scholarly journals Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 348
Author(s):  
Nak Cho ◽  
Virendra Kumar ◽  
V. Ravichandran
Keyword(s):  
Convex Functions ◽  
Univalent Functions ◽  
Higher Order ◽  
Sharp Bounds ◽  
Schwarzian Derivatives

Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction.

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.

Sharp coefficient bounds for starlike functions associated with the Bell numbers

2019 ◽  
Vol 69 (5) ◽  
pp. 1053-1064 ◽  
Author(s):  
Virendra Kumar ◽  
Nak Eun Cho ◽  
V. Ravichandran ◽  
H. M. Srivastava
Keyword(s):  
Starlike Functions ◽  
Higher Order ◽  
Positive Real Part ◽  
Bell Numbers ◽  
Positive Real ◽  
Sharp Bounds ◽  
Hankel Determinants ◽  
Coefficient Bounds ◽  
The Family ◽  
Schwarzian Derivatives

Abstract Let $\begin{array}{} \mathcal{S}^*_B \end{array}$ be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Bounds on the first three consecutive higher-order Schwarzian derivatives for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ are investigated.

scholarly journals Estimate for Schwarzian derivative of certain close-to-convex functions

2021 ◽  
Vol 6 (10) ◽  
pp. 10778-10788
Author(s):  
Zhenyong Hu ◽  
◽  
Xiaoyuan Wang ◽  
Jinhua Fan ◽  
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Schwarzian Derivative ◽  
Higher Order ◽  
Schwarzian Derivatives ◽  
Derivatives Of

<abstract><p>Let $ f(z) $ be analytic in the unit disk with $ f(0) = f'(0)-1 = 0 $. For the following close-to-convex subclasses: $ \Re \{(1-z)f'(z)\} &gt; 0, $ $ \Re \{(1-z^{2})f'(z)\} &gt; 0, $ $ \Re \{(1-z+z^{2})f'(z)\} &gt; 0 $ and $ \Re \{(1-z)^{2}f'(z)\} &gt; 0 $, we investigate the bounds for the first two consecutive derivatives of higher order Schwarzian derivatives of $ f(z) $.</p></abstract>

scholarly journals NORM ESTIMATES OF THE PRE-SCHWARZIAN DERIVATIVES FOR CERTAIN CLASSES OF UNIVALENT FUNCTIONS

2006 ◽  
Vol 49 (1) ◽  
pp. 131-143 ◽  
Author(s):  
Yong Chan Kim ◽  
Toshiyuki Sugawa
Keyword(s):  
Hardy Spaces ◽  
Convex Functions ◽  
Maximal Operator ◽  
Univalent Functions ◽  
Independent Interest ◽  
Norm Estimates ◽  
Norm Estimate ◽  
Schwarzian Derivatives ◽  
Derivatives Of

AbstractA sharp norm estimate will be given to the pre-Schwarzian derivatives of close-to-convex functions of specified type. In order to show the sharpness, we introduce a kind of maximal operator which may be of independent interest. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.

scholarly journals Pointwise estimates for higher order convexity preserving polynomial approximation

1994 ◽  
Vol 36 (2) ◽  
pp. 213-233 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Polynomial Approximation ◽  
Convex Functions ◽  
Higher Order ◽  
Second Order ◽  
Convolution Type ◽  
Shape Preservation ◽  
Moduli Of Continuity ◽  
Pointwise Estimates ◽  
Higher Order Convexity ◽  
Boolean Sum

AbstractDeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

scholarly journals Simpson type inequalities and applications

2021 ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Michael Th. Rassias ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Quadrature Formula ◽  
Convex Functions ◽  
Integral Identity ◽  
Higher Order ◽  
Auxiliary Result ◽  
First Order ◽  
New Class ◽  
Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

scholarly journals Higher order strongly general convex functions and variational inequalities

2020 ◽  
Vol 5 (4) ◽  
pp. 3646-3663 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Convex Functions ◽  
Higher Order ◽  
General Convex
Sign in / Sign up

Export Citation Format

Share Document