We consider a multiplier transformation and some subclasses of the class of meromorphic functions which was defined by means of the Hadamard product by N.E. Cho, O.S. Kwon and H.M. Srivastava in [N.E. Cho, O.S. Kwon, H.M. Srivastava, Inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations,

We define in this paper new distance generalizations of the Pell numbers and the companion Pell numbers. We give a graph interpretation of these numbers with respect to a special 3-edge colouring of the graph.

In [RUSCHEWEYH, S.-SHEIL-SMALL, T.: Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135] the authors proved the P`olya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely K ∗ K = K. They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class K. In this paper we consider similar convolution problems for some classes of functions. Especially we give a new applications of a result [SOKÓŁ, J.: Convolution and subordination in the convex hull of convex mappings, Appl. Math. Lett. 19 (2006), 303-306] on the subordinating relations in the convex hull of convex mappings under convolution. The paper deals with several ideas and techniques used in geometric function theory. Besides being an application to those results it provides interesting corollaries concerning special functions, regions and curves.

We consider the convolution operator d ζ f (z) = 1 z f (z) * z (1 − ζ z)(1 − z) on the class of analytic functions f (z) = z + a 2 z 2 + • • • , |z| < 1, in the complex plane, where ζ is complex, |ζ | ≤ 1. For ζ = 1, the operator becomes the derivative f (z), while for real ζ = q, 0 < q < 1 we obtain the Jackson's q-derivative d q f (z).

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