higher order convexity
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Author(s):  
Tamanna Yadav ◽  
S. K. Gupta

In the article, a semi-infinite fractional optimization model having multiple objectives is first formulated. Due to the presence of support functions in each numerator and denominator with constraints, the model so constructed is also non-smooth. Further, three different types of dual models viz Mond -Weir, Wolfe  and  Schaible  are presented and then usual duality results are proved using higher-order [[EQUATION]]   convexity assumptions. To show the existence of such generalized convex  functions, a nontrivial example has also been exemplified. Moreover, numerical examples have been  illustrated at suitable places to justify various results presented in the paper. The formulation and duality results discussed also generalize the well known results appeared in the literature.


2019 ◽  
Vol 93 (6) ◽  
pp. 1159-1176
Author(s):  
Mario Krnić ◽  
Rozarija Mikić ◽  
Josip Pečarić

2017 ◽  
Vol 26 (2) ◽  
pp. 137-143
Author(s):  
DAN BARBOSU

The goal of the paper is to present some results concerning the approximation of convex functions by linear positive operators. First, one recalls some results concerning the univariate real valued convex functions. Next, one presents the notion of higher order convexity introduced by Popoviciu [Popoviciu, T., Sur quelques propri´et´ees des fonctions d’une ou deux variable r´eelles, PhD Thesis, La Faculte des Sciences de Paris, 1933 (June)] . The Popoviciu’s famous theorem for the representation of linear functionals associated to convex functions of m−th order (with the proof of author) is also presented. Finally, applications of the convexity to study the monotonicity of sequences of some linear positive operators and also mean value theorems for the remainder term of some approximation formulas based on linear positive operators are presented.


Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska

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.


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