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)
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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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