Open Access

On analytic functions related with q-generalized bounded Mocanu variation


Let 𝒜 be the class of normalized functions which are analytic in the open unit disc. Jakson q-derivative represented by convolution operator Dqf(z)=1z{ f(z)z(1-qz)(1z) },f𝒜,q(0,1) {D_q}f\left( z \right) = {1 \over z}\left\{ {f\left( z \right)\star{z \over {\left( {1 - qz} \right)\left( {1z} \right)}}} \right\},\,\,\,\,f \in \mathcal{A},\,\,\,q \in \left( {0,1} \right) is used to introduce a unified class Mq(α, β, γ), α ≥ 0, β, γ ∈ [0, 1) and its various mapping properties are studied. For q → 1−1, β = γ = 0 and α = 1, this class reduces to the class K of close-to-convex univalent functions. A number of interesting results such as q-Bernardi integral operator and inclusion relations are included as part of this study. Applications are also pointed out as consequences of the main results.