Malaysian Journal of Mathematical Sciences, September 2026, Vol. 20, No. 3
Inclusion Results for Cardioid-Type Analytic Functions Based on the Multiplier Transformation
Bakare, N. G., Aijaz, A., Akinwunmi, S. A., and Ibrahim, G. R.
Corresponding Email: Aijaz.Bhatti@usindh.edu.pk
Received date: 4 August 2025
Accepted date: 4 January 2026
Abstract:
In this paper, we introduce new subclasses $\mathfrak{S}
_{\lambda ,\mu }^{\mathfrak{s}}(\gamma _{1};\phi _{c}) ,$ $%
\mathfrak{C}_{\lambda ,\mu }^{\mathfrak{s}}( \gamma _{1};\varphi
_{c}) \mathfrak{\ }$and $\mathfrak{K}_{\lambda ,\mu }^{\mathfrak{s}%
}( \gamma _{1},\gamma _{2};\varphi _{1},\varphi _{2}) \mathfrak{\
}$of normalized analytic functions defined on the open unit disk $\mathfrak{E%
}$, utilizing a generalized multiplier transformation $\mathfrak{I}_{\lambda
,\mu }^{\mathfrak{s}}$ that is based on the Hadamard product. These
subclasses are linked to Ma–Minda-type subordination, which involves
functions that map onto the cardioid domain. We investigate several
inclusion-type results concerning these subclasses under $\mathfrak{I}%
_{\lambda ,\mu }^{\mathfrak{s}}$ and examine inclusion results related to
the generalized Libera integral operator $\mathcal{F}_{\mathcal{C}}$.
Additionally, we emphasize several well-known subclasses of analytic
functions that arise as special cases when specific parameter values are
assigned.
Keywords: analytic functions; cardioid domain; Choi-Saigo-Srivastava fractional operator; multiplier transformation; integral operator.
