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Abstract
There are many results for sufficient conditions of functions f(z) which are analytic in the open unit disc U to be starlike and convex in U. The object of the present paper is to derive some interesting sufficient conditions for f(z) to be starlike of order α and convex of order α in U concerned with Jack's lemma. Some examples for our results are also considered with the help of Mathematica 5.2.
Results & Lemmas (4)
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Lemma 1.
Lemma 1. Let w(z) be analytic in U with w(0) = 0. Then if |w(z)| attains its maximum value on the circle |z| = r at a point z0 ∈U, then we…
Lemma 1. Let w(z) be analytic in U with w(0) = 0. Then if |w(z)| attains its maximum value on the circle |z| = r at a point z0 ∈U, then we have z0w′(z0) = kw(z0), where k ≧1 is a real number. 2. Main results Applying Lemma 1, we drive the following result for the class C.
Theorem 1.
Theorem 1. If f(z) ∈A satisfies Re 1 + zf ′′(z) f ′(z) < β + 1 2(β −1) (z ∈U) for some real 2 ≦β < 3, or Re 1 + zf ′′(z)
Theorem 1. If f(z) ∈A satisfies Re 1 + zf ′′(z) f ′(z) < β + 1 2(β −1) (z ∈U) for some real 2 ≦β < 3, or Re 1 + zf ′′(z)
Corollary 1.
Corollary 1. If f(z) ∈A satisfies Re 1 + zf ′′(z) f ′(z) < 3 2 (z ∈U), then zf ′(z) f(z) ≺2(1 −z) 2 −z (z ∈U)
Corollary 1. If f(z) ∈A satisfies Re 1 + zf ′′(z) f ′(z) < 3 2 (z ∈U), then zf ′(z) f(z) ≺2(1 −z) 2 −z (z ∈U)
Theorem 2.
Theorem 2. If f(z) ∈A satisfies Re 1 + zf ′′(z) f ′(z) > − β + 1 2β(β −1) (z ∈U) for some real β ≦−1, or Re 1 + zf ′′(z)
Theorem 2. If f(z) ∈A satisfies Re 1 + zf ′′(z) f ′(z) > − β + 1 2β(β −1) (z ∈U) for some real β ≦−1, or Re 1 + zf ′′(z)
Function classes studied:
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