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Jakšić, Rozarija; Pečarić, Josip; Rodić Lipanović, Mirna

Reverses of the Jensen-type inequalities for signed measures. (English) Zbl 1474.26111

Abstr. Appl. Anal. 2014, Article ID 626359, 11 p. (2014).
Summary: In this paper we derive refinements of the Jensen type inequalities in the case of real Stieltjes measure \(d \lambda\), not necessarily positive, which are generalizations of Jensen’s inequality and its reverses for positive measures. Furthermore, we investigate the exponential and logarithmic convexity of the difference between the left-hand and the right-hand side of these inequalities and give several examples of the families of functions for which the obtained results can be applied. The outcome is a new class of Cauchy-type means.

MSC:

26D15 Inequalities for sums, series and integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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\textit{R. Jakšić} et al., Abstr. Appl. Anal. 2014, Article ID 626359, 11 p. (2014; Zbl 1474.26111)
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References:

[1] Mitrinović, D. S.; Pečarić, J. E.; Fink, A. M., Uniform treatment of Jensen type inequalities, Mathematical Reports, 16(66), 2, 183-205 (1993)
[2] Boas, R. P., The Jensen-Steffensen inequality, Publikacije Elektrotehničkog fakulteta Univerziteta u Beogradu, Serija Matematika, 302-319, 1-8 (1970) · Zbl 0213.34601
[3] Pečarić, J. E.; Proschan, F.; Tong, Y. L., Convex Functions, Partial Orderings, and Statistical Applications. Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering, 187 (1992), Boston, Mass, USA: Academic Press, Boston, Mass, USA · Zbl 0749.26004
[4] Dragomir, S. S., Reverses of the Jensen inequality in terms of first derivative and applications, Acta Mathematica Vietnamica, 38, 3, 429-446 (2013) · Zbl 1280.26033
[5] Dragomir, S. S., Some reverses of the Jensen inequality with applications, Bulletin of the Australian Mathematical Society, 87, 2, 177-194 (2013) · Zbl 1275.26035
[6] Mitrinović, D. S.; J. E. Pečarić.; Fink, A. M., Classical and New Inequalities in Analysis (1993), Kluwer Academic Publishers · Zbl 0771.26009
[7] Gavrea, B.; Jakšetić, J.; Pečarić, J., On a global upper bound for Jenssen’s inequality, The ANZIAM Journal, 50, 2, 246-257 (2008) · Zbl 1182.26051
[8] Khan, A. R.; Pečarić, J.; Rodić Lipanović, M., n-exponential convexity for Jensen-type inequalities, Journal of Mathematical Inequalities, 7, 3, 313-335 (2013) · Zbl 1277.26039
[9] Klaričić Bakula, M.; Pečarić, J.; Perić, J., Extensions of the Hermite-Hadamard inequality with applications, Mathematical Inequalities and Applications, 15, 899-921 (2012) · Zbl 1253.26040
[10] Jakšetić, J.; Pečarić, J., Exponential convexity method, Journal of Convex Analysis, 20, 1, 181-197 (2013) · Zbl 1275.26038
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