Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
In this paper, we obtain some potentially useful conditions (or criteria) for the
Carathéodory functions as a certain class of analytic functions by applying
Nunokawa’s lemma. We also obtain several conditions for strong starlikeness and
close-to-convexity as special cases of the main results presented here.
MSC: Primary 30C45; secondary 30C80
Results & Lemmas (11)
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 (see [15] and [16]) Let the function p(z) given by p(z) = 1 + ∞ n=m cnzn (cm ̸= 0) be analytic in U with p(0) = 1 and p(z) ̸= 0…
Lemma 1.1 (see [15] and [16]) Let the function p(z) given by p(z) = 1 + ∞ n=m cnzn (cm ̸= 0) be analytic in U with p(0) = 1 and p(z) ̸= 0 (z ∈U). If there exists a point z0 (with |z0| < 1) such that arg
Theorem 2.1
Theorem 2.1 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality:…
Theorem 2.1 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality: p(z) 2 + zp′(z) p(z) < A(α) p(z) , (2.1) where A(α) = ⎧ ⎨
Corollary 2.1
Corollary 2.1 Let the function f ∈A, with f ′′(0) ̸= 0, satisfy the following inequality: 1 + zf ′′(z) f ′(z) + zf ′(z) f (z)
Corollary 2.1 Let the function f ∈A, with f ′′(0) ̸= 0, satisfy the following inequality: 1 + zf ′′(z) f ′(z) + zf ′(z) f (z)
Corollary 2.2
Corollary 2.2 If the function f ∈A, with f ′′(0) ̸= 0, satisfies the following inequality: f ′(z) 2 + zf ′(z) f (z) < 1 2 f…
Corollary 2.2 If the function f ∈A, with f ′′(0) ̸= 0, satisfies the following inequality: f ′(z) 2 + zf ′(z) f (z) < 1 2 f ′(z) , then f ∈C. We now state and prove the following result.
Theorem 2.2
Theorem 2.2 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality:…
Theorem 2.2 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality: p(z) + zp′(z) [p(z)]2 < B(α) p(z) , where B(α) = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 1 + α2 4 (α(1–2α)/2 +α(–1–2α)/2)2 + α(α(1–2α)/2 +α(–1–2α)/2)sin(πα)
Theorem 2.2.
Repeated OCR label — verify against source
Theorem 2.2. From the two above-discussed contradictions, it follows that arg p(z) < απ 2 (∀z ∈U). This completes the proof of…
Theorem 2.2. From the two above-discussed contradictions, it follows that arg p(z) < απ 2 (∀z ∈U). This completes the proof of Theorem 2.2. □
Corollary 2.3
Corollary 2.3 Let the function f ∈A, with f ′′(0) ̸= 0, satisfy the following inequality: f (z) zf ′(z)
Corollary 2.3 Let the function f ∈A, with f ′′(0) ̸= 0, satisfy the following inequality: f (z) zf ′(z)
Theorem 2.3
Theorem 2.3 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality:…
Theorem 2.3 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality: arg
Corollary 2.4
Corollary 2.4 Let the function f ∈A, with f ′′(0) ̸= 0, satisfy the following inequality: arg
Corollary 2.4 Let the function f ∈A, with f ′′(0) ̸= 0, satisfy the following inequality: arg
Theorem 2.4
Theorem 2.4 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality:…
Theorem 2.4 Let p be an analytic function in U, with p(0) = 1, p′(0) ̸= 0, and p(z) ̸= 0 for z ∈U, that satisfies the following inequality: arg
Corollary 2.5
Corollary 2.5 Suppose that the function f ∈A, with f ′′(0) ̸= 0, satisfies the following in- equality: arg
Corollary 2.5 Suppose that the function f ∈A, with f ′′(0) ̸= 0, satisfies the following in- equality: arg
Function classes studied:
Related Papers