EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Lemma 1. (see [1]) (i) A sufficient condition for a function q of the form (1) to be in the subclass SP(ϑ, ζ) is ∞ X ϵ=2 (2ϵ −ζ −cos ϑ) |βϵ| ≤cos ϑ −ζ (|ϑ| < π/2 ; 0 ≤ζ < 1), (11) and a necessary and sufficient condition for a function q of the form (2) to be in the subclass SPE(ϑ, ζ) is that the condition (11) is satisfied. In particular, when ζ = 0, we obtain a sufficient condition for a function q of the form (1) to be in the subclass SP(ϑ) is ∞ X ϵ=2

Lemma 2. [10] If q of the form (1) and q ∈Gτ(C1, C2), then |βϵ| ≤(C1 −C2) |τ| ϵ , ϵ ∈N −{1}. (15) The result is sharp for the function q(z) given by q(z) = zZ 0  1 + (C1 −C2)τtϵ−1 1 + C2tϵ−1  dt

Theorem 1. If k ∈N, then Υik (z) is in the subclass SPE(ϑ, ζ) if and only if 2(6 −ζ −cos ϑ) ln 2 ≤k (cos ϑ −ζ) . (22)

Theorem 2. If k ∈N, then Υik (z) is in the subclass CSPE(ϑ, ζ) if and only if 2(22 −3ζ −3 cos ϑ) ln 2 ≤8 + k(cos ϑ −ζ). (24)

Theorem 3. Let k ∈N. If q ∈Gτ(C1, C2), then Iik(z) is in the subclass SPE(ϑ, ζ) if (C1 −C2)|τ|(4 −ζ −cos ϑ) ln 2 ≤k (cos ϑ −ζ) . (27)

Theorem 4. Let k ∈N. If q ∈Gτ(C1, C2), then Iik(z) is in the subclass CSPE(ϑ, ζ) if 2(C1 −C2)|τ| (6 −ζ −cos ϑ) ln 2 ≤k (cos ϑ −ζ) . (28)

Theorem 5. Let k ∈N. The integral operator Lik(z) is in the subclass SPE(ϑ, ζ) if and only if the inequality (4 −ζ −cos ϑ) ln 2 ≤k(cos ϑ −ζ) (30) holds.

Theorem 6. Let k ∈N. The integral operator Lik(z) is in the subclass CSPE(ϑ, ζ) if and only if the inequality (22) holds.

Definition 1. A function q of the form (1) is said to be in the subclass SP(ϑ, ζ), if it satisfies the following condition: R  e−iϑ zq′(z) q(z)  ≥

Definition 2. [10] A function h ∈E is said to be in the class Gτ(C1, C2), τ ∈C\{0}, −1 ≤C2 < C1 ≤1, if it satisfies the condition