A subclass of \(\alpha \)-convex functions with respect to \((2j,k)\)-symmetric conjugate points. (English) Zbl 1409.30019
Summary: The theory of \((j,k)\)-symmetric functions has many applications in the investigation of fixed points, estimation of the absolute values of some integrals and in obtaining results of the type of Cartan’s uniqueness theorem. The concept of \((2j,k)\)-symmetric functions extends the idea of even, odd, \(k\), \(2k\), \((j,k) \)-symmetric and conjugate functions. In this paper, we introduce a new class \(\mathcal {M}_{\text{SCP}}^{j,k}(\alpha,\eta,\delta) \) of analytic functions using the notion of \((2j,k)\)-symmetric conjugate points. It unifies the classes \(\mathcal {S}_{\text{SCP}}^{j,k}(\eta,\delta) \) and \(\mathcal {C}_{\text{SCP}}^{j,k}(\eta,\delta) \) of starlike functions with respect to symmetric conjugate points and convex functions with respect to symmetric conjugate points, respectively. We also derive some inclusion results, integral representations and convolution conditions for functions belonging to the
general function class \(\mathcal {M}_{\text{SCP} }^{j,k}(\alpha,\eta,\delta) \). The various results presented in this paper may apply to yield the corresponding (new or known) results for a number of simpler known classes.
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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