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Hadamard type inequality for quasiconvex functions in higher dimensions. (English) Zbl 1010.26017

As was established in [S. S. Dragomir and C. E. M. Pearce, Bull. Aust. Math. Soc. 57, No. 3, 373-385 (1998; Zbl 0908.26015)] , for a quasiconvex function \(f\) defined on \([a,b]\) we have \[ f\left( \frac{a+b}{2}\right) \leq \frac{2}{b-a}\int_{a}^{b}f(x) dx. \] This Hadamard type inequality is extended here for quasiconvex functions in higher dimensions.

MSC:

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

Citations:

Zbl 0908.26015
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References:

[1] Dragomir, S. S.; Pečarić, J. E.; Persson, L. E., Some inequalities of Hadamard type, Soochow J. Math., 21, 335-341 (1995) · Zbl 0834.26009
[2] Dragomir, S. S.; Pearce, C. E.M., Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc., 57, 377-385 (1998) · Zbl 0908.26015
[3] Rubinov, A. M., Abstract Convexity and Global Optimization (2000), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0963.49014
[4] Pearce, C. E.M.; Rubinov, A. M., \(P\)-functions, quasiconvex functions and Hadamard-type inequalities, J. Math. Anal. Appl., 240, 92-104 (1999) · Zbl 0939.26009
[5] Kutateladze, S. S.; Rubinov, A. M., Minkowski duality and its applications, Russian Math. Surveys, 27, 137-192 (1972)
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