A Problem of Martin Concerning Strongly Convex Metrics on E 3

1974 ◽  
Vol 43 (2) ◽  
pp. 461
Author(s):  
E. D. Tymchatyn ◽  
B. O. Friberg
Keyword(s):  
Strongly Convex ◽  
Convex Metrics

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

2021 ◽  
Author(s):  
Young Jae Sim ◽  
Adam Lecko ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Invariance Property ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Second Hankel Determinant

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

A Continuous Extension Operator for Convex Metrics

2010 ◽  
Vol 53 (4) ◽  
pp. 719-729
Author(s):  
I. Stasyuk ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Extension ◽  
Extension Operator ◽  
Uniform Topology ◽  
Simultaneous Extension ◽  
Peano Continuum ◽  
Convex Metrics

AbstractWe consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.

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