TY - CHAP

T1 - Exponential taylor domination

AU - Friedland, Omer

AU - Goldman, Gil

AU - Yomdin, Yosef

N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020

Y1 - 2020

N2 - Let (Equation presented) be an analytic function in a disk DR of radius R > 0, and assume that f is p-valent in DR, i.e. it takes each value c∈C at most p times in DR. We consider its Borel transform (Equation presented) which is an entire function, and show that, for any R > 1, the valency of the Borel transform B(f) in DR is bounded in terms of p, R. We give examples, showing that our bounds, provide a reasonable envelope for the expected behavior of the valency of B(f). These examples also suggest some natural questions, whose expected answer will strongly sharper our estimates. We present a short overview of some basic results on multi-valent functions, in connection with “Taylor domination”, which, for (Equation presented), is a bound of all its Taylor coefficients ak through the first few of them. Taylor domination is our main technical tool, so we also discuss shortly some recent results in this direction.

AB - Let (Equation presented) be an analytic function in a disk DR of radius R > 0, and assume that f is p-valent in DR, i.e. it takes each value c∈C at most p times in DR. We consider its Borel transform (Equation presented) which is an entire function, and show that, for any R > 1, the valency of the Borel transform B(f) in DR is bounded in terms of p, R. We give examples, showing that our bounds, provide a reasonable envelope for the expected behavior of the valency of B(f). These examples also suggest some natural questions, whose expected answer will strongly sharper our estimates. We present a short overview of some basic results on multi-valent functions, in connection with “Taylor domination”, which, for (Equation presented), is a bound of all its Taylor coefficients ak through the first few of them. Taylor domination is our main technical tool, so we also discuss shortly some recent results in this direction.

UR - http://www.scopus.com/inward/record.url?scp=85091410603&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-44819-6_13

DO - 10.1007/978-3-030-44819-6_13

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AN - SCOPUS:85091410603

T3 - Operator Theory: Advances and Applications

SP - 377

EP - 386

BT - Operator Theory

PB - Springer Science and Business Media Deutschland GmbH

ER -