Unified approach to starlike and convex functions involving convolution between analytic functions. (English) Zbl 1388.30014
Summary: Using the idea of convolution between analytic functions, we define a class \(\mathcal {UM}(g,\gamma ,b,k)\) of analytic functions comprising of starlike and convex functions. These functions map the open unit disc on to the conic domains. We derive some sufficient conditions and then use them to define the class \(\mathcal {UM}^\ast(g,\gamma ,b,k)\). Making use of an increasing factor sequence, we discuss a subordination result. We may relate our findings with the previously known results.
MSC:
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |
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