Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
The goal of this manuscript to establish the best possible estimate on coefficient functionals like Hermitian-Toeplitz determinant of secoend order involving logarithmic coefficients, initial logarithmic inverse coefficients and initial order Schwarzian derivatives of the Ozaki close-to-convex functions.
Results & Lemmas (9)
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.
Lemma 1.1
Lemma 1.1. [ ](#page-12-16) Let w(z) = c1z + c2z <sup>2</sup> + c3z <sup>3</sup> + c4z <sup>4</sup> +... be a Schwarz function. Then,,.
Lemma 1.1. [\[6\]](#page-12-16) Let w(z) = c1z + c2z <sup>2</sup> + c3z <sup>3</sup> + c4z <sup>4</sup> + ... be a Schwarz function. Then
$$|c_1| \le 1$$
, $|c_2| \le 1 - |c_1|^2$ , $|c_3| \le 1 - |c_1|^2 - \frac{|c_2|^2}{1 + |c_1|}$ .
Lemma 1.2
Lemma 1.2. [ ](#page-12-17) Let P be the class of analytic functions having the Taylor series of the form satisfying the condition ℜ(p(z))…
Lemma 1.2. [\[14\]](#page-12-17) Let P be the class of analytic functions having the Taylor series of the form
$$p(z) = 1 + p_1 z + p_2 z^2 + p_3 z^3 + \cdots {(1.12)}$$
satisfying the condition ℜ(p(z)) > 0 (z ∈ D). Then
$$2p_{2} = p_{1}^{2} + t\xi,$$
$$4p_{3} = p_{1}^{3} + 2p_{1}t\xi - p_{1}t\xi^{2} + 2t(1 - |\xi|^{2})\eta,$$
$$8p_{4} = p_{1}^{4} + 3p_{1}^{2}t\xi + (4 - 3p_{1}^{2})t\xi^{2} + p_{1}^{2}t\xi^{3} + 4t(1 - |\xi|^{2})(1 - |\eta|^{2})\gamma$$
$$+ 4t(1 - |\xi|^{2})(p_{1}\eta - p_{1}\xi\eta - \bar{\xi}\eta^{2}),$$
for some ξ, η, γ ∈ D and t = (4 − p 2 1 ).
Theorem 2.1
Theorem 2.1. Let the function f(z) = z + P<sup>∞</sup> <sup>n</sup>=2 anz n be in the class F. Then The upper and lower bound are sharp for…
Theorem 2.1. Let the function f(z) = z + P<sup>∞</sup> <sup>n</sup>=2 anz n be in the class F. Then
$$-\frac{1}{16} \le T_{2,1}(F_f/\gamma) \le \frac{95}{256}.$$
The upper and lower bound are sharp for the functions f<sup>1</sup> and f<sup>2</sup> given by [\(1.1\)](#page-0-0) and [\(1.2\)](#page-0-1) respectively.
Theorem 2.2 · coeff
Theorem 2.2. Let the function be in the class. Then All inequalities are sharp for the function given in (1.1).
Theorem 2.2. Let the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{F}$ . Then
$$|\Gamma_1| \le \frac{3}{4}, \quad |\Gamma_2| \le \frac{11}{16}, \quad |\Gamma_3| \le \frac{7}{8}.$$
All inequalities are sharp for the function $f_1$ given in (1.1).
Theorem 2.3 · coeff
Theorem 2.3. Let the function be in the class. Then The inequality is sharp for the function given by (1.2).
Theorem 2.3. Let the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{F}$ . Then
The inequality is sharp for the function $f_2$ given by (1.2).
Theorem 2.4 · coeff
Theorem 2.4. Let the function be in the class. Then (a), (b). Both the inequalities are sharp for the function given by (1.1).
Theorem 2.4. Let the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{F}$ . Then
(a)
$$|A_3 - A_2| \le 4$$
,
(b)
$$|\Gamma_3 - \Gamma_2| \le \frac{25}{16}$$
.
Both the inequalities are sharp for the function given by (1.1).
Theorem 3.1 · coeff
Theorem 3.1. Let be in the class. Then The upper and lower bound are sharp for the function and given by (1.3) and (1.4) respectively.
Theorem 3.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{G}$ . Then
$$-\frac{1}{144} \le T_{2,1}(F_f/\gamma) \le \frac{15}{256}.$$
The upper and lower bound are sharp for the function $g_1$ and $g_2$ given by (1.3) and (1.4) respectively.
Theorem 3.2
Theorem 3.2. Let f(z) = z + P<sup>∞</sup> <sup>n</sup>=2 anz n be in the class G. Then All three inequalities are sharp for the function…
Theorem 3.2. Let f(z) = z + P<sup>∞</sup> <sup>n</sup>=2 anz n be in the class G. Then
$$|\Gamma_1| \le \frac{1}{4}, \quad |\Gamma_2| \le \frac{3}{16}, \quad |\Gamma_3| \le \frac{5}{24}.$$
All three inequalities are sharp for the function g<sup>1</sup> given by [\(1.3\)](#page-1-2).
Theorem 3.3 · coeff
Theorem 3.3. Let the function be in the class. Then and. The both the inequalities are sharp for the function given by (1.3).
Theorem 3.3. Let the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{G}$ . Then
$$|S_3| \le \frac{3}{2}$$
and $|S_4| \le 6$ .
The both the inequalities are sharp for the function given by (1.3).
Function classes studied:
Related Papers