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.
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:
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