Fekete-Szego Inequality For Analytic And Bi-univalent Functions Subordinate To (p; q)-Lucas Polynomials

Theorem 3.2. For $\lambda \geq 1$ , $\mu \geq 0$ and $\delta \geq 0$ , let $f \in \mathcal{A}$ belongs to the class $\mathfrak{B}^{\mu}_{\Sigma}(\lambda, \delta)$ . Then $$|a_2| \le \frac{2|p(x)|\sqrt{|p(x)|}}{\sqrt{\left|(\mu + 2\lambda)\left[1 + \mu + \frac{12\delta}{2\lambda + 1}\right]p^2(x) - 2(\mu + \lambda + 2\xi\delta)^2(p^2(x) + 2q(x))\right|}}$$ and <span id="page-3-1"></span><span id="page-3-0"></span> $$|a_3| \le \frac{p^2(x)}{(\mu + \lambda + 2\xi\delta)^2} + \frac{|p(x)|}{(\mu + 2\lambda + 2\xi\delta)}.$$

Corollary 3.3. If f belongs to the class $\mathfrak{B}_{\Sigma}(1) = \mathcal{S}_{\Sigma}^*$ of bi-starlike functions, then $$|a_2| \le \frac{2|p(x)|\sqrt{|p(x)|}}{\sqrt{|2p^2(x) - 2(p^2(x) + 2q(x))|}},$$ and $$|a_3| \le p^2(x) + |p(x)|$$ . <span id="page-4-3"></span>

Theorem 4.1. For $\lambda \geq 1$ , $\mu \geq 0$ and $\delta \geq 0$ , let $f \in \mathcal{A}$ belongs to the class $\mathfrak{B}^{\mu}_{\Sigma}(\lambda, \delta)$ . Then $$|a_3 - va_2^2| \le \begin{cases} \frac{|p(x)|}{(\mu + 2\lambda + 2\xi\delta)}, & |v - 1| \le \frac{1}{2(\mu + 2\lambda + 2\xi\delta)} \times |\Upsilon(x)| \\ \frac{2|p(x)|^3|1 - v|}{|p(x)\Upsilon(x)|}, & |v - 1| \ge \frac{1}{2(\mu + 2\lambda + 2\xi\delta)} \times |\Upsilon(x)|, \end{cases}$$ where $$\Upsilon(x) = (\mu + 2\lambda) \left[ 1 + \mu + \frac{12\delta}{2\lambda + 1} \right] p(x) - 2 \left( \mu + \lambda + 2\xi \delta \right)^2 \frac{p^2(x) + 2q(x)}{p(x)}$$ .

Corollary 4.2. If f belongs to the class $\mathcal{S}_{\Sigma}^*$ , then $$|a_3 - va_2^2| \le \begin{cases} |p(x)|, & |v - 1| \le \left| \frac{q(x)}{p(x)} \right| \\ \frac{2|p(x)|^3|1 - v|}{4|q(x)|}, & |v - 1| \ge \left| \frac{q(x)}{p(x)} \right| \end{cases}.$$ Putting v = 1 in Theorem 4.1, we conclude the following result:

Corollary 4.3. If f belongs to the class $\mathcal{S}_{\Sigma}^*$ , then $$|a_3 - a_2^2| \le \frac{|p(x)|}{(\mu + 2\lambda + 2\xi\delta)}.$$

Definition 2.1. (See [18]) 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 (1.4) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ . - Remark 2.2. In the following special cases of Definition 2.1; we show how the class of analytic bi-univalent functions $\mathfrak{B}^{\mu}_{\Sigma}(\alpha,\lambda,\delta)$ for suitable choices of $\lambda$ , $\mu$ and $\delta$ lead to certain new as well as known classes of analytic bi-univalent functions studied earlier in the literature. - (i) For $\delta = 0$ , we obtain the bi-univalent function class $\mathfrak{B}^{\mu}_{\Sigma}(\alpha, \lambda, 0) := \mathfrak{B}^{\mu}_{\Sigma}(\alpha, \lambda)$ introduced by Çağlar et al. [6]. - (iii) For $\delta = 0$ , $\mu = 1$ , and $\lambda = 1$ , we obtain the bi-univalent function class $\mathfrak{B}^1_{\Sigma}(\alpha, 1, 0) := \mathfrak{B}_{\Sigma}(\alpha)$ introduced by Srivastava et al. [12]. - (iv) For $\delta = 0$ , $\mu = 0$ , and $\lambda = 1$ , we obtain the well-known class $\mathfrak{B}^0_{\Sigma}(\alpha, 1, 0) := \mathcal{S}^*_{\Sigma}(\alpha)$ of bi-starlike functions of order $\alpha$ . - (iv) For $\mu = 1$ , we obtain the well-known class $\mathfrak{B}^1_{\Sigma}(\alpha, \lambda, \delta) := \mathfrak{B}_{\Sigma}(\alpha, \lambda, \delta)$ of bi-univalent functions.

Definition 3.1. For $\lambda \geq 1$ , $\mu \geq 0$ and $\delta \geq 0$ , a function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathfrak{B}^{\mu}_{\Sigma}(\lambda, \delta)$ if the following subordinations are satisfied: $$(1 - \lambda) \left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z) \left(\frac{f(z)}{z}\right)^{\mu - 1} + \xi \delta z f''(z) \prec A_{\{L_{p,q,k}(x)\}}(z) - 1$$ and $$(1 - \lambda) \left(\frac{f^{-1}(w)}{w}\right)^{\mu} + \lambda \left(f^{-1}(w)\right)' \left(\frac{f^{-1}(w)}{w}\right)^{\mu - 1} + \xi \delta z \left(f^{-1}(w)\right)'' \prec A_{\left\{L_{p,q,k}(x)\right\}}(w) - 1,$$ where $f^{-1}$ is given by (1.4).