Second Hankel determinant for certain subclasses of bi-univalent functions involving Chebyshev polynomials

Lemma 1. (see [26]) If the function $p \in \mathcal{P}$ is given by the following series: $$(1.11) p(z) = 1 + c_1 z + c_2 z^2 + c_3 z^3 + \dots,$$ then the sharp estimate given by $$|c_n| \le 2 \quad (n = 1, 2, 3, \dots)$$ holds true.

Lemma 2. [17] If the function $p \in \mathcal{P}$ is given by the series (1.11), then <span id="page-3-5"></span><span id="page-3-0"></span> $$2c_2 = c_1^2 + x(4 - c_1^2),$$ $$4c_3 = c_1^3 + 2(4 - c_1^2)c_1x - c_1(4 - c_1^2)x^2 + 2(4 - c_1^2)(1 - |x|^2)z$$ for some x and z with $|x| \le 1$ and $|y| \le 1$ . In the present investigation, we seek upper bound for the second Hankel determinant for functions $h_{\delta}$ belongs to the class $\mathcal{N}_{\sigma}^{\mu,\delta}(\lambda,t)$ by making use of the Chebyshev polynomials expansions and the Hadamard product. Also, we give some remarkable consequences related to the class $\mathcal{N}_{\sigma}^{\mu,\delta}(\lambda,t)$ .

Theorem 1. Let $h_{\delta} \in \sigma$ of the form (1.6) be in $\mathcal{N}_{\sigma}^{\mu,\delta}(\lambda;t)$ . Then $$|a_{2}a_{4} - a_{3}^{2}| \leq \begin{cases} \varphi(2^{-}, t), & \chi_{1} \geq 0 \text{ and } \chi_{2} \geq 0\\ \frac{36t^{2}}{(2\delta^{2} + 1)^{2}(2\lambda + \mu)^{2}}, & \chi_{1} \leq 0 \text{ and } \chi_{2} \leq 0\\ \max\left\{\frac{36t^{2}}{(2\delta^{2} + 1)^{2}(2\lambda + \mu)^{2}}, & \varphi(2^{-}, t)\right\}, & \chi_{1} > 0 \text{ and } \chi_{2} < 0\\ \max\left\{\varphi(c_{0}, t), \varphi(2^{-}, t)\right\}, & \chi_{1} < 0 \text{ and } \chi_{2} > 0 \end{cases}$$ where $$\varphi(2^{-},t) = \frac{9U_{1}^{2}(t)}{(2\delta^{2}+1)^{2}(2\lambda+\mu)^{2}} + \frac{\chi_{1}+9\chi_{2}}{6\delta(\delta^{3}+2\delta)(2\delta^{2}+1)^{2}(3\lambda+\mu)(2\lambda+\mu)^{2}(\lambda+\mu)^{4}},$$ $$\varphi(c_{0},t) = \frac{9U_{1}^{2}(t)}{(2\delta^{2}+1)^{2}(2\lambda+\mu)^{2}} - \frac{27\chi_{2}^{2}}{8\chi_{1}\delta(\delta^{3}+2\delta)(2\delta^{2}+1)^{2}(3\lambda+\mu)(2\lambda+\mu)^{2}(\lambda+\mu)^{4}}, \quad c_{0} = \sqrt{\frac{-18\chi_{2}}{\chi_{1}}}$$ and $$\chi_{1} = (2\lambda+\mu)^{2}U_{1}(t) |\Omega_{\lambda,\mu,\delta}(t)| + 18(\lambda+\mu)^{3} \left(3\delta(\delta^{3}+2\delta)(\lambda+\mu)(3\lambda+\mu) - (2\delta^{2}+1)^{2}(2\lambda+\mu)^{2}\right)U_{1}^{2}(t)$$ $$\chi_{1} = (2\lambda + \mu)^{2} U_{1}(t) |\Omega_{\lambda,\mu,\delta}(t)| + 18(\lambda + \mu)^{3} \left( 3\delta(\delta^{3} + 2\delta)(\lambda + \mu)(3\lambda + \mu) - (2\delta^{2} + 1)^{2}(2\lambda + \mu)^{2} \right) U_{1}^{2}(t)$$ $$- 9(\lambda + \mu)^{2} (2\lambda + \mu) U_{1}(t) \left( (3\lambda + \mu)(8\delta^{4} - 4\delta^{2} + 5)U_{1}^{2}(t) + 4(2\delta^{2} + 1)^{2}(2\lambda + \mu)(\lambda + \mu)U_{2}(t) \right),$$ $$\chi_{2} = \left[ (2\lambda + \mu)(3\lambda + \mu) \left( 8\delta^{4} - 4\delta^{2} + 5 \right) U_{1}^{3}(t) + 4(2\delta^{2} + 1)^{2}(\lambda + \mu)(2\lambda + \mu)^{2} U_{1}(t) U_{2}(t) \right.$$ $$+ (\lambda + \mu)U_{1}^{2}(t) \left( 2(2\delta^{2} + 1)^{2}(2\lambda + \mu)^{2} - 12\delta(\delta^{3} + 2\delta)(\lambda + \mu)(3\lambda + \mu) \right) \left[ (\lambda + \mu)^{2}, \right]$$ where, $$\Omega_{\lambda,\mu,\delta}(t) = 18(2\delta^2 + 1)^2(\lambda + \mu)^3U_3(t) - U_1^3(t)(3\lambda + \mu)\left(3(2\delta^2 + 1)^2(\mu^2 + 3\mu - 4) + 54\delta(\delta^3 + 2\delta)\right)$$ .

Corollary 1. Let $h_{\delta} \in \sigma$ of the form (1.6) be in $\mathcal{B}_{\sigma}^{\delta}(\lambda, t)$ . Then $$|a_{2}a_{4}-a_{3}^{2}| \leq \begin{cases} \varphi(2^{-},t), & \chi_{3} \geq 0 \text{ and } \chi_{4} \geq 0\\ \frac{36t^{2}}{(2\delta^{2}+1)^{2}(2\lambda+1)^{2}}, & \chi_{3} \leq 0 \text{ and } \chi_{4} \leq 0\\ \max\left\{\frac{36t^{2}}{(2\delta^{2}+1)^{2}(2\lambda+1)^{2}}, & \varphi(2^{-},t)\right\}, & \chi_{3} > 0 \text{ and } \chi_{4} < 0\\ \max\left\{\varphi(c_{0},t), \varphi(2^{-},t)\right\}, & \chi_{3} < 0 \text{ and } \chi_{4} > 0 \end{cases}$$ where where $$\varphi(2^-,t) = \frac{9U_1^2(t)}{(2\delta^2+1)^2(2\lambda+1)^2} + \frac{\chi_3+9\chi_4}{6\delta(\delta^3+2\delta)(2\delta^2+1)^2(3\lambda+1)(2\lambda+1)^2(\lambda+1)^4},$$ $$\varphi(c_0,t) = \frac{9U_1^2(t)}{(2\delta^2+1)^2(2\lambda+1)^2} - \frac{27\chi_4^2}{8\chi_3\delta(\delta^3+2\delta)(2\delta^2+1)^2(3\lambda+1)(2\lambda+1)^2(\lambda+1)^4}, \quad c_0 = \sqrt{\frac{-18\chi_4}{\chi_3}}$$ and $\chi_3 = \chi_1(\lambda,\mu=1,\delta;t), \; \chi_4 = \chi_2(\lambda,\mu=1,\delta;t).$ Taking $\lambda = 1$ in the Theorem 1, we get the following result.

Corollary 2. Let $h_{\delta} \in \sigma$ of the form (1.6) be in $\mathcal{B}_{\sigma}^{\mu,\delta}(t)$ . Then $$|a_{2}a_{4}-a_{3}^{2}| \leq \begin{cases} \varphi(2^{-},t), & \chi_{5} \geq 0 \text{ and } \chi_{6} \geq 0\\ \frac{36t^{2}}{(2\delta^{2}+1)^{2}(2+\mu)^{2}}, & \chi_{5} \leq 0 \text{ and } \chi_{6} \leq 0\\ \max\left\{\frac{36t^{2}}{(2\delta^{2}+1)^{2}(2+\mu)^{2}}, & \varphi(2^{-},t)\right\}, & \chi_{5} > 0 \text{ and } \chi_{6} < 0\\ \max\left\{\varphi(c_{0},t),\varphi(2^{-},t)\right\}, & \chi_{5} < 0 \text{ and } \chi_{6} > 0 \end{cases}$$ where $$\varphi(2^{-},t) = \frac{9U_{1}^{2}(t)}{(2\delta^{2}+1)^{2}(2+\mu)^{2}} + \frac{\chi_{5}+9\chi_{6}}{6\delta(\delta^{3}+2\delta)(2\delta^{2}+1)^{2}(3+\mu)(2+\mu)^{2}(1+\mu)^{4}},$$ $$\varphi(c_{0},t) = \frac{9U_{1}^{2}(t)}{(2\delta^{2}+1)^{2}(2+\mu)^{2}} - \frac{27\chi_{6}^{2}}{8\chi_{5}\delta(\delta^{3}+2\delta)(2\delta^{2}+1)^{2}(3+\mu)(2+\mu)^{2}(1+\mu)^{4}}, \quad c_{0} = \sqrt{\frac{-18\chi_{6}}{\chi_{5}}}$$ and $\chi_{5} = \chi_{1}(\lambda=1,\mu,\delta;t), \; \chi_{6} = \chi_{2}(\lambda=1,\mu,\delta;t).$ Putting $\lambda = 1$ and $\mu = 1$ in the Theorem 1, we find the following result. <span id="page-8-6"></span>Corollary 3. If $h_{\delta} \in \mathcal{B}_{\sigma}^{\delta}(t)$ is of the form (1.6). Then $$|a_2 a_4 - a_3^2| \le \begin{cases} \varphi(2^-, t), & \frac{1}{2} < t \le t_0 \\ \varphi(c_0, t), & t_0 < t < 1 \end{cases}$$ where $$\varphi(2^{-},t) = \frac{3t^{2} \left( (2\delta^{2}+1)^{2} - t(5\delta^{4}+2\delta^{2}+2) \right)}{(\delta^{2}+2)(2\delta^{3}+\delta)^{2}},$$ $$\varphi(c_{0},t) = \frac{4t^{2}}{(2\delta^{2}+1)^{2}} - \frac{t(\nabla_{(4,20,-3);(22,40,64)}^{(3,1,-1)})^{2}}{\delta(2\delta^{2}+1)^{2}(\delta^{3}+2\delta)\left(\nabla_{(-4,4,-3);(22,40,64)}^{(3,1,-1)} + 6t \left| t(5\delta^{4}+2\delta^{2}+2) - (2\delta^{2}+1)^{2} \right| \right)},$$ $\nabla^{(m,n,r)}_{(a,b,c);(d,e,f)}(\delta;t) = \nabla^{(m,n,r)}_{(a,b,c);(d,e,f)} = m(1+2\delta^2)^2 + nt(a+bt^2+ct^4) + rt^2(d+et^2+ft^4). \ \, \textit{Moreover, the value of $t_0$ is root of equation $\chi_1=0$ for $\lambda=\mu=1$ and $\frac{1}{2}< t<1$.}$ Setting $\delta = 1$ in the Corollary 3, we obtain the following result.

Corollary 4. If $h_{\delta} \in \mathcal{B}_{\sigma}(t)$ is of the form (1.6). Then $$\left| a_2 a_4 - a_3^2 \right| \le \begin{cases} t^2 (1 - t^2), & \frac{1}{2} < t \le t_{0_1} \\ \frac{t(260t^4 + 84t^3 - 139t^2 - 18t + 9)}{8(18t^3 + 42t^2 - 17t - 9)}, & t_{0_1} < t < 1 \end{cases},$$ where, the value of $t_{0_1}$ , which is approximately $t_{0_1} = 0.603615$ , is root of equation $\chi_1 = 0$ for $\lambda = \mu = \delta = 1$ and $\frac{1}{2} < t < 1$ . Next, taking $\lambda = 1$ and $\mu = 0$ in the Theorem 1, we arrive at the following result. <span id="page-8-7"></span>Corollary 5. If $h_{\delta} \in \mathcal{S}_{\sigma}^{\star,\delta}(t)$ is of the form (1.6), then $$|a_2 a_4 - a_3^2| \le \begin{cases} \varphi(2^-, t), & \frac{1}{2} < t \le t_{0_2} \\ \varphi(c_0, t), & t_{0_2} < t < 1 \end{cases}$$ where, $$\varphi(2^{-},t) = \frac{8t^{2} \left( (2\delta^{2}+1)^{2} - 6t^{2}(\delta^{2}-1)^{2} \right)}{(2\delta^{3}+\delta)^{2}(\delta^{2}+2)},$$ $$\varphi(c_{0},t) = \frac{9t^{2}}{(2\delta^{2}+1)^{2}} - \frac{t(\nabla^{(-2,-1,1)}_{(-2,10,1);(23,20,56)})^{2}}{\delta(2\delta^{2}+1)^{2}(\delta^{3}+2\delta)\left(\nabla^{(-4,1,2)}_{(4,-2,7);(23,20,56)} - 8t \left| 6t^{2}(\delta^{2}-1)^{2} - (2\delta^{2}+1)^{2} \right| \right)}.$$ Moreover, the value of $t_{0_2}$ is root of equation $\chi_1 = 0$ for $\lambda = 1$ , $\mu = 0$ and $\frac{1}{2} < t < 1$ . Now, taking $\delta = 1$ in the Corollary 5, we attain the following result.

Corollary 6. If $h_{\delta} \in \mathcal{S}_{\sigma}^{\star}(t)$ is of the form (1.6), then $$\left|a_2a_4 - a_3^2\right| \le \begin{cases} \frac{8t^2}{3}, & \frac{1}{2} < t \le \frac{7+\sqrt{401}}{44} \\ t^2 + \frac{t(2+t-11t^2)^2}{3(-4-7t+22t^2)}, & \frac{7+\sqrt{401}}{44} < t < 1 \end{cases}.$$ When $\delta = 1$ , we note that the results given the above coincide with the results in [23].

Definition 1. For $\lambda \geq 1$ , $\mu \geq 0$ , $\delta \geq 1$ and $t \in (1/2,1]$ , a function $h_{\delta} \in \sigma$ given by (1.6) is said to be in class $\mathcal{N}_{\sigma}^{\mu,\delta}(\lambda,t)$ if the following subordinations hold for all $z, w \in \mathbb{U}$ : $$(1.9) (1-\lambda)(\frac{h_{\delta}(z)}{z})^{\mu} + \lambda h_{\delta}'(z)(\frac{h_{\delta}(z)}{z})^{\mu-1} \prec G(z,t)$$ and $$(1.10) \qquad (1-\lambda)\left(\frac{k_{\delta}(w)}{w}\right)^{\mu} + \lambda k_{\delta}'(w)\left(\frac{k_{\delta}(w)}{w}\right)^{\mu-1} \prec G(w,t),$$ where the function $k_{\delta} = h_{\delta}^{-1}$ is defined by (1.2). Obviously, for $\delta = 1$ , we get that $\mathcal{N}_{\sigma}^{\mu,1}(\lambda,t) = \mathcal{N}_{\sigma}^{\mu}(\lambda,t)$ . It is important to mention that the class $\mathcal{N}_{\sigma}^{\mu}(\lambda,t)$ was introduced and investigated by Bulut et al. [7]. Also, they discussed initial coefficient estimates and Fkete-Szegö bounds for the class $\mathcal{N}_{\sigma}^{\mu}(\lambda,t)$ and its subclasses given in the following remark. Remark 1. (i) For $\delta = 1$ and $\mu = 1$ , we get the class $\mathcal{N}_{\sigma}^{1,1}(\lambda,t) = \mathcal{B}_{\sigma}(\lambda,t)$ consist of functions $f \in \sigma$ satisfying the condition $$(1-\lambda)\frac{f(z)}{z} + \lambda f'(z) \prec G(z,t)$$ and $$(1-\lambda)\frac{g(w)}{w} + \lambda g'(w) \prec G(w,t)$$ where the function $g = f^{-1}$ is defined by (1.2). This class was introduced and studied by Bulut et al [8] (see also [20]). (ii) For $\delta = 1$ and $\lambda = 1$ , we obtain the class $\mathcal{N}_{\sigma}^{\mu,1}(1,t) = \mathcal{B}_{\sigma}^{\mu}(t)$ consist of bi-Bazilevič functions: $$f'(z) \left(\frac{f(z)}{z}\right)^{\mu-1} \prec G(z,t)$$ and $$g'(w) \left(\frac{g(w)}{w}\right)^{\mu-1} \prec G(w,t),$$ where the function $g = f^{-1}$ is defined by (1.2). This class was introduced and studied by Altinkaya and Yalçın [6]. (iii) For $\delta = 1$ , $\mu = 1$ and $\lambda = 1$ , we have the class $\mathcal{N}_{\sigma}^{1,1}(1,t) = \mathcal{B}_{\sigma}(t)$ consist of functions f satisfying the condition $$f'(z) \prec G(z,t)$$ and $$q'(w) \prec G(w,t)$$ where the function $g = f^{-1}$ is defined by (1.2). (iv) For $\delta = 1$ , $\lambda = 1$ and $\mu = 0$ , we have the class $\mathcal{N}_{\sigma}^{0,1} = \mathcal{S}_{\sigma}^{\star}(t)$ satisfying the condition $$\frac{zf'(z)}{f(z)} \prec G(z,t)$$ and $$\frac{wg'(z)}{g(w)} \prec G(w,t)$$ where the function $g = f^{-1}$ is defined by (1.2). Let us take a look some lemmas which are very useful in building our main results. Let $\mathcal{P}$ denote the class of analytic functions p in $\mathbb{U}$ such that p(0) = 1 and Re(p(z)) > 0, $z \in \mathbb{U}$ . Also, we know that this class is usually called the Carathéodory class.