On strongly convex sets and strongly convex functions

2000 ◽  
Vol 100 (6) ◽  
pp. 2633-2681 ◽  
Author(s):  
E. S. Polovinkin
Keyword(s):  
Convex Functions ◽  
Convex Sets ◽  
Strongly Convex Functions ◽  
Strongly Convex

scholarly journals On strongly E-convex sets and strongly E-convex cone sets

2019 ◽  
Vol 11 (1) ◽  
Author(s):  
Saba Naser Majeed
Keyword(s):  
Convex Hull ◽  
Convex Functions ◽  
Convex Cone ◽  
Convex Sets ◽  
Important Class ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Generalized Classes

              -convex sets and -convex functions, which are considered as an important class of generalized convex sets and convex functions, have been introduced and studied by Youness [5] and other researchers. This class has recently extended, by Youness, to strongly -convex sets and strongly -convex functions. In these generalized classes, the definitions of the classical convex sets and convex functions are relaxed and introduced with respect to a mapping . In this paper, new properties of strongly -convex sets are presented. We define strongly -convex hull, strongly -convex cone, and strongly -convex cone hull and we proof some of their properties.  Some examples to illustrate the aforementioned concepts and to clarify the relationships between them are established.

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.

Bounds having Riemann type quantum integrals via strongly convex functions

2017 ◽  
Vol 54 (2) ◽  
pp. 221-240 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Gabriela Cristescu ◽  
Muhammad Uzair Awan
Keyword(s):  
Convex Functions ◽  
Strongly Convex Functions ◽  
Strongly Convex
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