A Comprehensive Subclass of Bi-Univalent Functions Associated with Chebyshev Polynomials of the Second Kind

Theorem 2.1. Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\lambda, \delta, t)$ . Then $$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(\lambda + \mu + 2\xi\delta)^2 - 2[2(\lambda + \mu + 2\xi\delta)^2 - (2\lambda + \mu)(\mu + 1) - 12\xi\delta]t^2|}}$$ (2.1) and $$|a_3| \le \frac{4t^2}{(\lambda + \mu + 2\xi\delta)^2} + \frac{2t}{2\lambda + \mu + 6\xi\delta}.$$ (2.2)

Corollary 2.2. [10] Let the function f(z) given by (1.1) be in the class $\mathcal{B}_{\Sigma}(t)$ . Then $$|a_2| \le \frac{t\sqrt{2t}}{\sqrt{1-t^2}},$$ and $$|a_3| \le t^2 + \frac{2}{3}t.$$ Taking $\mu = 1$ and $\delta = 0$ in Theorem 2.1, we get the following consequence.

Corollary 2.3. [10] Let the function f(z) given by (1.1) be in the class $\mathscr{B}_{\Sigma}(\lambda, t)$ . Then $$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(\lambda+1)^2 - 4\lambda^2 t^2|}}$$ and $$|a_3| \le \frac{4t^2}{(\lambda+1)^2} + \frac{2t}{2\lambda+1}.$$ Taking $\delta = 0$ in Theorem 2.1, we get the following consequence.

Corollary 2.4. [11] Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\lambda, t)$ . Then $$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(\lambda + \mu)^2 - 2[2(\lambda + \mu)^2 - (2\lambda + \mu)(\mu + 1)]t^2|}}$$ and $$|a_3| \le \frac{4t^2}{(\lambda + \mu)^2} + \frac{2t}{2\lambda + \mu}.$$ Taking $\mu = 1$ in Theorem 2.1, we get the following consequence.

Corollary 2.5. [30] Let the function f(z) given by (1.1) be in the class $\mathscr{B}_{\Sigma}(\lambda, \delta, t)$ . Then $$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{\left|(1+\lambda+2\delta)^2-4\left[(\lambda+2\delta)^2-2\delta\right]t^2\right|}}$$ and $$|a_3| \le \frac{4t^2}{(1+\lambda+2\delta)^2} + \frac{2t}{1+2\lambda+6\delta}.$$

Theorem 3.1. Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\lambda, \delta, t)$ . Then for some $\eta \in \mathbb{R}$ , $$|a_{3} - \eta a_{2}^{2}| \leq \begin{cases} \frac{2t}{2\lambda + \mu + 6\xi\delta}, & |\eta - 1| \leq M\\ \frac{8|\eta - 1|t^{3}}{|(\lambda + \mu + 2\xi\delta)^{2} - 2[2(\lambda + \mu + 2\xi\delta)^{2} - ((2\lambda + \mu)(\mu + 1) + 12\xi\delta)]t^{2}|}, & |\eta - 1| \geq M \end{cases}$$ $$(3.1)$$ where $$M := \frac{\left| (\lambda + \mu + 2\xi \delta)^2 - 2 \left[ 2(\lambda + \mu + 2\xi \delta)^2 - ((2\lambda + \mu)(\mu + 1) + 12\xi \delta) \right] t^2 \right|}{4(2\lambda + \mu + 2\xi \delta) t^2}.$$

Corollary 3.2. Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\lambda, \delta, t)$ . Then $$|a_3 - a_2^2| \le \frac{2t}{2\lambda + \mu + 6\xi\delta}.$$ Taking $\lambda = 1$ , $\mu = 1$ and $\delta = 0$ in Theorem 3.1, we get the following corollary.

Corollary 3.3. [11] Let the function f(z) given by (1.1) be in the class $\mathcal{B}_{\Sigma}(t)$ . Then for some $\eta \in \mathbb{R}$ , $$|a_3 - \eta a_2^2| \le \begin{cases} \frac{2}{3}t, & |\eta - 1| \le \frac{1 - t^2}{3t^2} \\ \frac{2|\eta - 1|t^3}{1 - t^2}, & |\eta - 1| \ge \frac{1 - t^2}{3t^2} \end{cases}$$ Taking $\eta = 1$ in Corollary 3.3, we get the following corollary.

Corollary 3.4. [30] Let the function f(z) given be (1.1) be in the class $\mathcal{B}_{\Sigma}(t)$ . Then $$|a_3 - a_2^2| \le \frac{2}{3}t.$$ Taking $\mu = 1$ and $\delta = 0$ in Theorem 3.1, we get the following corollary.

Corollary 3.5. [11] Let the function f(z) given by (1.1) be in the class $\mathcal{B}_{\Sigma}(\lambda, t)$ . Then for some $\eta \in \mathbb{R}$ , $$|a_{3} - \eta a_{2}^{2}| \leq \begin{cases} \frac{2t}{1+2\lambda}, & |\eta - 1| \leq \frac{|(1+\lambda)^{2} - 4\lambda^{2}t^{2}|}{4(1+2\lambda)t^{2}} \\ \frac{8|\eta - 1|t^{3}}{|(1+\lambda)^{2} - 4\lambda^{2}t^{2}|}, & |\eta - 1| \geq \frac{|(1+\lambda)^{2} - 4\lambda^{2}t^{2}|}{4(1+2\lambda)t^{2}} \end{cases}$$ (3.2) Taking $\eta = 1$ in Corollary 3.5, we get the following corollary.

Corollary 3.6. [30] Let the function f(z) given by (1.1) be in the class $\mathscr{B}_{\Sigma}(\lambda, t)$ . Then $$|a_3 - a_2^2| \le \frac{2t}{1 + 2\lambda}.$$ Taking $\delta = 0$ in Theorem 3.1, we get the following corollary.

Corollary 3.7. [11] Let the function f(z) given by (1.1) be in the class $\mathscr{B}^{\mu}_{\Sigma}(\lambda, t)$ . Then for some $\eta \in \mathbb{R}$ , $$|a_{3} - \eta a_{2}^{2}| \leq \begin{cases} \frac{2t}{2\lambda + \mu}, & |\eta - 1| \leq \frac{|(\lambda + \mu)^{2} - 2[2(\lambda + \mu)^{2} - (2\lambda + \mu)(\mu + 1)]t^{2}|}{4(2\lambda + \mu)t^{2}} \\ \frac{8|\eta - 1|t^{3}}{|(\lambda + \mu)^{2} - 2[2(\lambda + \mu)^{2} - (2\lambda + \mu)(\mu + 1)]t^{2}|}, & |\eta - 1| \geq \frac{|(\lambda + \mu)^{2} - 2[2(\lambda + \mu)^{2} - (2\lambda + \mu)(\mu + 1)]t^{2}|}{4(2\lambda + \mu)t^{2}} \end{cases}$$ (3.3) Taking $\mu = 1$ in Theorem 3.1, we get the following corollary.

Corollary 3.8. [30] Let the function f(z) given by (1.1) be in the class $\mathcal{B}_{\Sigma}(\lambda, \delta, t)$ . Then for some $\eta \in \mathbb{R}$ , $$|a_{3} - \eta a_{2}^{2}| \leq \begin{cases} \frac{2t}{1+2\lambda+6\delta}, & |\eta - 1| \leq \frac{|(1+\lambda+2\delta)^{2} - 4[(\lambda+2\delta)^{2} - 2\delta]t^{2}|}{4(1+2\lambda+6\delta)t^{2}} \\ \frac{8|\eta - 1|t^{3}}{|(1+\lambda+2\delta)^{2} - 4[(\lambda+2\delta)^{2} - 2\delta]t^{2}|}, & |\eta - 1| \geq \frac{|(1+\lambda+2\delta)^{2} - 4[(\lambda+2\delta)^{2} - 2\delta]t^{2}|}{4(1+2\lambda+6\delta)t^{2}} \end{cases}$$ $$(3.4)$$

Definition 1.1. For $\lambda \geq 1$ , $\mu \geq 0$ , $\delta \geq 0$ and $0 \leq \beta < 1$ , a function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathscr{B}^{\mu}_{\Sigma}(\beta, \lambda, \delta)$ if the following conditions hold for all $z, w \in \mathbb{U}$ : $$\operatorname{Re}\left((1-\lambda)\left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z)\left(\frac{f(z)}{z}\right)^{\mu-1} + \xi \delta z f''(z)\right) > \beta \tag{1.6}$$ and $$\operatorname{Re}\left((1-\lambda)\left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w)\left(\frac{g(w)}{w}\right)^{\mu-1} + \xi \delta w g''(w)\right) > \beta,\tag{1.7}$$ where the function $g(w) = f^{-1}(w)$ is defined by (1.4) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ . This work is concerned with the coefficient bounds for the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ and the Fekete-Szegö inequality for functions belonging to the class $\mathscr{B}^{\mu}_{\Sigma}(\lambda, \delta, t)$ defined as follows:

Definition 1.2. For $\lambda \ge 1$ , $\mu \ge 0$ , $\delta \ge 0$ and $t \in \left(\frac{1}{2}, 1\right)$ , a function $f \in \Sigma$ given by (1.1) is said to be in the class $\mathscr{B}^{\mu}_{\Sigma}(\lambda, \delta, t)$ if the following subordinations hold for all $z, w \in \mathbb{U}$ : $$(1 - \lambda) \left(\frac{f(z)}{z}\right)^{\mu} + \lambda f'(z) \left(\frac{f(z)}{z}\right)^{\mu - 1} + \xi \delta z f''(z) < H(z, t) := \frac{1}{1 - 2tz + z^2}$$ (1.8) and $$(1 - \lambda) \left(\frac{g(w)}{w}\right)^{\mu} + \lambda g'(w) \left(\frac{g(w)}{w}\right)^{\mu - 1} + \xi \delta w g''(w) < H(w, t) := \frac{1}{1 - 2tw + w^2},\tag{1.9}$$ where the function $g(w) = f^{-1}(w)$ is defined by (1.4) and $\xi = \frac{2\lambda + \mu}{2\lambda + 1}$ . The following special cases of Definitions 1.2 are worthy of note: Remark 1. Note that for $\lambda=1, \mu=1$ and $\delta=0$ , the class of functions $\mathscr{B}^1_\Sigma(1,0,t):=\mathscr{B}_\Sigma(t)$ have been introduced and studied by Altinkaya and Yalçin [2], for $\mu=1$ and $\delta=0$ , the class of functions $\mathscr{B}^1_\Sigma(\lambda,0,t):=\mathscr{B}_\Sigma(\lambda,t)$ have been introduced and studied by Bulut et al. [10], for $\delta=0$ , the class of functions $\mathscr{B}^\mu_\Sigma(\lambda,0,t):=\mathscr{B}^\mu_\Sigma(\lambda,t)$ have been introduced and studied by Bulut et al. [11], and for $\mu=1$ , the class of functions $\mathscr{B}^1_\Sigma(\lambda,\delta,t):=\mathscr{B}_\Sigma(\lambda,\delta,t)$ have been introduced and studied by Yousef et al. [30].