Notes on Convex Functions of Order (Formula presented.)

Toshiyuki Sugawa, Li Mei Wang

Research output: Contribution to journalArticlepeer-review

Abstract

Marx and Strohhäcker showed in 1933 that f(z) / z is subordinate to (Formula presented.) for a normalized convex function f on the unit disk (Formula presented.) In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved that f(z) / z is subordinate to (Formula presented.) if f is convex of order (Formula presented.) for (Formula presented.) and conjectured that this is true also for (Formula presented.) Here, (Formula presented.) is the standard extremal function in the class of normalized convex functions of order (Formula presented.) and (Formula presented.) We prove the conjecture and study geometric properties of convex functions of order (Formula presented.) In particular, we prove that (Formula presented.) is starlike whenever both f and g are convex of order 3 / 5.

Original languageEnglish
Pages (from-to)79-92
Number of pages14
JournalComputational Methods and Function Theory
Volume16
Issue number1
DOIs
Publication statusPublished - 2016 Mar 1

Keywords

  • Convex functions of order (Formula presented.)
  • Hypergeometric function
  • Subordination

ASJC Scopus subject areas

  • Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics

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