$(s,p)$Valent Functions
Abstract
We introduce the notion of $(\mathcal F,p)$valent functions. We concentrate in our investigation on the case, where $\mathcal F$ is the class of polynomials of degree at most $s$. These functions, which we call $(s,p)$valent functions, provide a natural generalization of $p$valent functions (see~\cite{Ha}). We provide a rather accurate characterizing of $(s,p)$valent functions in terms of their Taylor coefficients, through "Taylor domination", and through linear nonstationary recurrences with uniformly bounded coefficients. We prove a "distortion theorem" for such functions, comparing them with polynomials sharing their zeroes, and obtain an essentially sharp Remeztype inequality in the spirit of~\cite{Y3} for complex polynomials of one variable. Finally, based on these results, we present a Remeztype inequality for $(s,p)$valent functions.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 arXiv:
 arXiv:1503.00325
 Bibcode:
 2015arXiv150300325F
 Keywords:

 Mathematics  Classical Analysis and ODEs
 EPrint:
 arXiv admin note: text overlap with arXiv:1102.2580