The great number of (integral, but also some discrete) inequalities for concave functions (and some applications) in this paper presents such an ”embarras de richesse” that we reluctantly confine ourselves to just one: Let \(a\leq b\), \(\lambda\in] 0,1[\), \[ A=\lambda a+(1-\lambda)b,\qquad B=(b- a)\min(\lambda,1-\lambda), \] \(f:[a,b]\to\mathbb{R}\) concave and \(g:[a,b]\to[0,\infty[\) integrable, \(g(A-x)=g(A+x)\) for all \(x\in[0,B]\). Then, for all \(y\in[0,B]\), \[ (\lambda f(a)+(1-\lambda)f(b)) \int^{A+y}_{A-y}g(t)dt\leq \int^{A+y}_{A-y}f(t)g(t)dt\leq f(A)\int^{A+y}_{A-y}g(t)dt. \] (For \(\lambda=1/2\), \(y=(b-a)/2\) this result is due to L. Fejér [Math.-Natur. Anz. Ungar. Akad. Wiss. 24, 269-390 (1906)], who applied it to determine the sign of Fourier coefficients.)
The authors give several counterexamples showing that their inequalities do not hold for non-concave functions. They state also open problems. The list of references contains 42 items, each of which is indeed quoted in the body of the paper. This reviewer was particularly impressed by the author’s familiarity with two papers which have been published only in Hungarian.
{Note: The authors’ definition of concavity implies continuity on the interior, so their ”continuous and concave” functions could be described as ”concave on the interval and continuous at its boundaries”.}.