On the Jensen's inequality for convex functions on the coordinates in a rectangle from the plane
type: Item
instance of: scholarly article
title: On the Jensen's inequality for convex functions on the coordinates in a rectangle from the plane
zbMATH Open document ID: 1114.26019
author: N. E. Zubov
published in: Taiwanese Journal of Mathematics
publication date: 2007-01-29T00:00:00Z
review text: Several inequalities of Jensen's type for functions convex on the coordinates are considered and investigated. A function \(f : [a, b]\times [c, d]\to\mathbb R\) is called convex on the coordinates if the mapping \(f_y : [a, b] \to\mathbb R\), defined as \(f_y (u) := f (u, y)\), and \(f_x : [c, d] \to\mathbb R\), defined as \(f_x (v) := f (x, v)\), are convex for all \(y\in [c, d]\) and all \(x\in [a, b]\). The following inequalities were proved by \textit{S. S. Dragomir} [Taiwanese J. Math. 5, No. 4, 775--788 (2001; Zbl 1002.26017)]: \[ \begin{aligned} f\left(\frac{a+b}2, \frac{c+d}2\right)&\leq \frac1{2(b-a)}\int_a^b f\left(x,\frac{c+d}2\right) dx+\frac1{2(d-c)}\int_c^d f\left(\frac{a+b}2,y\right) dy\\ &\leq \frac1{(b-a)(d-c)}\int_a^b\int_c^d f(x,y)dydx\tag{1}\\ &\leq \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}4. \end{aligned} \] The authors give a generalization of the inequalities (1) and few other results from \textit{S. S. Dragomir} (loc. cit.), a generalization of the inequalities (1) which involves weight functions and nonlinear transformations of the base intervals, and some results for functions whose second partial derivatives are convex on the coordinates. Using the results, some other interesting Jensen-type inequalities for convex functions on the coordinates, and two functions which are closely connected with the integral Jensen's inequality are introduced and investigated.
reviewed by: Choonkil Park
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26A51
zbMATH DE Number: 5119522
zbMATH Keywords: Hadamard's inequality
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500557302
OpenAlex ID: W1794654463
DOI: 10.11650/TWJM/1500557302
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description: scientific article; zbMATH DE number 5119522