Coefficient Conditions for Starlikeness of Nonnegative Order

and Applied Analysis 3 Theorem 1.3 see 12 . Let c2k c2k 1 μ k/k!, μ ∈ 0, 1 . For any positive integer n and 0 < θ < π , then i ∑n k 0 ck cos kθ > 0 if and only if 0 < μ ≤ μ0, ii ∑2n 1 k 1 ck sin kθ > 0 if and only if 0 < μ ≤ μ0, iii ∑2n k 1 ck sin kθ > 0 if 0 < μ ≤ 1/2. Here μ0 0.691556 · · · is the unique root in 0, 1 of ∫3π/2 0 cos t t1−μ dt 0. 1.6 2. Main Results For our purpose, it will be more expedient to allow the terms of the sequence in Theorem 1.3 to consist of nonnegative numbers. Thus as a prelude to the main results, Theorem 1.3 is first appropriately adapted to yield the following two preliminary results. Lemma 2.1. Let {bk} be a decreasing sequence of nonnegative numbers satisfying b0 > 0 and kb2k ≤ k μ − 1 b2k−1, k ≥ 1, μ ∈ 0, 1 . For any positive integer n and 0 < θ < π , then n ∑ k 0 bk cos kθ > 0 iff 0 < μ ≤ μ0. 2.1 Proof. Let the sequence {ck} be given by c2k c2k 1 μ k/k!. It is evident from Theorem 1.3 i that