\lambda \geq 1$ , $\mu \geq 0$ , and $\delta \geq 0$ , let the function $f \in \mathfrak{B}_{\Sigma}(\mu, \lambda, \Phi, \xi)$ be given by (1.1). Then
$$|a_2| \le \min \left\{ \frac{2}{\mu + \lambda + 2\xi\delta}, \sqrt{\frac{8}{(\mu + 2\lambda + 6\xi\delta)(\mu + 1)}} \right\}$$
and
$$|a_3| \le \min \left\{ \begin{array}{c} \frac{1}{\left(1 + \frac{6\delta}{2\lambda + 1}\right)} \left[ \frac{4}{(\mu + \lambda + 2\xi\delta)^2} + \frac{2}{(\mu + 2\lambda + 6\xi\delta)} \right], \\ \frac{1}{\left(1 + \frac{6\delta}{2\lambda + 1}\right)} \left[ \frac{8}{(\mu + 2\lambda + 6\xi\delta)(\mu + 1)} + \frac{2}{(\mu + 2\lambda + 6\xi\delta)} \right] \end{array} \right\}.$$
Theorem 3.2 · coeff
Theorem 3.2. Let. If with, then (3.11)
Theorem 3.2. Let $\mathfrak{B}_{\Sigma}(\mu, \lambda, \Phi, \xi)$ . If $a_m = 0$ with $2 \leq m \leq n-1$ , then
$$|a_n| \le \frac{2}{\mu + (n-1)\lambda + n(n-1)\xi\delta} \quad (n \ge 4).$$
(3.11)
Definitions (2)
Def 2.1
Definition 2.1. For,, and, a function given by (1.1) is said to be in the class if the following conditions hold for all: and where the…
Definition 2.1. For $\lambda \geq 1$ , $\mu \geq 0$ , $\delta \geq 0$ and $0 \leq \alpha < 1$ , a function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathfrak{B}^{\mu}_{\Sigma}(\alpha, \lambda, \delta)$ if the following conditions hold for all $z, w \in \mathbb{U}$ :
$$\operatorname{Re}\left((1-\lambda)\left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z)\left(\frac{f(z)}{z}\right)^{\mu-1} + \xi \delta z f''(z)\right) > \alpha \tag{2.1}$$
and
$$\operatorname{Re}\left((1-\lambda)\left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w)\left(\frac{g(w)}{w}\right)^{\mu-1} + \xi \delta w g''(w)\right) > \alpha, \tag{2.2}$$
where the function $g(w) = f^{-1}(w)$ is defined by (1.2) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ .
Using the Faber polynomial expansion of functions $f \in \mathcal{A}$ of the form (1.1), the coefficients of its inverse map $g = f^{-1}$ may be expressed as in [1]:
$$g(w) = f^{-1}(w) = w + \sum_{n=2}^{\infty} \frac{1}{n} K_{n-1}^{-n} (a_2, a_3, ...) w^n,$$
(2.3)
where
$$K_{n-1}^{-n} = \frac{(-n)!}{(-2n+1)! (n-1)!} a_2^{n-1} + \frac{(-n)!}{(2(-n+1))! (n-3)!} a_2^{n-3} a_3 + \frac{(-n)!}{(-2n+3)! (n-4)!} a_2^{n-4} a_4$$
$$+ \frac{(-n)!}{(2(-n+2))! (n-5)!} a_2^{n-5} \left[ a_5 + (-n+2) a_3^2 \right] + \frac{(-n)!}{(-2n+5)! (n-6)!} a_2^{n-6}$$
$$\left[ a_6 + (-2n+5) a_3 a_4 \right] + \sum_{i \ge 7} a_2^{n-j} V_j,$$
$$(2.4)$$
such that $V_j$ with $7 \le j \le n$ is a homogeneous polynomial in the variables $a_2, a_3, ..., a_n$ [2].
In particular, the first three terms of $K_{n-1}^{-n}$ are
$$K_1^{-2} = -2a_2, \ K_2^{-3} = 3\left(2a_2^2 - a_3\right), \ K_3^{-4} = -4\left(5a_2^3 - 5a_2a_3 + a_4\right).$$
(2.5)
In general, for any $p \in \mathbb{N} := \{1, 2, 3, ...\}$ , an expansion of $K_n^p$ is as in [1],
$$K_n^p = pa_n + \frac{p(p-1)}{2}D_n^2 + \frac{p!}{(p-3)!3!}D_n^3 + \dots + \frac{p!}{(p-n)!n!}D_n^n,$$
(2.6)
where $D_n^p = D_n^p(a_2, a_3, ...)$ , and by [22], $D_n^m(a_1, a_2, ..., a_n) = \sum_{n=1}^{\infty} \frac{m!}{i_1!...i_n!} a_1^{i_1} ... a_n^{i_n}$ while $a_1 = 1$ , and the sum is taken over all non-negative integers $i_1, ..., i_n$ satisfying $i_1 + i_2 + \cdots + i_n = m$ , $i_1 + 2i_2 + \cdots + ni_n = n$ , it is clear that $D_n^m(a_1, a_2, ..., a_n) = a_1^n$ .
Now, we are ready to establish a new subclass of analytic and bi-univalent functions based on subordination.
Def 2.2
Definition 2.2. For,, and, A function is said to be in the class, if the following subordinations are satisfied: <span…
Definition 2.2. For $\lambda \geq 1$ , $\mu \geq 0$ , and $\delta \geq 0$ , A function $f \in \Sigma$ is said to be in the class $\mathfrak{B}_{\Sigma}(\mu, \lambda, \Phi, \xi)$ , if the following subordinations are satisfied:
<span id="page-3-0"></span>
$$(1 - \lambda) \left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z) \left(\frac{f(z)}{z}\right)^{\mu - 1} + \xi \delta z f''(z) \prec \Phi(z) \tag{2.7}$$
and
<span id="page-3-1"></span>
$$(1 - \lambda) \left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w) \left(\frac{g(w)}{w}\right)^{\mu - 1} + \xi \delta w g''(w) \prec \Phi(w)$$
(2.8)
where the function $g(w) = f^{-1}(w)$ is defined by (1.2) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ .
3. Coefficient bounds for the function class $\mathfrak{B}_{\Sigma}(\mu,\lambda,\Phi,\xi)$
Function classes studied:
Coefficient bounds & claims (4)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
|a_2| ≤ min(2/(mu+lambda+2*xi*delta), sqrt(8/((mu+2*lambda+6*xi*delta)*(mu+1)))) for class B_Sigma(mu, lambda, Phi, xi) [Theorem 3.1]
coefficient_bound
B_Sigma(mu, lambda, Phi, xi): |a3| <= min of two expressions involving mu, lambda, xi, delta [Theorem 3.1]
coefficient_bound
|a_n| (general coefficient, a_m=0 for 2<=m<=n-1) ≤ 2/(mu+(n-1)*lambda+n*(n-1)*xi*delta) for class B_Sigma(mu, lambda, Phi, xi) [Theorem 3.2]
function_family
Class B_Sigma(mu, lambda, Phi, xi): f in Sigma: (1-lambda)(f(z)/z)^mu + lambda*f'(z)*(f(z)/z)^{mu-1} + xi*delta*z*f''(z) subordinate to Phi(z), and same for inverse g=f^{-1}; xi=(2lambda+mu)/(2lambda+1)
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