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Abstract
The Chebyshev polynomials are utilized in this study to define the subclass of the bi-univalent function. Also, Chebyshev polynomial bounds and Fekete-Szego inequalities for functions defined in the classes are established.
Results & Lemmas (1)
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Theorem 2.1 · coeff
Theorem 2.1. Let the function f of the form (1.1) be in the class. Then (2.3) (2.4) and where
Theorem 2.1. Let the function f of the form (1.1) be in the class $\in G_{\Sigma}(\delta, t, m)$ . Then
$$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{4t^2[(1-\delta)3^m + \delta 3^{m+1}]^2 - (8t^2 - 2)[(1-\delta)2^m + \delta 2^{m+1}]}}$$
(2.3)
$$|a_3| \le \frac{2t}{(1-\delta)3^m + \delta 3^{m+1}} + \frac{16t^2}{[(1-\delta)2^m + \delta 2^{m+1}]^2}$$
(2.4)
and
$$\left|a_{3}-ra_{2}^{2}\right| = \begin{cases} \frac{\left|U_{1}(t)\right|}{\left[(1-\delta)3^{m}+\delta3^{m+1}\right]} & \text{if} \quad 0 \leq \left|\sigma(r,t)\right| \leq \frac{1}{2\left[(1-\delta)3^{m}+\delta3^{m+1}\right]} \\ \frac{2\left|U_{1}(t)\right|\left|\sigma(r,t)\right|}{\left[(1-\delta)3^{m}+\delta3^{m+1}\right]} & \text{if} \quad \left|\sigma(r,t)\right| \geq \frac{1}{2\left[(1-\delta)3^{m}+\delta3^{m+1}\right]} \end{cases}$$
where
$$\sigma(r,t) = \frac{(1-r)[U_1(t)]^2}{2\left[ [U_1(t)]^3 [(1-\delta)3^m + \delta 3^{m+1}]^2 - U_2(t) [(1-\delta)2^m + \delta 2^{m+1}] \right]}.$$
Definitions (1)
Def 2.1
Definition 2.1. A function of the form (1.1) belongs to the class,, and if the following conditions are satisfied: (2.1) and (2.2) where.
Definition 2.1. A function $f \in \Sigma$ of the form (1.1) belongs to the class $G_{\Sigma}(\delta, t, m)$ , $\delta \geq 1$ , $t \in (\frac{1}{2}, 1)$ and $z, w \in E$ if the following conditions are satisfied:
$$\frac{(1-\delta)D^{m}f(z) + \delta D^{m+1}f(z)}{z} \prec H(z,t) = \frac{1}{1 - 2tz - z^{2}}, \quad z \in E$$
(2.1)
and
$$\frac{(1-\delta)D^{m}g(w) + \delta D^{m+1}g(w)}{w} \prec H(w,t) = \frac{1}{1-2tw-w^{2}}, \quad w \in E$$
(2.2)
where $q = f^{-1}$ .
Function classes studied:
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