g−Loewner chain associated with Muir-type extension operator

Abstract

In this paper, we will give two sufficient conditions that the Muir-type extension operator ${\mathrm{\Psi }}_{n,\frac{1}{p_2},\dots ,\frac{1}{p_n}}$ preserves the first element of a $g-$Loewner chain on the Reinhardt domain ${\mathrm{\Omega }}_{n,p_2,\dots ,p_n}$, where $g(\zeta )=\frac{1+\zeta }{1+(2\alpha -1)\zeta }$ or $g(\zeta )=\frac{1+\mathrm{c\zeta }}{1-\mathrm{c\zeta }}$. Using these conclusions, it can be easily proved that the Muir-type extension operator ${\mathrm{\Psi }}_{n,\frac{1}{p_2},\dots ,\frac{1}{p_n}}$ preserves starlikeness of order α and strong starlikeness.

Keywords:

• Loewner chain
• Muir-type extension operator
• starlike mappings of order α
• strongly starlike mappings

AMS Subject Classifications:

• 32H02
• 30C45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 12101185], [grant number 11701307], the Key Scientific Research Projects in Universities of Henan Province [grant number 22B110009], the Doctoral Foundation of Pingdingshan University [grant number PXY-BSQD-2015005], the Foster Foundation of Pingdingshan University (No.PXY-PYJJ2016007).