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The sharp bound of the third Hankel determinant for functions of bounded turning

Bogumiła Kowalczyk · 2021 📄 DOI →

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Abstract

We find the sharp bound for the third Hankel determinant H3;1ðfÞ :¼ a1 a2 a3 a2 a3 a4 a3 a4 a5   for analytic functions f with an :¼ f ðnÞð0Þ=n!; n 2 N; a1 :¼ 1; such that Re f 0ðzÞ [ 0; z 2 D :¼ fz 2 C : jzj\1g:

Results & Lemmas (2)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

Lemma 1 Lemma 1 If p 2 P is of the form (4) with c1  0; then c1 ¼ 2f1; ð5Þ c2 ¼ 2f2 1 þ 2ð1  f2 1Þf2; ð6Þ c3 ¼ 2f3 1 þ 4ð1  f2 1Þf1f2  2ð1  f2…
Lemma 1 If p 2 P is of the form (4) with c1  0; then c1 ¼ 2f1; ð5Þ c2 ¼ 2f2 1 þ 2ð1  f2 1Þf2; ð6Þ c3 ¼ 2f3 1 þ 4ð1  f2 1Þf1f2  2ð1  f2 1Þf1f2 2 þ 2ð1  f2 1Þð1  jf2j2Þf3 ð7Þ
Theorem 1 Theorem 1 maxfjH3;1ðfÞj: f 2 P0g ¼ 1 4 ð9Þ with extreme function f0 2 P0 given by f 0 0ðzÞ:¼ 1  z3 1 þ z3; z 2 D: ð10Þ
Theorem 1 maxfjH3;1ðfÞj : f 2 P0g ¼ 1 4 ð9Þ with extreme function f0 2 P0 given by f 0 0ðzÞ :¼ 1  z3 1 þ z3 ; z 2 D: ð10Þ

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