Bol. Soc. Paran. Mat.

Lemma 2.1. [18] Let p ∈P and of the form (2.1). Then |p2 −νp2 1| ≤      −4ν + 2, ν < 0 2, 0 ≤ν ≤1, 4ν −2, ν > 1.

Lemma 2.2. [18] Let p ∈P be of the form (2.1), then for any complex number ν, |p2 −νp2 1| ≤2 max(1, |2ν −1|). (2.2) This result is sharp for the functions p(z) = 1 + z2 1 −z2 , p(z) = 1 + z 1 −z .

Lemma 2.3. ([16], [17, p. 254]) Let the function p ∈P be given by the power series (2.1). Then 2p2 = p2 1 + x(4 −p2 1) (2.3) and 4p3 = p3 1 + 2(4 −p2 1)p1x −(4 −p2 1)p1x2 + 2(4 −p2 1)(1 −|x|2)z (2.4) for some complex numbers x, z satisfying |x| ≤1 and |z| ≤1. 3. Main Results

Theorem 3.1. Let the function f given by (1.1) be in the class ML∗ λ. Then for real µ, we have |a3 −µa2 2| ≤        1−3λ2−2λ−2λµ−4µ 8(2+λ)(1+λ)2 ,

Theorem 3.4. Let the function f given by (1.1) be in the class ML∗ λ. Then, for a complex number µ, we have |a3 −µa2 2| ≤ 1 2(2 + λ) max  1,

Corollary 3.6. [27] If the function f, given by (1.1) belongs to the class SL∗, then |a3 −a2 2| ≤1 4. Further, putting λ = µ = 1 and λ = 1, µ = 0 in Theorem 3.4, we have the following results due to Sahoo and Patel [28].

Corollary 3.7. [28, Corollary 2.1] If the function f, given by (1.1) belongs to the class ¯R, then |a3 −a2 2| ≤1 6 and |a3| ≤1 6. (3.18) The estimates are sharp. Now, we determine the sharp upper bound to the second Hankel determinant |H2(1)| for the class ML∗ λ.

Theorem 3.8. Let f ∈A given by (1.1) be in the class ML∗ λ. Assume that its coefficients a2, a3 and a4 are given by (3.13), (3.14) and (3.15), with p1 > 0. Then |a2a4 −a2 3| ≤ 1 4(2 + λ)2 . (3.19) The estimate in (3.19) is sharp.

Theorem 3.10. Let the function f given by (1.1) be in the class ML∗ L. Then |a4| ≤ 1 2(3 + λ) (0 ≤λ ≤1). (3.26)

Lemma 2.2 in (3.15) assuming that (1 −4λ −3λ2) > 0, it follows that |a4| ≤ 1 16(3 + λ)  1 + 5λ2 8(1 + λ)(2 + λ)p3 + (1 −4λ −3λ2) 2(1 + λ)(2 + λ)(4 −p2)py +(4 −p2)py2 + 2(4 −p2)(1 −y2)  = T (p, y; λ) (say) . (3.27) Now we maximize the function T (p, y; λ) on the closed rectangle [0, 2] × [0, 1]. Suppose that the maximum of T occurs at the interior point of [0, 2] × [0, 1]. Dif- ferentiating (3.27) with respect to y, we obtain