In this paper, subclasses of the function class ∑ analytic and bi-univalent functions associated with operator L_q^(k, λ) are introduced defined in open unit disk △ by applying quasi-subordination. We obtain some results about corresponding bound estimations coefficients a_(2 ) a_(3 ).

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Extension of two known facts concerning subordination is made. The first fact is that, in subordination of 1-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on R,...

It has been shown by Voiculescu and Biane that the analytic subordination property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of ‘freeness with subordination’, or ‘s-free independence’, and ‘orthogonal independence’, introduced recently in th...