Sharp coefficients bounds for Starlike functions associated

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Abstract

In this paper we introduced the class $\mathcal{S}_{G}^{\ast }$ of analytic functions which is related with starlike functions and generating function of Gregory coefficients. By using bounds on some coefficient functionals for the family of functions with positive real part, we obtain for functions in the class $\mathcal{S}_{G}^{\ast }$ several sharp coefficient bounds on the first six coeffcients and also further sharp bounds on the corresponding Hankel determinants.

Results & Lemmas (8)

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Lemma 2.1 Lemma 2.1. [28] If, then for some with and.

Lemma 2.1. [28] If $p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \cdots \in \mathcal{P}$ $(p_1 \ge 0)$ , then $$2p_2 = p_1^2 + x(4 - p_1^2) (2.1)$$ $$4p_3 = p_1^3 + 2(4 - p_1^2)p_1x - p_1(4 - p_1^2)x^2 + 2(4 - p_1^2)\left(1 - |x|^2\right)y, \quad (2.2)$$ for some $x, y \in \mathbb{C}$ with $|x| \leq 1$ and $|y| \leq 1$ .

Lemma 2.2 Lemma 2.2. If, then and if and, then Also = (2.5) The inequalities (2.3), (2.4) and (2.5) are taken from [8, 28] and [36], respectively.

Lemma 2.2. If $p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \cdots \in \mathcal{P}$ $(p_1 \ge 0)$ , then $$|p_n| \le 2 \quad (n \ge 1) \,, \tag{2.3}$$ and if $Q \in [0,1]$ and $Q(2Q-1) \le R \le Q$ , then $$\left| p_3 - 2Qp_1p_2 + Rp_1^3 \right| \le 2. (2.4)$$ Also $$|p_{n+k} - \mu p_n p_k| \le 2 \max\{1, |2\mu - 1|\}$$ = $2 \begin{cases} 1, & \text{for } 0 \le \mu \le 1, \\ |2\mu - 1|, & \text{otherwise} \end{cases}$ (2.5) The inequalities (2.3), (2.4) and (2.5) are taken from [8, 28] and [36], respectively.

Lemma 2.3 Lemma 2.3. (see [37]). Let,, and satisfy the inequalities and (2.6) 4 ERHAN DENIZ1, SERCAN KAZIMOĞLU1 AND H. M.…

Lemma 2.3. (see [37]). Let $\tau$ , $\psi$ , $\rho$ and $\varsigma$ satisfy the inequalities $0 < \tau < 1, \ 0 < \varsigma < 1$ and $$8\varsigma (1 - \varsigma) \left[ (\tau \psi - 2\rho)^{2} + (\tau (\varsigma + \tau) - \psi)^{2} \right] + \tau (1 - \tau) (\psi - 2\varsigma \tau)^{2}$$ $$\leq 4\varsigma \tau^{2} (1 - \tau)^{2} (1 - \varsigma).$$ (2.6) 4 ERHAN DENIZ1, SERCAN KAZIMOĞLU1 AND H. M. SRIVASTAVA2,3,4,5,6 If $$p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \dots \in \mathcal{P}$$ $(p_1 \ge 0)$ , then $$\left| \rho p_1^4 + \varsigma p_2^2 + 2\tau p_1 p_3 - \frac{3}{2} \psi p_1^2 p_2 - p_4 \right| \le 2. \tag{2.7}$$

Lemma 2.4 Lemma 2.4. [34] Let. Also, for any real numbers a, b and c, let the quantity. If, then Furthermore, if ac < 0, then where

Lemma 2.4. [34] Let $\overline{\mathcal{U}} = \{z : |z| \le 1\}$ . Also, for any real numbers a, b and c, let the quantity $Y(a,b,c) = \max_{z \in \overline{\mathcal{U}}} \{|a+bz+cz^2|+1-|z|^2\}$ . If $ac \ge 0$ , then $$Y(a,b,c) = \begin{cases} |a| + |b| + |c| & |b| \ge 2(1 - |c|) \\ 1 + |a| + \frac{b^2}{4(1 - |c|)} & |b| < 2(1 - |c|) \end{cases}.$$ Furthermore, if ac < 0, then $$Y(a,b,c) = \begin{cases} 1 - |a| + \frac{b^2}{4(1-|c|)} & (-4ac(c^{-2}-1) \le b^2; \ |b| < 2(1-|c|)) \\ 1 + |a| + \frac{b^2}{4(1+|c|)} & b^2 < \min\left\{4(1+|c|)^2, -4ac(c^{-2}-1)\right\} \\ R(a,b,c) & (otherwise) \end{cases},$$ where $$R(a,b,c) = \begin{cases} |a| + |b| - |c| & (|c| (|b| + 4 |a|) \le |ab|)) \\ -|a| + |b| + |c| & (|ab| \le |c| (|b| - 4 |a|)) \\ (|a| + |c|) \sqrt{1 - \frac{b^2}{4ac}} & (otherwise) \end{cases}.$$

Theorem 3.1 · coeff Theorem 3.1. Let. Then All of the above estimates, except that on, are sharp.

Theorem 3.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_G^*$ . Then $$|a_n| \le \frac{1}{2(n-1)} \quad (n=2,3,4,5)$$ $|a_6| \le \frac{13}{48}.$ All of the above estimates, except that on $a_6$ , are sharp.

Theorem 3.3 · coeff Theorem 3.3. Let. Then, the following sharp estimates holds: and

Theorem 3.3. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_G^*$ . Then, the following sharp estimates holds: $$\left| a_3 - \mu a_2^2 \right| \le \frac{1}{4} \max\{1, \left| \mu - \frac{1}{3} \right| \}, \quad (\mu \in \mathbb{C})$$ $$\left| a_2 a_3 - a_4 \right| \le \frac{1}{6}$$ and $$\left| a_2 a_4 - a_3^2 \right| \le \frac{1}{16}.$$

Theorem 3.5 · coeff Theorem 3.5. If and has the series representation then (3.24) These bounds are sharp and can be obtained from the extremal functions i = 1,…

Theorem 3.5. If $f \in \mathcal{S}_G^*$ and has the series representation $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ then $$|\beta_n| \le \frac{1}{4n} \ (n = 1, 2, 3, 4).$$ (3.24) These bounds are sharp and can be obtained from the extremal functions $f_i(z)$ i = 1, 2, 3, 4 given by (3.10)-(3.13).

Theorem 3.6 · coeff Theorem 3.6. If and has the series representation then These bounds are sharp, except.

Theorem 3.6. If $f \in \mathcal{S}_G^*$ and has the series representation $f^{-1}(w) = w + A_2 w^2 + A_3 w^3 + \cdots$ then $$|A_2| \le \frac{1}{2},$$ $|A_3| \le \frac{5}{12},$ $|A_4| \le \frac{31}{72}.$ $$(3.33)$$ These bounds are sharp, except $A_4$ .

Function classes studied: