Fekete-Szego Inequality for Analytic and Bi-univalent Functions Subordinate to Chebyshev Polynomials

Theorem 3.1. Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(\lambda,\mu,t)$ . Then $$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(1+\lambda+2\mu)^2-4t^2[(\lambda+2\mu)^2-2\mu]|}}$$ (3.1) and $$|a_3| \le \frac{4t^2}{(1+\lambda+2\mu)^2} + \frac{2t}{1+2\lambda+6\mu}. (3.2)$$

Corollary 3.2. [7] Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(t)$ . Then $$|a_2| \le \frac{t\sqrt{2t}}{\sqrt{1-t^2}},$$ and $$|a_3| \le t^2 + \frac{2}{3}t.$$ For Corollary 3.2, it's worthy to mention that Altinkaya and Yalçin [5] have obtained a remarkable result for the coefficient $|a_2|$ , as shown in the following corollary.

Corollary 3.3. Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(t)$ . Then $$|a_2| \le \frac{t\sqrt{2t}}{\sqrt{1+2t-t^2}}.$$ Taking $\mu = 0$ in Theorem 3.1, we get the following corollary.

Corollary 3.4. [7] Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(\lambda,t)$ . Then $$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(1+\lambda)^2 - 4t^2\lambda^2|}}$$ and $$|a_3| \le \frac{4t^2}{(1+\lambda)^2} + \frac{2t}{1+2\lambda}.$$ 4. Fekete-Szegő inequality for the function class $\mathscr{B}_{\Sigma}(\lambda,\mu,t)$ Now, we are ready to find the sharp bounds of Fekete-Szegö functional $a_3 - \eta a_2^2$ defined for $f \in \mathcal{B}_{\Sigma}(\lambda, \mu, t)$ given by (2.1).

Theorem 4.1. Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(\lambda, \mu, t)$ . Then for some $\eta \in \mathbb{R}$ , $$|a_3 - \eta a_2^2| \le \begin{cases} \frac{2t}{1+2\lambda+6\mu}, & |\eta - 1| \le M\\ \frac{8|\eta - 1|t^3}{[(1+\lambda+2\mu)^2 - 4t^2[(\lambda+2\mu)^2 - 2\mu]]}, & |\eta - 1| \ge M \end{cases}$$ $$(4.1)$$ where $$M := \frac{|(1+\lambda+2\mu)^2 - 4t^2 [(\lambda+2\mu)^2 - 2\mu]|}{4(1+2\lambda+6\mu)t^2}.$$

Corollary 4.2. Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(\lambda,\mu,t)$ . Then $$|a_3 - a_2^2| \le \frac{2t}{1 + 2\lambda + 6\mu}.$$ Taking $\lambda = 1$ and $\mu = 0$ in Theorem 4.1, we get the following corollary.

Corollary 4.3. Let the function f(z) given by (2.1) 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 4.3, we get the following corollary.

Corollary 4.4. Let the function f(z) given be (2.1) be in the class $\mathscr{B}_{\Sigma}(t)$ . Then $$|a_3 - a_2^2| \le \frac{2}{3}t.$$ Taking $\mu = 0$ in Theorem 4.1, we get the following corollary.

Corollary 4.5. Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(\lambda, t)$ . Then for some $\eta \in \mathbb{R}$ , $$|a_{3} - \eta a_{2}^{2}| \leq \begin{cases} \frac{2t}{1+2\lambda}, & |\eta - 1| \leq \frac{\left|(1+\lambda)^{2} - 4t^{2}\lambda^{2}\right|}{4(1+2\lambda)t^{2}} \\ \frac{8|\eta - 1|t^{3}}{\left|(1+\lambda)^{2} - 4t^{2}\lambda^{2}\right|}, & |\eta - 1| \geq \frac{\left|(1+\lambda)^{2} - 4t^{2}\lambda^{2}\right|}{4(1+2\lambda)t^{2}} \end{cases}$$ $$(4.2)$$ Taking $\eta = 1$ in Corollary 4.5, we get the following corollary.

Corollary 4.6. Let the function f(z) given by (2.1) be in the class $\mathscr{B}_{\Sigma}(\lambda,t)$ . Then $$|a_3 - a_2^2| \le \frac{2t}{1 + 2\lambda}.$$

Definition 2.1. The Chebyshev polynomials of the first kind are defined by the following three-terms recurrence relation: $$T_0(t) = 1,$$ $T_1(t) = t,$ $T_{n+1}(t) := 2tT_n(t) - T_{n-1}(t).$ The first few of the Chebyshev polynomials of the first kind are $$T_2(t) = 2t^2 - 1$$ , $T_3(t) = 4t^3 - 3t$ , $T_4(t) = 8t^4 - 8t^2 + 1$ , ... (2.3) The generating function for the Chebyshev polynomials of the first kind, $T_n(t)$ , is given by: $$F(z,t) = \frac{1 - tz}{1 - 2tz + z^2} = \sum_{n=0}^{\infty} T_n(t)z^n \quad (z \in \mathbb{U}).$$

Definition 2.2. The Chebyshev polynomials of the second kind are defined by the following three-terms recurrence relation: $$U_0(t) = 1,$$ $U_1(t) = 2t,$ $U_{n+1}(t) := 2tU_n(t) - U_{n-1}(t).$ The first few of the Chebyshev polynomials of the second kind are $$U_2(t) = 4t^2 - 1, \ U_3(t) = 8t^3 - 4t, \ U_4(t) = 16t^4 - 12t^2 + 1, \cdots$$ (2.4) The generating function for the Chebyshev polynomials of the second kind, $U_n(t)$ , is given by: $$H(z,t) = \frac{1}{1 - 2tz + z^2} = \sum_{n=0}^{\infty} U_n(t)z^n \quad (z \in \mathbb{U}).$$ The Chebyshev polynomials of the first and second kinds are connected by the following relations: $$\frac{dT_n(t)}{dt} = nU_{n-1}(t); \ T_n(t) = U_n(t) - tU_{n-1}(t); \ 2T_n(t) = U_n(t) - U_{n-2}(t).$$

Definition 2.3. For $\lambda \geq 1$ , $\mu \geq 0$ and $t \in (1/2, 1)$ , a function $f \in \Sigma$ given by (2.1) is said to be in the class $\mathscr{B}_{\Sigma}(\lambda, \mu, t)$ if the following subordinations hold for all $z, w \in \mathbb{U}$ : $$(1 - \lambda)\frac{f(z)}{z} + \lambda f'(z) + \mu z f''(z) \prec H(z, t) := \frac{1}{1 - 2tz + z^2}$$ (2.5) and $$(1 - \lambda)\frac{g(w)}{w} + \lambda g'(w) + \mu w g''(w) \prec H(w, t) := \frac{1}{1 - 2tw + w^2},$$ (2.6) where the function $g(w) = f^{-1}(w)$ is defined by (2.2). Remark 2.4. (1) For $\lambda = 1$ and $\mu = 0$ , we have the class $\mathscr{B}_{\Sigma}(1,0,t) := \mathscr{B}_{\Sigma}(t)$ of functions $f \in \Sigma$ given by (2.1) and satisfying the following subordination conditions for all $z, w \in \mathbb{U}$ : $$f'(z) \prec H(z,t) = \frac{1}{1 - 2tz + z^2}$$ and $$g'(w) \prec H(w,t) = \frac{1}{1 - 2tw + w^2}.$$ This class of functions have been introduced and studied by Altinkaya and Yalçin [5]. (2) For $\mu = 0$ , we have the class $\mathscr{B}_{\Sigma}(\lambda, 0, t) := \mathscr{B}_{\Sigma}(\lambda, t)$ of functions $f \in \Sigma$ given by (2.1) and satisfying the following subordination conditions for all $z, w \in \mathbb{U}$ : $$(1 - \lambda)\frac{f(z)}{z} + \lambda f'(z) \prec H(z, t) = \frac{1}{1 - 2tz + z^2}$$ and $$(1 - \lambda) \frac{g(w)}{w} + \lambda g'(w) \prec H(w, t) = \frac{1}{1 - 2tw + w^2}$$ This class of functions have been introduced and studied by Bulut et al. [7]. 3. Coefficient bounds for the function class $\mathscr{B}_{\Sigma}(\lambda,\mu,t)$ We begin with the following result involving initial coefficient bounds for the function class $\mathscr{B}_{\Sigma}(\lambda, \mu, t)$ .