Abstract
In this paper, we establish the sharp estimates of the pre-Schwarzian
norm of functions f in the Ma-Minda type starlike and convex classes S∗(φ) and
C(φ), respectively, whenever φ(z) = 3/
3 + (α −3)z −αz2
with −3 < α ≤1,
φ(z) = (1+z)(1−sz) with −1/3 ≤s ≤1/3 and φ(z) = 1+z/ ((1 −z)(1 + αz)) with
0 ≤α ≤1/2.
Results & Lemmas (5)
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Theorem 3.1
Theorem 3.1. Let. Then the pre-Schwarzian norm satisfies the following sharp inequality
Theorem 3.1. Let $f \in \mathcal{S}_{con}^*$ . Then the pre-Schwarzian norm satisfies the following sharp inequality
$$||P_f|| \le \frac{2(\alpha+6)}{3+\alpha}.$$
Theorem 3.2
Theorem 3.2. Let. Then the pre-Schwarzian norm satisfies the following sharp inequality where is the unique root of the equation
Theorem 3.2. Let $f \in \mathcal{S}_{lim}^*$ . Then the pre-Schwarzian norm satisfies the following sharp inequality
$$||P_f|| \le \frac{(1+t_s)(1-s+2st_s)}{1+st_s} + (1-t_s^2)(1-s+st_s),$$
where $t_s \in (0,1)$ is the unique root of the equation
$$-3s^3t^4 + \left(2s^3 - 8s^2\right)t^3 + \left(s^3 + 6s^2 - 7s\right)t^2 + \left(2s^2 + 6s - 2\right)t + s^2 + s + 1 = 0.$$
Theorem 3.3
Theorem 3.3. Let. Then the pre-Schwarzian norm satisfies the following inequality
Theorem 3.3. Let $f \in \mathcal{S}_{cs}^*$ . Then the pre-Schwarzian norm satisfies the following inequality
$$||P_f|| \le \frac{2(2-\alpha)}{(1+\alpha)(1-2\alpha)}.$$
Theorem 3.4
Theorem 3.4. For any, the pre-Schwarzian norm satisfies the following sharp inequality
Theorem 3.4. For any $f \in \mathcal{C}_{con}$ , the pre-Schwarzian norm satisfies the following sharp inequality
$$||P_f|| \le \frac{6}{3+\alpha}.$$
Theorem 3.5
Theorem 3.5. For any, the pre-Schwarzian norm satisfies the following sharp inequality where is the unique positive root of the equation in…
Theorem 3.5. For any $f \in C_{lim}$ , the pre-Schwarzian norm satisfies the following sharp inequality
$$||P_f|| \le \begin{cases} (1 - r_s^2)(1 - s + sr_s) & for \quad s \in (0, 1/3], \\ 1 - s & for \quad s \in [-1/3, 0], \end{cases}$$
where $r_s$ is the unique positive root of the equation $3sr^2 + 2(1-s)r - s = 0$ in (0,1).
Function classes studied:
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