Abstract
For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $φ(f,n,m;λ):=|λa_n a_m -a_{n+m-1}|$. For all real parameters $λ$ and $ β<1$, we provide the sharp upper bound of $φ(f,n,m;λ)$ for functions $f$ satisfying $\operatorname{Re}f'(z) > β$ and hence settles the open problem of estimating $φ(f,n,m;λ)$ recently proposed by Agrawal and Sahoo [S. Agrawal and S. k.
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. [20, Lemma 2.3, p. 507] If, then for all, The result is sharp.
Lemma 1.1. [20, Lemma 2.3, p. 507] If $p(z) = 1 + \sum_{k=1}^{\infty} c_k z^k \in \mathcal{P}$ , then for all $n, m \in \mathbb{N}$ ,
$$|\mu c_n c_m - c_{n+m}| \le \begin{cases} 2, & 0 \le \mu \le 1; \\ 2|2\mu - 1|, & elsewhere. \end{cases}$$
The result is sharp.
Lemma 1.2
Lemma 1.2. [12, Theorem 4(b), p. 678] A function if and only if for every sequence of complex numbers which satisfy.
Lemma 1.2. [12, Theorem 4(b), p. 678] A function $p(z) = 1 + \sum_{k=1}^{\infty} c_k z^k \in \mathcal{P}$ if and only if
$$\sum_{j=0}^{\infty} \left\{ \left| 2z_j + \sum_{k=1}^{\infty} c_k z_{k+j} \right|^2 - \left| \sum_{k=0}^{\infty} c_{k+1} z_{k+j} \right|^2 \right\} \ge 0$$
for every sequence $\{z_k\}$ of complex numbers which satisfy $\limsup_{k\to\infty} |z_k|^{1/k} < 1$ .
Lemma 1.3
Lemma 1.3. [12, Theorem 4(f), p. 678] A function if and only if for every sequence of complex numbers which satisfy.
Lemma 1.3. [12, Theorem 4(f), p. 678] A function $f(z) = z + \sum_{k=2}^{\infty} a_k z^k \in T$ if and only if
$$\sum_{j=0}^{\infty} \left\{ \left| 2z_j + \sum_{k=1}^{\infty} (a_{k+1} - a_{k-1}) z_{k+j} \right|^2 - \left| \sum_{k=0}^{\infty} (a_{k+2} - a_k) z_{k+j} \right|^2 \right\} \ge 0$$
for every sequence $\{z_k\}$ of complex numbers which satisfy $\limsup_{k\to\infty} |z_k|^{1/k} < 1$ .
Lemma 1.4
Lemma 1.4. Let be a probability measure on. Then for all,
Lemma 1.4. Let $\nu(t)$ be a probability measure on $[0, 2\pi]$ . Then for all $n, m \in \mathbb{N}$ ,
$$\left|\lambda \int_0^{2\pi} e^{int} \, d\nu(t) \int_0^{2\pi} e^{imt} \, d\nu(t) - \int_0^{2\pi} e^{i(n+m)t} \, d\nu(t)\right| \leq \begin{cases} 1, & 0 \leq \lambda \leq 2; \\ |\lambda - 1|, & elsewhere. \end{cases}$$
Theorem 2.1 · coeff
Theorem 2.1. If, then for all n, m = 2, 3,..., where is given by (2.2). The second inequality is sharp for the function and its rotations…
Theorem 2.1. If $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in HS^*(\alpha) \ (\alpha < 1)$ , then for all n, m = 2, 3, ...,
$$|\lambda a_{n}a_{m} - a_{n+m-1}| \leq \begin{cases} A_{n+m-1}, & 0 \leq \lambda \leq \frac{2A_{n+m-1}}{A_{n}A_{m}}; \\ |\lambda A_{n}A_{m} - A_{n+m-1}|, & elsewhere, \end{cases}$$
where $A_n$ is given by (2.2). The second inequality is sharp for the function $f_1$ and its rotations where $f_1$ is given by the equation (2.1).
Theorem 3.1 · coeff
Theorem 3.1. If, then for all n, m = 2, 3,..., The result is sharp.
Theorem 3.1. If $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{R}(\beta) \ (\beta < 1)$ , then for all n, m = 2, 3, ...,
$$|\lambda a_n a_m - a_{n+m-1}| \le \begin{cases} \frac{2(1-\beta)}{n+m-1}, & 0 \le \lambda \le \frac{nm}{(1-\beta)(n+m-1)}; \\ \left| \frac{4\lambda(1-\beta)^2}{nm} - \frac{2(1-\beta)}{n+m-1} \right|, & elsewhere. \end{cases}$$
The result is sharp.
Theorem 3.3 · coeff
Theorem 3.3. If and, then - (i) if n = 2 and m is even, the upper bound of is (a) for, - (b) - (ii) if m=2 and n is even, the upper bound…
Theorem 3.3. If $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in T$ and $\lambda \geq 1$ , then
- (i) if n = 2 and m is even, the upper bound of $|\lambda a_n a_m a_{n+m-1}|$ is (a) $3 + (2\lambda - 1)(m - 2)$ for $1 \le \lambda \le 3/2$ ,
- (b) $2\lambda m m 1 \text{ for } \lambda > 3/2;$
- (ii) if m=2 and n is even, the upper bound of $|\lambda a_n a_m a_{n+m-1}|$ is
- (a) $3 + (2\lambda 1)(n 2)$ for $1 < \lambda < 3/2$ ,
- (b) $2\lambda n n 1$ for $\lambda \ge 3/2$ ;
- (iii) in the other cases, we have
$$|\lambda a_n a_m - a_{n+m-1}| \le \lambda mn - n - m + 1.$$
The bounds given by (i)(b), (ii)(b) and (iii) are sharp whereas the bounds in (i)(a) and (ii)(a) are sharp for $\lambda = 1$ or the case when n = 2 and m = 2.
Theorem 4.1 · coeff
Theorem 4.1. If and, then for all n, m = 2, 3,..., where (4.2) The inequality is sharp.
Theorem 4.1. If $\mu \ge \max\{nm/((n+m-1)(1-\beta)), nm/(n+m-1)\}$ and $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{F}_1(\beta)$ $(\beta < 1)$ , then for all n, m = 2, 3, ...,
$$|\mu a_n a_m - a_{n+m-1}| \le \mu B_n B_m - B_{n+m-1},$$
where
(4.2)
$$B_n = \frac{1 + 2(n-1)(1-\beta)}{n} \quad (n \ge 2).$$
The inequality is sharp.
Theorem 4.3 · coeff
Theorem 4.3. If and, then for all except when both n and m are even, where, for, (4.6) The result is sharp.
Theorem 4.3. If $\mu \geq \max\{nm/((n+m-1)(1-\beta)), nm/(n+m-1)\}$ and $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{F}_2(\beta)$ $(\beta < 1)$ , then for all $n, m = 2, 3, \ldots$ except when both n and m are even,
$$|\mu a_n a_m - a_{n+m-1}| \le \mu C_n C_m - C_{n+m-1}$$
where, for $n \geq 2$ ,
(4.6)
$$C_n = \begin{cases} \frac{1 + (n-1)(1-\beta)}{n}, & \text{if } n \text{ is odd;} \\ 1 - \beta, & \text{if } n \text{ is even.} \end{cases}$$
The result is sharp.
Function classes studied:
Coefficient bounds & claims (14)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
HS*(alpha): If f(z) = z + sum a_n z^n in HS*(alpha) (alpha < 1), then for all n,m = 2,3,..., |lambda*a_n*a_m - a_{n+m-1}| <= A_{n+m-1} if 0 <= lambda <= 2*A_{n+m-1}/(A_n*A_m); |lambda*A_n*A_m - A_{n+m-1}| elsewhere, where A_n = prod_{j=0}^{n-2}(2(1-alpha)+j)/(n-1)!. The second inequality is sharp for f1 and its rotations. (sharp) [Theorem 2.1]
coefficient_bound
HS*(alpha): If f(z) in HS*(alpha) (alpha < 1), then |lambda*a_n^2 - a_{2n-1}| <= A_{2n-1} if 0 <= lambda <= 2*A_{2n-1}/A_n^2; |lambda*A_n^2 - A_{2n-1}| elsewhere. Second inequality sharp for f1 and rotations. (sharp) [Corollary 2.2]
coefficient_bound
R(beta): If f(z) in R(beta) (beta < 1), then |lambda*a_n*a_m - a_{n+m-1}| <= 2(1-beta)/(n+m-1) if 0 <= lambda <= nm/((1-beta)(n+m-1)); |sqrt(4*lambda*(1-beta)^2/(nm)) - 2(1-beta)/(n+m-1)| elsewhere. The result is sharp. (sharp) [Theorem 3.1]
coefficient_bound
|lambda*a_n*a_m - a_{n+m-1}| for T (lambda >= 1) ≤ lambda*m*n - n - m + 1 for class T (sharp) [Theorem 3.3]
coefficient_bound
|lambda*a_n*a_m - a_{n+m-1}| for S_R (lambda >= 1) ≤ lambda*m*n - n - m + 1 for class S_R (sharp) [Corollary 3.4]
coefficient_bound
F1(beta): If mu >= max{nm/((n+m-1)(1-beta)), nm/(n+m-1)} and f in F1(beta), then |mu*a_n*a_m - a_{n+m-1}| <= mu*B_n*B_m - B_{n+m-1} where B_n = 1 + 2(n-1)(1-beta)/n. The inequality is sharp. (sharp) [Theorem 4.1]
coefficient_bound
|mu*a_{2n+1}^2 - a_{4n+1}| for F2(0) ≤ mu - 1 for class F2(beta) (sharp) [Corollary 4.4]
function_family
Class HS*(alpha): closed convex hull of S*(alpha), starlike functions of order alpha (Re(zf'(z)/f(z)) > alpha)
function_family
Class HK(alpha): closed convex hull of K(alpha), convex functions of order alpha (Re(1+zf''(z)/f'(z)) > alpha)
function_family
Class R(beta): f in A satisfying Re(f'(z)) > beta; for 0<=beta<1 a subclass of S
function_family
Class T: typically real functions: f in A with real values on real axis and non-real values elsewhere
function_family
Class S_R: subclass of S with real coefficients
function_family
Class F1(beta): f in A satisfying Re((1-z)*f'(z)) > beta; close-to-convex of order beta w.r.t. starlike g(z)=z/(1-z)
function_family
Class F2(beta): f in A satisfying Re((1-z^2)*f'(z)) > beta; close-to-convex of order beta w.r.t. starlike g(z)=z/(1-z^2)
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