Criteria for univalence, starlikeness and convexity

Theorem 1.1. If f(z) = z + P∞ n=2 anzn ∈U(λ), then f ∈S∗for 0 < λ ≤λ∗, where λ∗= −a + √ 2 −a2 2 with a = |f ′′(0)|/2. This result was originally stated as a conjecture in [3] and was proved in [4]. In this article, we discuss the relationship between U(λ) and S∗(δ), as well as between P(2λ) and K(δ). As a consequence, we improve certain coefficient results due to Reade, Silverman and Todorov [7]. We now state our first result which gives a condition for functions in U(λ) to be starlike of order δ(λ

Theorem 1.2. If f ∈U(λ) and a = |f ′′(0)|/2 ≤1, then f ∈S∗(δ) whenever 0 < λ ≤λ(δ), where λ(δ) =        p (1 −2δ)(2 −a2 −2δ) −a(1 −2δ) 2(1 −δ) if 0 ≤δ < 1 + a 3 + a,

Corollary 1.3. If f ∈U(λ) with f ′′(0) = 0, then f ∈S∗(δ) whenever 0 < λ ≤λ(δ) =          s 1 −2δ 2(1 −δ) if 0 ≤δ ≤1/3,

Theorem 1.5. Let f ∈A with f ′′(0) = 0 and suppose that (1.2)

Theorem 1.6. A function of the form (1.4) is in K if any one of the following conditions holds: (i) there exist p, q > 0 with 1/p + 1/q ≤1 such that (2p + 1)|b1| + max n ∞ X k=1 (2kp + 1)|bk|, ∞ X k=1 (k −1)(kq + 1)|bk| o ≤1. (ii)

Theorem 1.6(i) is due to [7] while Theorem 1.6(ii) has been obtained recently by Obradovi´c et al. [4]. Our next result improves Theorem 1.6.

Theorem 1.7. Let 0 < λ ≤1/ √ 2. If f ∈A is of the form (1.4) and satisfies the coefficient condition (1.5) ∞ X k=2 (k −1)k|bk| ≤2λ, then f ∈K(β), where β = β(λ) is defined by (1.3). A comparison of the λ-values of Theorems 1.7 and 1.6 shows that The- orem 1.7 improves Theorem 1.6. Indeed, for the case b1 = 0, it suffices to note that (7 − √ 33)/8 < 3 −2

Theorem 1.8. Let 0 ≤µ ≤1, 0 < λ ≤1 and f ∈A. (i) If f satisfies (1.6) 1 + 1 µ + 1 zf ′′(z) f ′(z) −zf ′(z) f(z) < log(1 + λ) µ + 1 , z ∈∆, then

Theorem 1.8 improves the result of Obradovi´c and Tuneski [5]. In particular, for each 0 < λ ≤1 and f ∈A, one has 1 + zf ′′(z) f ′(z) −zf ′(z) f(z) < log(1 + λ) ⇒

Theorem 1.5, we note that Lemma 2.1 may be used to state a more general result.

Lemma 2.1. Let 0 < λ < 1, α > −2 and let g ∈H, the class of all analytic functions in the unit disc ∆, satisfy the condition g(z) ≺1 + λz for z ∈∆with g(0) = 1. Suppose that Re φ(z) ≥δ in ∆. If p ∈H, p(0) = 1 and (2.1) |g(z)(β + (1 −β)p(z) + 1 −α −2φ(z)) −α| < λ(α + 2), z ∈∆,

Lemma 2.2. Let φ(z) = 1+P∞ n=1 bnzn be a nonvanishing analytic func- tion in ∆and f(z) = z/φ(z) and 0 < λ ≤1. If any one of the following coefficient conditions holds: (i) ∞ X n=2 (n −1)|bn| ≤λ, (ii) ∞ X n=2 n(n −1)|bn| ≤2λ, then f ∈U(λ).