On the Fekete-Szegö problem associated with Libera type close-to-convex functions

Theorem 1. For 0 ≤β < 1, let f (z) given by (1.1) belongs to the function class S∗(β). Then for any real number µ, a3 −µa2 2 ≤(1 −β) max {1, |3 −2β −4µ (1 −β)|} .

Theorem 2. If f ∈C and if µ is real, then a3 −µa2 2 ≤           

Theorem 3. Let f ∈C(α, β) and be given by (1.1). Then for 0 ≤α, β < 1, 3 a3 −µa2 2 ≤          

Lemma 1. [11] Let p ∈P with p (z) = 1 + c1z + c2z2 + · · · . Then |cn| ≤2 (n ≥1) .

Lemma 2. [17] Let p ∈P with p (z) = 1 + c1z + c2z2 + · · · . Then for any complex number ν c2 −νc2 1 ≤2 max {1, |2ν −1|} , and the result is sharp for the functions given by p (z) = 1 + z2 1 −z2 and p (z) = 1 + z 1 −z.

Lemma 3. [18] Let the function g defined by g(z) = z + ∞ X k=2 bkzk (z ∈U) (2.1) belongs to the function class S∗(β) (0 ≤β < 1). Then we have |b2| ≤2 (1 −β) and |b3| ≤(1 −β) (3 −2β) .

Lemma 4. Let the function f given by (1.1) belongs to the function class Kλ,δ (α, β) . Then (1 + λ −δ + 2λδ) |a2| ≤2 −α −β (2.2) and 3 (1 + 2λ −2δ + 6λδ) |a3| ≤(3 −2α −β) (3 −2β) . (2.3)

Theorem 4. Let f (z) given by (1.1) belongs to the function class Kλ,δ (α, β) . Then, for any complex number µ, 3 (1 + 2λ −2δ + 6λδ) a3 −µa2 2

Corollary 1. Let f (z) given by (1.1) belongs to the function class Uβ λ,α. Then, for any complex number µ, 3 (1 + 2λ) a3 −µa2 2

Corollary 2. Let f (z) given by (1.1) belongs to the function class Uβ λ . Then, for any complex number µ, 3 (1 + 2λ) a3 −µa2 2

Corollary 3. Let f (z) given by (1.1) belongs to the function class C(α, β). Then, for any complex number µ, 3 a3 −µa2 2

Theorem 5. Let f (z) given by (1.1) belongs to the function class Kλ,δ (α, β) . Then 3 (1 + 2λ −2δ + 6λδ) a3 −µa2 2 ≤          

Theorem 3.1] with a different way.

Corollary 4. In Theorem 5, letting δ = 0; δ = 0, α = 0; δ = λ = 0; or δ = λ = 0, α = β = 0, we have [3, Theorem 1], [2, Theorem 3.1], Theorem 3 and Theorem 2, respectively. References [1] H.R. Abdel-Gawad and D.K. Thomas, The Fekete-Szeg¨o problem for strongly close-to-convex functions, Proc. Amer. Math. Soc. 114 (1992), 345–349. [2] M.H. Al-Abbadi and M. Darus, The Fekete-Szeg¨o theorem for a certain class of analytic functions, Sains Malays. 40 (4) (2011), 385–389. [3] M.H. Al-Abbadi and M. Da