Abstract
By use of a modified Nunokawa's lemma, we obtain some new conditions for univalence. Also, some sharp inequalities concerning univalent functions are presented.
Results & Lemmas (5)
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Lemma 1.1.
Lemma 1.1. (Simple generalization of Nunokawa’s lemma [9]) Let p(z) be an analytic function in |z| < 1 of the form p(z) = 1 + ∞ X n=m cnzn…
Lemma 1.1. (Simple generalization of Nunokawa’s lemma [9]) Let p(z) be an analytic function in |z| < 1 of the form p(z) = 1 + ∞ X n=m cnzn (cm ̸= 0), with p(z) ̸= 0 in |z| < 1. If there exists a point z0, |z0| < 1, such that Re{p(z)} > 0 for |z| < |z0| and Re{p(z)} = 0, a = |p(z0)| ̸= 0,
Theorem 2.1.
Theorem 2.1. Let f be of the form (1.1). If f satisfies (2.1) Re f ′(z) + zf ′′(z) ≤1 (z ∈∆), or (2.2) Re f ′(z) + zf ′′(z) > 1 (z ∈∆), then…
Theorem 2.1. Let f be of the form (1.1). If f satisfies (2.1) Re {f ′(z) + zf ′′(z)} ≤1 (z ∈∆), or (2.2) Re {f ′(z) + zf ′′(z)} > 1 (z ∈∆), then Re 1 f ′(z) > 1
Theorem 2.2.
Theorem 2.2. Let f be of the form (1.1) and satisfies (2.9) Re zf ′(z) f(z) < 3 2 (z ∈∆). Then Re z f(z)
Theorem 2.2. Let f be of the form (1.1) and satisfies (2.9) Re zf ′(z) f(z) < 3 2 (z ∈∆). Then Re z f(z)
Theorem 2.3.
Theorem 2.3. Let f be of the form (1.1) and satisfies Re 1 + zf ′′(z) f ′(z) < 3 2 (z ∈∆). Then Re 1 f ′(z)
Theorem 2.3. Let f be of the form (1.1) and satisfies Re 1 + zf ′′(z) f ′(z) < 3 2 (z ∈∆). Then Re 1 f ′(z)
Theorem 2.4.
Theorem 2.4. Let f ∈A be starlike function of order 1/2. Then Re f(z) z > 1 2 (z ∈∆) and
Theorem 2.4. Let f ∈A be starlike function of order 1/2. Then Re f(z) z > 1 2 (z ∈∆) and