Schwarzian Norm Estimates for Some Classes of Analytic Functions

Ali, Md Firoz; Pal, Sanjit

doi:10.1007/s00009-023-02496-x

Cited by 3 publications

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“…For B > 0 and any z 0 ∈ S c ∩ D with −1 < z 0 ≤ 0, we have 2 . Again, for B < 0 and any z 0 ∈ S c ∩ D with 0 ≤ z 0 < 1, we have 2 .…”

Section: 12)mentioning

confidence: 99%

“…For B > 0 and any z 0 ∈ S c ∩ D with −1 < z 0 ≤ 0, we have 2 . Again, for B < 0 and any z 0 ∈ S c ∩ D with 0 ≤ z 0 < 1, we have 2 . This shows that the second inequality in (2.4) is sharp in such cases.…”

Section: 12)mentioning

confidence: 99%

“…Recently, the present authors [2] considered two classes of functions G(β) with β > 0 and F(α) with − 1 2 ≤ α ≤ 0, consisting of functions in A that satisfy the relations Re 1 + zf (z)…”

Section: Introductionmentioning

confidence: 99%

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The Schwarzian norm estimates for Janowski convex functions

Ali,

Pal

2024

Proceedings of the Edinburgh Mathematical Society

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For $-1\leq B \lt A\leq 1$ , let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z| \lt 1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$ . In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class $\mathcal{C}(A,B)$ . The Dieudonné’s lemma which gives the exact region of variability for derivatives at a point of bounded functions, plays the key role in this study, and we also use this lemma to construct the extremal functions for the sharpness by a new method.

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“…For B > 0 and any z 0 ∈ S c ∩ D with −1 < z 0 ≤ 0, we have 2 . Again, for B < 0 and any z 0 ∈ S c ∩ D with 0 ≤ z 0 < 1, we have 2 .…”

Section: 12)mentioning

confidence: 99%

Section: 12)mentioning

confidence: 99%