A Class of Non-Bazilevic Functions Subordinate to Gegenbauer Polynomials

Lemma 2.1. Let the Schwarz function w(z) be given by: $$w(z) = w_1 z + w_2 z^2 + w_3 z^3 + \cdots$$ where $z \in \mathbb{D}$ , then $|w_1| \leq 1$ and for $t \in \mathbb{C}$ $$|w_2 - tw_1^2| \le 1 + (|t| - 1)|w_1|^2 \le \max\{1, |t|\}.$$ The results is sharp for the functions w(z) = z and $w(z) = z^2$ . The primary goal of this article is determining the estimates for the initial Taylor-Maclarin coefficients $|a_2|$ and $|a_3|$ for functions belonging to the class $P_{\Sigma}(\alpha, \beta, \lambda, x, y)$ . Furthermore, we examine the corresponding Fekete-Szegö functional problem for functions in the presented class.

Theorem 3.1. Let the function f given by (1.1) be in the class $P_{\Sigma}(\alpha, \beta, \lambda, x, y)$ . Then $$|a_2| \le \frac{2\lambda t}{|2 - \alpha\beta(x + y)|}\tag{3.1}$$ and $$|a_3| \le \frac{2\lambda t}{|3 - \alpha\beta(x^2 + xy + y^2)|} \max\left\{1, \frac{\left|\Delta(\alpha, \beta, \lambda)t^2 - 1\right|}{2t}\right\},\tag{3.2}$$ where $$\Delta(\alpha,\beta,\lambda) = \frac{2(\lambda+1)(2-\alpha\beta\delta_2)^2 + 2\lambda\alpha\beta\left(4\delta_2 - (\beta+1)\delta_2^2\right)}{(2-\alpha\beta\delta_2)^2}.$$

Corollary 3.1. Let the function f given by (1.1) be in the class $P_{\Sigma}^{*}(\beta, \lambda)$ . Then $$|a_2| \le \frac{2\lambda t}{|2 - \beta|}$$ and $$|a_3| \le \frac{2\lambda t}{|3-\beta|} \max \left\{ 1, \frac{\left| \Delta(\beta,\lambda)t^2 - 1 \right|}{2t} \right\},$$ where $$\Delta(\beta,\lambda) = \frac{2(\lambda+1)(2-\beta)^2 + 2\lambda\beta(3-\beta)}{(2-\beta)^2}.$$ If $\alpha = 0$ and $\beta = 0$ , then we get the following corollary.

Corollary 3.2. Let the function f given by (1.1) be in the class $P_{\Sigma}(\alpha, \beta, \lambda, x, y)$ with $\alpha = \beta = 0$ . Then $$|a_2| \leq \lambda t$$ , and $$|a_3| \leq \frac{2\lambda t}{3} \max\left\{1, \frac{\left|2(\lambda+1)t^2-1\right|}{2t}\right\}.$$

Theorem 4.1. Let the function f given by (1.1) be in the class $P_{\Sigma}(\alpha, \beta, \lambda, x, y)$ . Then for some $\zeta \in \mathbb{R}$ , $$|a_{3} - \zeta a_{2}^{2}| \leq \begin{cases} \frac{2\lambda t}{|A|}, & \text{if } \zeta \in [\zeta_{1}, \zeta_{2}]\\ \frac{\lambda|2B(\lambda+1)t^{2} - B - 2\lambda Rt^{2} - 4\lambda \zeta At^{2}|}{|AB|}, & \text{if } \zeta \notin [\zeta_{1}, \zeta_{2}], \end{cases}$$ (4.1) where $$\zeta_1 = \frac{[2\lambda(B-R) + 2B]t^2 - (2t+1)B}{4\lambda At^2},$$ $$\zeta_2 = \frac{[2\lambda(B-R) + 2B]t^2 + (2t-1)B}{4\lambda At^2},$$ $$A=3-\alpha\beta\delta_3$$ , $B=(2-\alpha\beta\delta_2)^2$ , and $R=\alpha\beta(4-(\beta+1)\delta_2)\delta_2$ . Proof. In view of (3.4) and (3.5) we have $$a_3 = \frac{C_1^{\lambda}(t)}{A}p_2 + \frac{C_2^{\lambda}(t)}{A}p_1^2 + \frac{R[C_1^{\lambda}(t)]^2}{2AB}p_1^2$$ (4.2) Now, for any real number $\zeta$ , using equation (3.4) we can write the last equation as $$a_3 - \zeta a_2^2 = \frac{C_1^{\lambda}(t)}{A} \left\{ p_2 + \left( \frac{C_2^{\lambda}(t)}{C_1^{\lambda}(t)} + \frac{RC_1^{\lambda}(t)}{2B} - \frac{\zeta A C_1^{\lambda}(t)}{B} \right) p_1^2 \right\}.$$ Using the initial values (2.2) and Lemma 2.1, we obtain $$|a_3 - \zeta a_2^2| \le \frac{2\lambda t}{|A|} \max\left\{1, \left| \frac{2(\lambda + 1)t^2 - 1}{2t} - \frac{\lambda tR}{B} - \frac{2\lambda \zeta At}{B} \right| \right\}$$ (4.3) Since t > 0, we have $$\left| \frac{2(\lambda+1)t^2 - 1}{2t} - \frac{\lambda tR}{B} - \frac{2\lambda \zeta At}{B} \right| \le 1.$$ Hence, solving for $\zeta$ we get: $$\frac{[2\lambda(B-R)+2B]t^2-(2t+1)B}{4\lambda At^2}\leq \zeta \leq \frac{[2\lambda(B-R)+2B]t^2+(2t-1)B}{4\lambda At^2}$$ Therefore, equation (4.3) becomes $$|a_3 - \zeta a_2^2| \le \begin{cases} \frac{2\lambda t}{|A|}, & \text{if } \zeta \in [\zeta_1, \zeta_2] \\ \frac{2\lambda t}{|A|} \left| \frac{2(\lambda + 1)t^2 - 1}{2t} - \frac{\lambda tR}{B} - \frac{2\lambda \zeta A t}{B} \right|, & \text{if } \zeta \notin [\zeta_1, \zeta_2], \end{cases}$$ $$(4.4)$$ This completes the Theorem's proof. The following corollaries are just consequences of Theorem 4.1.

Corollary 4.1. Let the function f given by (1.1) be in the class $P_{\Sigma}^*(\beta, \lambda)$ . Then for some $\zeta \in \mathbb{R}$ , $$|a_{3} - \zeta a_{2}^{2}| \leq \begin{cases} \frac{2\lambda t}{|3-\beta|}, & \text{if } \zeta \in [\zeta_{1}, \zeta_{2}] \\ \lambda \left| \frac{2(\lambda+1)t^{2}-1}{3-\beta} - \frac{2\lambda(\beta+2\zeta)t^{2}}{(2-\beta)^{2}} \right|, & \text{if } \zeta \notin [\zeta_{1}, \zeta_{2}], \end{cases}$$ $$(4.5)$$ where $$\zeta_1 = \frac{2\lambda(\beta^2 - 7\beta + 4)t^2 + (2t^2 - 2t - 1)(2 - \beta)^2}{4\lambda(3 - \beta)t^2},$$ $$\zeta_2 = \frac{2\lambda(\beta^2 - 7\beta + 4)t^2 + (2t^2 + 2t - 1)(2 - \beta)^2}{4\lambda(3 - \beta)t^2}.$$ If $\alpha = 0$ and $\beta = 0$ , then we get the following corollary.

Corollary 4.2. Let the function f given by (1.1) be in the class $P_{\Sigma}(\alpha, \beta, \lambda, x, y)$ with $\alpha = \beta = 0$ . Then for some $\zeta \in \mathbb{R}$ , $$|a_{3} - \zeta a_{2}^{2}| \leq \begin{cases} \frac{2\lambda t}{3}, & \text{if } \zeta \in [\zeta_{1}, \zeta_{2}] \\ \frac{\lambda}{3} \left| ((2\lambda - 3\zeta + 2)t^{2} - 1) \right|, & \text{if } \zeta \notin [\zeta_{1}, \zeta_{2}], \end{cases}$$ (4.6) where $$\zeta_1 = \frac{(2\lambda + 2)t^2 - 2t - 1}{3\lambda t^2},$$ $$\zeta_2 = \frac{(2\lambda + 2)t^2 + 2t - 1}{3\lambda t^2}.$$