Coefficient Inequalities for Classes of Univalent Functions Defined by q − Derivatives

where ≺ denotes the usual subordination ( see [7], [3], [2]). For different choices of q, b, λ, in (1.3), the class H q,b(κ), generalizes many classes studied earlier, for example ( see Seoudy and Aouf [10], [11], Ravichandran et al. [9], Ali et al. [1] with p = 1 ∗ The third-named author was supported by the Basic Science Research Program through the National Research Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2019R1I1A3A01050861). 2010 Mathematics Subject Classification: 30C45. Submitted June 09, 2018. Published November 07, 2018

Denote by P the class of analytic functions φ of positive real part on U with φ(0) = 1, R{φ(z)} > 0.
where ≺ denotes the usual subordination ( see [7], [3], [2]). For different choices of q, b, λ, in (1.3), the class H λ q,b (κ), generalizes many classes studied earlier, for example ( see Seoudy and Aouf [10], [11], Ravichandran et al. [9], Ali et al. [1] The following known lemma is needed to establish our results. Lemma 1.1 [6]. If p(z) = 1 + r 1 z + r 2 z 2 + ... ∈ P and δ is a complex number, then The result is sharp for the functions given by Also, we note that or one of its rotations. If γ = 1, the equality holds if and only if p is the reciprocal of one of the functions such that equality holds in the case of ξ = 0.
Also the above upper bound is sharp, and it can be improved as follows when 0 < ξ < 1:

Main results
We assume in the reminder of this paper that f ∈ A, κ ∈ P, 0 < q < 1, 0 ≤ λ ≤ 1 and b ∈ C * . .

(2.7)
Our result now follows by an application of (1.4). The result is sharp for the functions The proof of Theorem 1 is completed. (2.11) The result is sharp.