Coefficient bounds for certain subclasses of starlike functions

Lemma 1.1 ([10, 11, Libera and Zlotkiewicz]) If p ∈P has the form given by (2) with p1 ≥0, then 2p2 = p2 1 + x  4 – p2 1  (3) and 4p3 = p3 1 + 2p1  4 – p2 1

Lemma 1.2 ([21, Ravichandran and Verma]) Let α, β, γ and a satisfy the inequalities 0 < α < 1, 0 < a < 1 and 8a(1 – a)  (αβ – 2γ )2 +  α(a + α) – β 2 + α(1 – α)(β – 2aα)2 ≤4aα2(1 – α)2(1 – a). If p ∈P has the form given by (2), then γ p4 1 + ap2 2 + 2αp1p3 – (3/2)βp2 1p2 – p4  ≤2.

Lemma 1.3 ([17, Prokhorov and Szynal]) If w ∈B, then for any real numbers μ and ν the following sharp estimate Ψ (w) ≤Φ(μ,ν) holds: Φ(μ,ν) = ⎧ ⎨ ⎩ |ν|, if (μ,ν) ∈A, 2 3(|μ| + 1)( |μ|+1 3(|μ|+1+ν))1/2, if (μ,ν) ∈B.

Lemma 1.4 ([14, Ohno and Sugawa]) For any real numbers a, b and c, let the quantity Y(a,b,c) be given by Y(a,b,c) = max z∈D a + bz + cz2 + 1 – |z|2 , where D := {z ∈C : |z| ≤1}. If ac ≥0, then Y(a,b,c) = ⎧ ⎨ ⎩ |a| + |b| + |c|, if |b| ≥2(1 – |c|), 1 + |a| + b2

Theorem 2.1 Let f ∈S∗ q with the form given by (1). Then the following inequalities hold:

Lemma 1.1 again, we get p1 = 0 and p2 = 2 and p3 = 0. Thus, we get the function p ∈P defined by p(z) = (1 + z2)/(1 – z2) and the corresponding function for which equality holds in the result is f3, given by (11). □ The function (6) suggests the following conjecture. Conjecture 2.2 Let f ∈S∗ q. Then |a5| ≤13/54. 3 Coefficient bounds for the class S∗ lλ In this section, the work of Sarfraz and Malik [22] has been generalized for the class S∗ lλ. In addition to that a sharp upper bound for |a5| is al

Theorem 3.1 Let f ∈S∗ lλ, λ ∈(0,1] with the form given by (1). Then the following inequal- ities hold: (1) |a2| ≤λ/2, |a3| ≤λ/4, |a4| ≤λ/6, |a5| ≤λ/8; and for any complex number μ a3 – μa2 2  ≤λ 4 max  1; |4μ – 1| 4

Theorem 3.3 Let f ∈S∗ lλ. Then ∞  k=2  k2 – λ – 1  |ak|2 ≤1.