Certain results on $q$-starlike and $q$-convex error functions

Lemma 1.1. [3] If $w \in \Omega$ , then (1.12) $$|w_2 - tw_1^2| \le \begin{cases} -t & if \quad t < -1, \\ 1 & if \quad -1 \le t \le 1, \\ t & if \quad t > 1. \end{cases}$$ When t < -1 or t > 1, the equality holds if and only if w(z) = z or one of its rotations. If -1 < t < 1, then equality holds if and only if $w(z) = z^2$ or one of its rotations. Equality holds for t = -1 if and only if (1.13) $$w(z) = z \frac{\lambda + z}{1 + \lambda z} \qquad (0 \le \lambda \le 1)$$ or one of its rotations, while for t = 1 the equality holds if and only if $$(1.14) w(z) = -z \frac{\lambda + z}{1 + \lambda z} (0 \le \lambda \le 1)$$ or one of its rotations. Although the above upper bound is sharp, it can be improved in the case, when -1 < t < 1 (1.15) $$|w_2 - tw_1^2| + (1+t)|w_1|^2 \le 1 \qquad (-1 < t \le 0),$$ $$|w_2 - tw_1^2| + (1-t)|w_1|^2 \le 1 \qquad (0 < t < 1).$$

Theorem 2.1. Let $0 \le k < \infty$ , $0 \le \alpha < 1$ , and let $p_{k,\alpha}(z) = 1 + p_1 z + p_2 z^2 + \cdots$ . Set $$\sigma_1 = \frac{10(p_1 + p_2)\varrho\vartheta_2^2 + 10\vartheta_2bp_1^2}{9\vartheta_3bp_1^2}, \quad \sigma_2 = \frac{10(p_1 - p_2)\varrho\vartheta_2^2 - 10\vartheta_2bp_1^2}{9\vartheta_3bp_1^2},$$ and $$\sigma_3 = \frac{10\varrho \vartheta_2^2 p_2 + 10\vartheta_2 b p_1^2}{9\vartheta_3 b p_1^2}.$$ If f given by (1.2) belongs to $\mathcal{ES}_{q,b}^{\beta}(p_{k,\alpha})$ , then <span id="page-4-4"></span> $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{10bp_1}{\varrho\vartheta_3} \left(\frac{p_2}{p_1} + \frac{9\mu\vartheta_3 - 10\vartheta_2}{10\varrho\vartheta_2^2}bp_1\right) & if \quad \mu < \sigma_1, \\ \frac{10bp_1}{\varrho\vartheta_3} & if \quad \sigma_1 \le \mu \le \sigma_2, \\ \frac{10bp_1}{\varrho\vartheta_3} \left(\frac{p_2}{p_1} + \frac{10\vartheta_2 - 9\mu\vartheta_3}{\varrho\vartheta_2}bp_1\right) & if \quad \mu > \sigma_2, \end{cases}$$ where $\varrho = 1 + i \tan \beta$ , $\vartheta_2 = [2]_q - 1$ and $\vartheta_3 = [3]_q - 1$ . Further, if $\sigma_1 \le \mu \le \sigma_3$ , then $$\left| a_3 - \mu a_2^2 \right| + \frac{10\varrho \vartheta_2^2}{9\vartheta_3 b p_1} \left( p_1 + p_2 + \frac{10\vartheta_2 - 9\mu \vartheta_3}{10\vartheta_2^2} b p_1^2 \right) \left| a_2 \right|^2 \le \frac{10b p_1}{\varrho \vartheta_3},$$ and, if $\sigma_3 \leq \mu \leq \sigma_2$ , then $$\left| a_3 - \mu a_2^2 \right| + \frac{10\varrho \vartheta_2^2}{9\vartheta_3 b p_1} \left( p_1 - p_2 - \frac{10\vartheta_2 - 9\mu \vartheta_3}{10\varrho \vartheta_2^2} b p_1^2 \right) \left| a_2 \right|^2 \le \frac{10b p_1}{\varrho \vartheta_3} b p_1^2$$ For any complex number $\mu$ , <span id="page-4-5"></span> $$\left| a_3 - \mu a_2^2 \right| \le \frac{10bp_1}{\varrho \vartheta_3} \max \left\{ 1, \left| \frac{p_2}{p_1} + \frac{10\vartheta_2 - 9\mu \vartheta_3}{\varrho \vartheta_2} b p_1 \right| \right\}.$$

Theorem 2.2. Let $0 \le k < \infty$ , $0 \le \alpha < 1$ , and let $p_{k,\alpha}(z) = 1 + p_1 z + p_2 z^2 + \cdots$ . For $f \in \mathcal{EC}^{\beta}_{a,b}(p_{k,\alpha})$ , it holds $$\begin{split} \left|a_{3}-\mu a_{2}^{2}\right| &\leq \left\{ \begin{array}{ll} \frac{5bp_{1}}{\varrho[3]_{q}}\left(\frac{p_{2}}{p_{1}}+\frac{bp_{1}}{\varrho}-\frac{9\mu[3]_{q}bp_{1}}{5[2]_{q}^{2}\varrho}\right) & if \quad \mu<\sigma_{1}, \\ \\ \frac{5bp_{1}}{\varrho[3]_{q}} & if \quad \sigma_{1}\leq\mu\leq\sigma_{2}, \\ \\ -\frac{5bp_{1}}{\varrho[3]_{q}}\left(\frac{p_{2}}{p_{1}}+\frac{bp_{1}}{\varrho}-\frac{9\mu[3]_{q}bp_{1}}{5[2]_{q}^{2}\varrho}\right) & if \quad \mu>\sigma_{2}, \end{array} \end{split}$$ where $$\sigma_1 = \frac{5[2]_q^2 \varrho}{9[3]_q b p_1^2} \left( p_2 - p_1 + \frac{b p_1^2}{\varrho} \right), \quad \sigma_2 = \frac{5[2]_q^2 \varrho}{9[3]_q b p_1^2} \left( p_1 + p_2 + \frac{b p_1^2}{\varrho} \right),$$ and $$\sigma_3 = \frac{5[2]_q^2 \varrho}{9[3]_a b p_1^2} \left( p_2 + \frac{b p_1^2}{\varrho} \right).$$ Further, if $\sigma_1 < \mu < \sigma_3$ , then $$\left|a_3 - \mu a_2^2\right| + \frac{5[2]_q^2 \varrho}{9\mu[3]_q b p_1} \left(p_1 - p_2 - \frac{bp_1^2}{\varrho} + \frac{9\mu[3]_q b p_1}{5[2]_q^2 \varrho}\right) \left|a_2\right|^2 \le \frac{5bp_1}{\varrho[3]_q},$$ and, if $\sigma_3 \leq \mu \leq \sigma_2$ , then $$\left|a_3 - \mu a_2^2\right| + \frac{5[2]_q^2 \varrho}{9\mu[3]_q b p_1} \left(p_1 + p_2 + \frac{b p_1^2}{\varrho} - \frac{9\mu[3]_q b p_1}{5[2]_q^2 \varrho}\right) \left|a_2\right|^2 \le \frac{5b p_1}{\varrho[3]_q}$$ For any complex number $\mu$ $$\left|a_3 - \mu a_2^2\right| \le \frac{5bp_1}{\varrho[3]_q} \max\left\{1, \left|\frac{p_2}{p_1} + \frac{bp_1}{\varrho} - \frac{9\mu[3]_q bp_1}{5[2]_q^2 \varrho}\right|\right\}$$

Theorem 3.1. Let $-\frac{\pi}{2} < \beta < \frac{\pi}{2}$ , 0 < q < 1, $b \neq 0$ and let $\varrho = 1 + i \tan \beta$ . If f of the form (1.2) belongs to $\widetilde{\mathcal{ES}}_{q,b}^{\beta}(\phi)$ , then $$|a_2| \le \frac{3bc_1}{\varrho(1-[2]_q)},$$ $$|a_3| \le \frac{10b}{\varrho([3]_q - 1)} \left( c_1 + \max \left\{ c_1, \left| \frac{bc_1^2}{\varrho([2]_q - 1)} \right| + |c_2| \right\} \right),$$ and for any complex number $\mu$ $$(3.2) \left| a_3 - \mu a_2^2 \right| \le \frac{10b}{\varrho([3]_q - 1)} \left( c_1 + \max \left\{ c_1, \left| \frac{10(1 - [2]_q) + 9\mu b([3]_q - 1)}{10\varrho(1 - [2]_q)^2} \right| bc_1^2 + |c_2| \right\} \right).$$

Theorem 3.2. Let $-\frac{\pi}{2} < \beta < \frac{\pi}{2}$ , 0 < q < 1 and $b \neq 0$ . If f given by (1.2) belongs to $\widetilde{\mathcal{ES}}_{q,b}^{\beta}(\phi)$ , then $$|a_2| \le \frac{3bc_1}{\varrho[2]_q},$$ $$|a_3| \le \frac{5b}{\varrho[3]_q} \left( c_1 + \max\left\{ c_1, \left| \frac{bc_1^2}{\varrho} \right| + |c_2| \right\} \right),$$ and for any complex number $\mu$ , $$|a_3 - \mu a_2^2| \le \frac{5b}{\varrho[3]_q} \left( c_1 + \max \left\{ c_1, \left| \frac{\varrho^2[2]_q^2 + 9\mu b}{\varrho^2[2]_q^2} \right| bc_1^2 + |c_2| \right\} \right).$$

Definition 1.1. Let $0 \le k < \infty$ , $0 \le \alpha < 1$ , $-\frac{\pi}{2} < \beta < \frac{\pi}{2}$ , 0 < q < 1, $b \ne 0$ , and let $p_{k,\alpha}(z)$ be defined as above. A function $f \in \mathcal{A}$ is in the class $\mathcal{ES}_{a,b}^{\beta}(p_{k,\alpha})$ if $$(1.11) 1 + \frac{1}{b} \left( (1 + i \tan \beta) \left( \frac{z D_q \mathcal{F}(z)}{\mathcal{F}(z)} \right) - i \tan \beta - 1 \right) \prec p_{k,\alpha}(z) (z \in \mathbb{U}).$$ A function $f \in \mathcal{A}$ is in the class $\mathcal{EC}_{ab}^{\beta}(p_{k,\alpha})$ if $$1 + \frac{1}{b} \left( (1 + i \tan \beta) \left( \frac{(z D_q \mathcal{F}(z))'}{D_q(\mathcal{F}(z))} \right) - i \tan \beta - 1 \right) \prec p_{k,\alpha}(z) \qquad (z \in \mathbb{U}).$$ Let $\phi(z) = 1 + c_1 z + c_2 z^2 + \cdots$ $(c_1 > 0)$ be an analytic function with positive real part on $\mathbb{U}$ which maps the open unit disk $\mathbb{U}$ onto a region starlike with respect to 1 and symmetric with respect to the real axis.

Definition 1.2. Let $0 \le k < \infty$ , $0 \le \alpha < 1$ , $-\frac{\pi}{2} < \beta < \frac{\pi}{2}$ , 0 < q < 1, $b \ne 0$ . By $\widetilde{\mathcal{ES}}_{q,b}^{\beta}(\phi)$ we mean a family that consist of the functions $f \in \mathcal{A}$ satisfying the quasi-subordination $$1 + \frac{1}{b} \left( (1 + i \tan \beta) \left( \frac{z D_q \mathcal{F}(z)}{\mathcal{F}(z)} \right) - i \tan \beta - 1 \right) \prec_q \phi(z) - 1 \quad (z \in \mathbb{U}),$$ and let the class $\widetilde{\mathcal{EC}}_{q,b}^{\beta}(\phi)$ consist of the functions $f \in \mathcal{A}$ satisfying the quasi-subordination $$1 + \frac{1}{b} \left( (1 + i \tan \beta) \left( \frac{(z D_q \mathcal{F}(z))'}{D_q(\mathcal{F}(z))} \right) - i \tan \beta - 1 \right) \prec_q \phi(z) - 1 \quad (z \in \mathbb{U}).$$ The principal significance of the sharp bounds of the coefficients is the information about geometric properties of the functions. For instance, the sharp bounds of the second coefficient of normalized univalent functions readily yields the growth and distortion bounds. Also, sharp bounds of the coefficient functional $|a_3 - \mu a_2^2|$ obviously help in the investigation of univalence of analytic functions. Apart from these n-th coefficient bounds were used to determine the extreme points of the classes of analytic functions. Estimates of Fekete-Szegö functional for various subclasses of univalent and multivalent functions were given, among other, in [2, 6, 35, 36]. In this paper, we obtain coefficient estimates for the functions in the above defined class for q-difference operator associated with subordination and quasi subordination. The following lemma is needed to prove our main results. Lemma 1.1 is a reformulation of the corresponding result for functions with positive real part due to Ma and Minda [29].