Integral majorization type inequalities for the functions in the sense of strong convexity. (English) Zbl 1429.26029
The authors establish several integral majorization estimates which generalize J. Favard’s inequalities [Bull. Sci. Math., II. Ser. 57, 54–64 (1933; Zbl 0007.06104)] for strongly convex functions.
One of their main results is the following: Let \(c > 0\), \(\Psi: [0,\infty)\rightarrow \mathbb R\) be a continuous strongly convex function with modulus \(c\), and let \(f\), \(g\), and \(\Omega\) be three positive and integrable functions, defined on \([a, b]\), such that \(\int_a^xf(r)\Omega(r)\,dr\le\int_a^xg(r)\Omega(r)\,dr\), for all \(x\in[a,b]\) and \(\int_a^bf(r)\Omega(r)\,dr=\int_a^bg(r)\Omega(r)\,dr\). Then, if \(f\) is decreasing, \[ \int_a^b\Psi(g(r))\Omega(r)\,dr\ge\int_a^b\Psi(f(r))\Omega(r)\,dr+ c\int_a^b(f(r)-g(r))^2\Omega(r)\,dr. \] Similar results hold assuming \(g\) to be an increasing function.
One of their main results is the following: Let \(c > 0\), \(\Psi: [0,\infty)\rightarrow \mathbb R\) be a continuous strongly convex function with modulus \(c\), and let \(f\), \(g\), and \(\Omega\) be three positive and integrable functions, defined on \([a, b]\), such that \(\int_a^xf(r)\Omega(r)\,dr\le\int_a^xg(r)\Omega(r)\,dr\), for all \(x\in[a,b]\) and \(\int_a^bf(r)\Omega(r)\,dr=\int_a^bg(r)\Omega(r)\,dr\). Then, if \(f\) is decreasing, \[ \int_a^b\Psi(g(r))\Omega(r)\,dr\ge\int_a^b\Psi(f(r))\Omega(r)\,dr+ c\int_a^b(f(r)-g(r))^2\Omega(r)\,dr. \] Similar results hold assuming \(g\) to be an increasing function.
Reviewer: Javier Soria (Madrid)
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 26D15 | Inequalities for sums, series and integrals |
Citations:
Zbl 0007.06104References:
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