The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

Lemma 1  [5, 6] If p ∈P and is given by (5) with c1 ≥0 , then for some complex valued 휁 with |휁| ≤1 , and some complex valued 휂 with |휂| ≤1,

Lemma 2  [3] Let 𝔻∶= {z ∈ℂ∶|z| ≤1} , and for real numbers A, B, C, let If AC ≥0, then Hq,n(f) ∶= ||||||||| an an+1 ⋯ an+q−1 an+1 an+2 ⋯ an+q ⋮ ⋮ ⋮ ⋮ an+q−1 an+q ⋯an+2(q−1)

Theorem 1  Let 훽∈(0, 1] and 휆∈[1∕2, 1] . If f ∈FO(휆, 훽) , then Y(A, B, C) = ⎧ ⎪ ⎨ ⎪⎩ A + B + C, B ≥2(1 −C), 1 + A + B2 4(1 −C), B < 2(1 −C). (8) Y(A, B, C) = ⎧

Proposition 1  Let t ∈(2, 11] and u ∈(1, 3]. Define H ∶[0, 4] →ℝ by where p2(z) ∶= 1 −z2 1 −bz + z2 , z ∈픻, (25) b ∶= √ 휏=

Theorem 2  Let 훽∈(0, 1] and 휆∈[1∕2, 1] . If f ∈FO(휆, 훽) , then Both inequalities are sharp.

Theorem 3  If f ∈C훽 , then for 0 < 𝛽≤1, When 훽= 1 , we deduce the following [4].

Corollary 1  If f ∈C , then We note that the proof of the weaker result above used Lemma 1, which alone was not strong enough to give the sharp estimate, whereas the additional use of Lemma 2 produces the correct estimates given in Theorem 3.

Corollary 2  If f ∈FO(휆) with 휆= 1 18(8 + √ 73) , then Both inequalities are sharp. Acknowledgements  The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP; Ministry of Science, ICT & Future Planning) (No. NRF-2017R1C1B5076778). Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as

Definition 1  Let 0 < 𝛽≤1 and 1∕2 ≤휆≤1 . Then, f ∈A is called a strongly Ozaki close- to-convex if, and only if, for z ∈픻, We denote this class of functions by FO(휆, 훽) , noting that when 훽= 1 this reduces to (3), and when 휆= 1∕2 we obtain the class C훽 of strongly convex functions considered in [9]. Re {zf (z) f(z) } > 0, (2)