Abstract
Let A be the class of analytic functions in the unit disc U which are of the form f(z) = z +
P∞
n=2 anzn. For 0 ≤α < 1, let Cα, be the class of all functions f ∈A satisfying the condition Re{f′(z) +
αzf′′(z)} > 0. We consider the Toeplitz matrices whose elements are the coefficients an of the function f in the
class Cα. In this paper we obtain upper bounds for the Toeplitz determinants.
Results & Lemmas (6)
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. Let p(z) = 1 + P∞ n=1 cnzn ∈P with c1 ≥0. Then for some complex valued x with |x| ≤1 and some complex valued z with |z| ≤1, we…
Lemma 1.1. Let p(z) = 1 + P∞ n=1 cnzn ∈P with c1 ≥0. Then for some complex valued x with |x| ≤1 and some complex valued z with |z| ≤1, we have 2c2 = c2 1 + x(4 −c2 1), 4c3 = c3 1 + 2(4 −c2 1)c1x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)z.
Theorem 2.1.
Theorem 2.1. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T2(2)| ≤5(1 + α)2 + 9α(2 + 3α) 9(1 + α)2(1 + 2α)2. (2.4) The…
Theorem 2.1. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T2(2)| ≤5(1 + α)2 + 9α(2 + 3α) 9(1 + α)2(1 + 2α)2 . (2.4) The bound is sharp.
Theorem 2.2.
Theorem 2.2. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T2(3)| ≤ 4 9(1 + 2α)2. (2.7) The bound is sharp.
Theorem 2.2. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T2(3)| ≤ 4 9(1 + 2α)2 . (2.7) The bound is sharp.
Theorem 2.3.
Theorem 2.3. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T3(1)| ≤13 + 36α + 36α2 9(1 + 2α)2. (2.10) The bound is…
Theorem 2.3. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T3(1)| ≤13 + 36α + 36α2 9(1 + 2α)2 . (2.10) The bound is sharp.
Theorem 2.3.
Theorem 2.3. □
Theorem 2.3. □
Theorem 2.4.
Theorem 2.4. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T3(2)| ≤ 4(1 + 5α) 9(1 + α)(1 + 2α)2(1…
Theorem 2.4. Let f given by (1.1) be in the class ˜Rα. Then we have the bound |T3(2)| ≤ 4(1 + 5α) 9(1 + α)(1 + 2α)2(1 + 3α)
Definitions (3)
Def 1.1.
Definition 1.1. ([4]) For α ≥0, a function f ∈A with f(z)f ′(z) z ̸= 0 is said to be an alpha- close-to-convex function if for a starlike…
Definition 1.1. ([4]) For α ≥0, a function f ∈A with f(z)f ′(z) z ̸= 0 is said to be an alpha- close-to-convex function if for a starlike function ϕ(z), it satisfies the condition Re (1 −α)zf ′(z) ϕ(z) + α(zf ′(z))′ ϕ′(z)
Def 1.2.
Definition 1.2. ([4]) Let ˜Rα be the class of all functions f ∈A which satisfy Re(f ′(z) + αzf ′′(z)) > 0, for all z ∈U. For α = 0, ˜Rα ≡R0…
Definition 1.2. ([4]) Let ˜Rα be the class of all functions f ∈A which satisfy Re(f ′(z) + αzf ′′(z)) > 0, for all z ∈U. For α = 0, ˜Rα ≡R0 ≡R = {f(z) ∈A : Re(f ′(z)) > 0, for all z ∈U}. These classes have been studied by many authors [4, 9, 10, 12, 13] in various viewpoints.
Def 1.3.
Definition 1.3. The q-th Toeplitz determinant of f(z) for q ≥1 and n ≥1 is defined as Tq(n) =
Definition 1.3. The q-th Toeplitz determinant of f(z) for q ≥1 and n ≥1 is defined as Tq(n) =
Function classes studied:
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