Results & Lemmas (5)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
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:
Coefficient bounds & claims (12)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
||Pf|| pre-Schwarzian norm ≤ 2*(alpha+6)/(3+alpha) for class S*_con (sharp) [Theorem 3.1]
coefficient_bound
||Pf|| pre-Schwarzian norm ≤ (1+t_s)*(1-s+2*s*t_s)/(1+s*t_s) + (1-t_s**2)*(1-s+s*t_s) for class S*_lim (sharp) [Theorem 3.2]
coefficient_bound
||Pf|| pre-Schwarzian norm ≤ 2*(2-alpha)/((1+alpha)*(1-2*alpha)) for class S*_cs [Theorem 3.3]
coefficient_bound
||Pf|| pre-Schwarzian norm ≤ 6/(3+alpha) for class C_con (sharp) [Theorem 3.4]
coefficient_bound
C_lim: For any f in C_lim, the pre-Schwarzian norm satisfies the following sharp inequality ||Pf|| <= (1-r_s^2)(1-s+sr_s) for s in (0,1/3], and 1-s for s in [-1/3,0], where r_s is the unique positive root of 3sr^2+2(1-s)r-s=0 in (0,1). (sharp) [Theorem 3.5]
coefficient_bound
||Pf|| pre-Schwarzian norm ≤ 2/(1+alpha) for class C_cs (sharp) [Theorem 3.6]
function_family
Class S*_con: f in A: zf'(z)/f(z) subordinate to 3/(3+(alpha-3)z-alpha*z^2), -3 < alpha <= 1
function_family
Class S*_lim: f in A: zf'(z)/f(z) subordinate to (1+z)(1-sz), -1/3 <= s <= 1/3
function_family
Class S*_cs: f in A: zf'(z)/f(z) subordinate to 1 + z/((1-z)(1+alpha*z)), 0 <= alpha <= 1/2
function_family
Class C_con: f in A: 1 + zf''(z)/f'(z) subordinate to 3/(3+(alpha-3)z-alpha*z^2), -3 < alpha <= 1
function_family
Class C_lim: f in A: 1 + zf''(z)/f'(z) subordinate to (1+z)(1-sz), -1/3 <= s <= 1/3
function_family
Class C_cs: f in A: 1 + zf''(z)/f'(z) subordinate to 1+z/((1-z)(1+alpha*z)), 0 <= alpha <= 1/2
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