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On Hadamard’s inequality on a disk. (English) Zbl 0948.26012

The following Hadamard-type inequalities for a convex function of two variables is offered: Let \(D= D(C,r)\) be a disc centered at the point \(C\) and having radius \(r>0\) in the plane. If \(f:D\to \mathbb{R}\) is convex on \(D\) then \[ f(C)\leq{1\over \pi r^2} \iint_D f(x,y) dx dy\leq {1\over 2\pi r} \int_L f(\gamma) d\ell(\gamma), \] where \(L\) is the circle centered at \(C\) with radius \(r\). Some mappings connected with this inequality and related results are also obtained.

MSC:

26D15 Inequalities for sums, series and integrals
26B25 Convexity of real functions of several variables, generalizations
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