Coefficient Inequalities for Starlikeness and Convexity

Theorem 1.1 ([7, Theorem 3.3, p. 334]). If the function f(z) = z + P∞ n=2 anzn satisfies the inequality P∞ n=2 n(n −1)|an| ≤1/(k + 2), (0 ≤k < ∞), then f ∈k −UCV. The bound 1/(k + 2) cannot be replaced by a larger number. Goodman [3, Theorem 6] proved the result in the case k = 1 for functions to be uniformly convex. Theorem 1.1 in the special case k = 0 shows that the corresponding constant is 1/2 for functions f to be convex. A function f ∈A is parabolic starlike of order α [1] if

Theorem 1.2 (Ali [1, Theorem 3.1, p. 564]). If the function f(z) = z +P∞ n=2 anzn satisfies the inequality P∞ n=2(n −1)|an| ≤(1 −α)/(2 −α), then the function f is parabolic starlike of order α. The bound (1 −α)/(2 −α) cannot be replaced by a larger number. Motivated by Theorems 1.1 and 1.2, the range of β is determined for the analytic function f(z) = z + P∞ n=2 anzn satisfying the inequality P∞ n=2 n(n −1)|an| ≤β to be either starlike or convex of order α. Similar problems were investigated for

Theorem 1.3 ([8, Theorem 2, p. 961], and [10, Theorem 1 and Corollary, p. 110]). 1. If the function f(z) = z + P∞ n=2 anzn satisfies the inequality ∞ X n=2 (n −α)|an| ≤1 −α, (3) then f ∈S∗ α. If an ≤0, then the condition (3) is a necessary condition for f ∈S∗(α). 2. Similarly, if the function f satisfies the inequality ∞ X n=2 n(n −α)|an| ≤1 −α,

Theorem 2.1. Let α ∈[0, 1). If the function f ∈A given by (1) satisfies the inequality ∞ X n=2 n(n −1)|an| ≤β < 1, (5) then the following holds. (1) The function f belongs to the class Cα for β ≤(1−α)/(2−α), and the bound (1−α)/(2−α) cannot be replaced by a larger number. (2) The function f belongs to the class S∗ α for β ≤2(1 −α)/(2 −α), and the bound 2(1 − α)/(2 −α) cannot be replaced by a larger number.

Corollary 2.2. [7, Theorem 3.3, p. 334] If f ∈A given by (1) satisfies the inequality ∞ X n=2 n(n −1)|an| ≤ 1 k + 2, then f ∈k −UCV. Further, the bound 1/(k + 2) cannot be replaced by a larger number.

Corollary 2.3. Let α ∈[0, 1). If f ∈A is given by (1) and ∞ X n=2 (n −1)|an| ≤1 −α 2 −α, then f ∈S∗ α. Further, the bound (1 −α)/(2 −α) cannot be replaced by a larger number.

Theorem 2.5. Let α ∈[0, 1) and f ∈A be given by (1). (1) If the inequality P∞ n=2 n|an| ≤1 −α holds, then f ∈S∗ α. (2) If the inequality P∞ n=2 n2|an| ≤1 −α holds, then f ∈Cα. (3) If the inequality P∞ n=2 n2|an| ≤4(1 −α)/(2 −α) holds, then f ∈S∗ α and the bound 4(1 −α)/(2 −α) is sharp.

Lemma 3.1. [6, cf. Theorem 6, p. 412] Let β < 1, and α ∈R. If f ∈A satisfies the inequality ∞ X n=2 αn2 + (1 −α)n −β
|an| ≤1 −β, (9) then f ∈R(α, β). It should be remarked that Lemma 3.1 reduces to Theorem 1.3(1) in the special case α = 0. The following theorem provides sufficient coefficient conditions for functions to belong to either R(α, β) ∩S∗η or R(α, β) ∩Cη.

Theorem 3.2. Let β < 1 and α > 0. If the function f ∈A satisfies the inequality (9), then the following holds. (1) The function f is in the class S∗η for η ≤(2α+β)/(2α+1) and the bound (2α+β)/(2α+1) is sharp. (2) The function f is in the class Cη for η ≤(α −1 + β)/α, β > 0.

Theorem 3.3. Let β < 1, α ∈R and f ∈A. (1) If the function f satisfies the inequality P∞ n=2 n(n −1)|an| ≤2(1 −β)/(2α + 2 −β), then f ∈R(α, β). The bound 2(1 −β)/(2α + 2 −β) is sharp. (2) Let α ≤1 and η ∈R be defined by η = ( 4(1 −β)/(3α + 1), α + β > 1, 4(1 −β)/(2α + 2 −β), α + β ≤1. If the function f satisfies the inequality P∞ n=2 n2|an| ≤η, then f ∈R(α, β). When α + β ≤1, the result is sharp.

Lemma 3.1 then shows that f ∈R(α, β). Now, let α + β < 1 and the function f satisfy P∞ n=2 n2|an| ≤4(1 −β)/(2α + 2 −β). In this case, the inequality 4 αn2 + (1 −α)n −β
≤n2(2α + 2 −β) (n ≥2) shows that ∞ X n=2 αn2 + (1 −α)n −β

Theorem 4.1. Let α ∈[0, 1). If f ∈T is given by (10), then f ∈T S∗(α) ⇐⇒f ∈T S∗ α ⇐⇒ ∞ X n=2 (n −α)an ≤1 −α, and f ∈T C(α) ⇐⇒f ∈T Cα ⇐⇒ ∞ X n=2 n(n −α)an ≤1 −α. For functions with negative coefficients, the next theorem proves the equivalence of the inequalities P∞

Theorem 4.2. Let β > 0. If the function f ∈T is given by (10), then |f ′′(z)| ≤β ⇐⇒ ∞ X n=2 n(n −1)an ≤β. 7

Theorem 4.4. If the function f ∈T C(α), 0 ≤α < 1, then the following holds: (1) The inequality P∞ n=2 nan ≤(1 −α)/(2 −α) holds and the bound (1 −α)/(2 −α) is sharp. (2) The inequality P∞ n=2 n(n −1)an ≤1 −α holds. (3) The inequality P∞ n=2(n −1)an ≤(1 −α)/2(2 −α) holds and the bound (1 −α)/2(2 −α) is sharp. (4) The inequality P∞ n=2 n2an ≤2(1 −α)/(2 −α) holds and the bound 2(1 −α)/(2 −α) is sharp.

Corollary 4.5. If the function f ∈T S∗(α), 0 ≤α < 1, then the following holds. (1) The inequality P∞ n=2 an ≤(1 −α)/(2 −α) holds and the bound (1 −α)/(2 −α) is sharp. (2) The inequality P∞ n=2(n −1)an ≤1 −α holds. (3) The inequality P∞ n=2 nan ≤2(1−α)/(2−α) holds and the bound 2(1−α)/(2−α) is sharp. In the remaining part of this section, the properties of functions with negative coefficients belonging to the class R(α, β) are investigated. The class of all functions with negative coeffi- cients belo

Lemma 4.6. [6, Theorem 8, p.414] If β < 1, α ∈R . If f ∈T , then f ∈T R(α, β) ⇐⇒ ∞ X n=2 αn2 + (1 −α)n −β
an ≤1 −β.

Corollary 4.7. If f ∈T R(α, β) with β < 1, α > 0, then the following holds: (1) The function f ∈T S∗η for η ≤(2α + β)/(2α + 1) and the bound (2α + β)/(2α + 1) is sharp. 8

Theorem 4.8. Let 0 ≤β < 1, and α > 0. If η ≥(2α+3β−2)/(2α+β), then T C(η) ⊆T R(α, β).

Theorem 4.9. Let β < 1, and α ∈R. If f ∈T R(α, β), then (1) P∞ n=2 n(n −1)an ≤(1 −β)/α when α > 0. (2) P∞ n=2(n −1)an ≤η where η = ( (1 −β)/(1 −α), β < 3α + 1, 0 ≤α < 1 (1 −β)/(2α + 2 −β), β ≥3α + 1, 0 ≤α. The result for β > 3α + 1 is sharp. (3) For 0 ≤α ≤1, P∞

Theorem 5.1. Let a, b ∈C and c ∈R satisfy either F(|a|, |b|; c; 1)  (|a|)2(|b|)2 (c −|a| −|b| −2)2 + 2|ab| c −|a| −|b| −1  ≤ 2(1 −β) 2α + 2 −β , for c > |a| + |b| + 2, α ≥0, β < 1, or F(|a|, |b|; c; 1)