Preservation of Mobius-Type α-Starlikeness of Order beta Under Bernardi Integral Operators

Author(s)	Dr. Saibah Siregar, Ms. Arni Sabita Lina Mat Jusoh, Dr. Mohd Nazran Mohamed Pauzi
Country	Malaysia
Abstract	The purpose of this paper is to establish a corrected and explicit preservation theorem for the Bernardi integral operator, A central point is that the source class must be formulated in the same Mobius geometry as the target class; replacing it by the ordinary starlike class of order δ does not follow from the boundary argument. We introduce the sharp boundary-admissibility constant. The proof uses an auxiliary Schwarz function, Jack’s lemma, and an exact boundary-modulus estimate. The result contains several classical preservation statements as special cases and is also connected with Briot–Bouquet differential subordination. A generalized Bernardi-type operator is treated by an exact boundary infimum, avoiding an unjustified claim about an extremal boundary point.
Keywords	analytic functions; starlike functions; Mobius subordination; α-starlikeness; Bernardi operator; differential subordination; Jack’s lemma.
Field	Mathematics
Published In	Volume 8, Issue 5, September-October 2026
Published On	2026-10-10
DOI	https://doi.org/10.36948/ijfmr.2026.v08i05.89311

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