On the Jensen's inequality for convex functions on the coordinates in a rectangle from the plane
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.
scientific article; zbMATH DE number 5119522