Abstract
Using the principal of subordination and the qderivative, we obtain sharp bounds for some classes of univalent functions.
Highlights
We obtain the new class Hqλ,,θα(κ) for b α) cos θ, 0
Remark 2.1. (i) Putting λ = 0 in Theorem 1, we obtain the result of Seoudy and Aouf [10, Theorem 1]; (ii) Putting λ = 1 in Theorem 1, we obtain the result of Seoudy and Aouf [10, Theorem 2]; (iii) Theorem 1 for b = 1, corrects the result of Ramachandram et al [8, Theorem 2, α = 0, β = 1]
Remark 2.2 (i) Taking q → 1− and λ = α, in the above results, we obtain the results of [12, with λ = 0]; (ii) Theorem 2 for b = 1, corrects the result of Ramachandram et al [8, Theorem 1, α = 0, β = 1]; (iii) Putting λ = 0 in Theorem 2, we obtain the result of Seoudy and Aouf [10, Theorem 3]; (iv) Putting λ = 1 in Theorem 2, we obtain the result of Seoudy and Aouf [10, Theorem 3]; (v) Taking b = e−iθ(1 − α) cos θ in the above results, we obtain results for the class Hqλ,θα(κ)
Summary
We obtain the new class Hqλ,,θα(κ) for b α) cos θ, 0 The following known lemma is needed to establish our results. The result is sharp for the functions given by p(z) z2 z2 and p(z)
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