EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Lemma 1. Let K, L ∈R and p, q ∈C. If |p| < R and |q| < R, |(K + L)p + (K −L)q| ≤ ( 2|K|R, if |K| ≥|L| 2|L|R, if |K| ≤|L|

Theorem 1. Let the function f given by (1) be in the class F(n, α, β). Then |a2| ≤ 2αx p x(n!) p |(3α(n + 2)!(1 + 2β)x2 −4(1 + β)2(n + 1)(n + 1)!{(2 + 2α)x2 −1}| (8) and |a3| ≤ 4αx(n!) 3(1 + 2β)(n + 2)! + α2x2 (1 + β)2(n + 1)2 (9)

Corollary 1. Let the function f given by (1) be in the class F(n, 1, β). Then |a2| ≤ 2x p x(n!) p |(3(n + 2)!(1 + 2β)x2 −4(1 + β)2(n + 1)(n + 1)!(4x2 −1)| , and |a3| ≤ 4x(n!) 3(1 + 2β)(n + 2)! + x2 (1 + β)2(n + 1)2 . On the other hand, taking β = 1, we get the following corollary.

Corollary 2. Let the function f given by (1) be in the class F(n, α, 0). Then |a2| ≤ 2αx p x(n!) p |(9α(n + 2)!x2 −16(n + 1)(n + 1)!{(2 + 2α)x2 −1}| , and |a3| ≤4αx(n!) 9(n + 2)! + α2x2 4(n + 1)2 . 4. Fekete-Szeg¨o problem for the function class F(n, α, β)

Theorem 2. Let the function f given by (1) be in the class F(n, α, β). Then for some ζ ∈R, |a3 −ζa2 2| ≤ (4αx B , if |1 −ζ| ≤∆(α,n,β) 4Bα2x2 16α3x3|1−ζ| ∆(α,n,β) , if |1 −ζ| ≥∆(α,n,β) 4Bα2x2 , (21) where ∆(α, n, β) = 4α[B −4(1 + β)2(n + 1)2(α + 1)]x2 −8α(1 + β)2(n + 1)2,

Corollary 3. Let the function f given by (1) be in the class F(n, 1, β). Then for some ζ ∈R, |a3 −ζa2 2| ≤ (4x B , if |1 −ζ| ≤G 16x3|1−ζ| 4B(n+2)(n+1)x2−8(1+β)(n+1)2(4x2−1), if |1 −ζ| ≥G, (22) where G = 4B(n + 2)(n + 1)x2 −8(1 + β)(n + 1)2(4x2 −1) 2Bx2 .

Corollary 4. Let the function f given by (1) be in the class F(n, α, 0). Then for some ζ ∈R, |a3 −ζa2 2| ≤ ( 4(n!)αx 9(n+2)!, if |1 −ζ| ≤ (n!)H(n,α) 36(n+2)!α2x2 16α3x3|1−ζ| H(n,α) , if |1 −ζ| ≥ (n!)H(n,α) 36(n+2)!α2x2 ,