On some simple sufficient conditions for univalence

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2. Conditions on f ′′ (z)<br />

Theorem 1. If f(z) ∈ A and |f ′′ (z)| k, z ∈ U, 0

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2. Conditions on f ′′ (z) Theorem 1. If f(z) ∈ A and |f ′′ (z)| k, z ∈ U, 0

ÊÑÖ 1. For α =0(k = 1) in Corollary 1 we get Theorem 1 from [4]. Corollary 1.1. Let f(z) ∈ A. Then (i) |f ′′ (z)| 4/5 implies f(z) ∈ S ∗ (1/3); (ii) |f ′′ (z)| 2/3 implies f(z) ∈ S ∗ (1/2); and (iii) |f ′′ (z)| 1/2 implies f(z) ∈ S ∗ (2/3). Corollary 2. If f(z) ∈ A and |f ′′ (z)| 2sin(αÔ/2)/(1 + sin(αÔ/2)) = k, z ∈ U, 0

Page 1 and 2: Mathematica Bohemica Nikola Tuneski
Page 3: In addition to these classes we wil
Page 7 and 8: ÊÑÖ 3. For α = 0, i.e., k =1/2,
Page 9: We can rewrite Theorem 3 in the fol

On some simple sufficient conditions for univalence

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