## Inequalities for the Riemann-Stieltjes integral of under the chord functions with applications.(English)Zbl 1310.26019

Summary: We say that the function $$f:[a,b] \to \mathbb{R}$$ is under the chord if
$\frac{( b-t) f(a) +( t-a) f(b)}{b-a}\geq f(t)$
for any $$t\in [a,b]$$.
In this paper we prove amongst other that
$\int _a^b u(t) df(t) \geq \frac{f(b) -f(a)}{b-a}\int _a^bu(t) dt$
provided that $$u: [ a,b] \to \mathbb{R}$$ is monotonic nondecreasing and $$f: [a,b] \to \mathbb{R}$$ is continuous and under the chord.
Some particular cases for the weighted integrals in connection with the Fejér inequalities are provided. Applications for continuous functions of selfadjoint operators on Hilbert spaces are also given.

### MSC:

 26D15 Inequalities for sums, series and integrals 47A63 Linear operator inequalities
Full Text:

### References:

 [1] Azpeitia, A. G.: Convex functions and the Hadamard inequality. Rev. Colombiana Mat. 28, 1 (1994), 7-12. · Zbl 0832.26015 [2] Beckenbach, E. F., Bellman, R.: Inequalities. 4th Edition, Springer-Verlag, Berlin, 1983. · Zbl 0513.26003 [3] Cerone, P., Dragomir, S. S., Roumeliotis, J., Šunde, J.: A new generalization of the trapezoid formula for $$n$$-time differentiable mappings and applications. Demonstratio Math. 33, 4 (2000), 719-736. · Zbl 0974.26010 [4] Dragomir, S. S.: A mapping in connection to Hadamard’s inequalities. An. Öster. Akad. Wiss. Math.-Natur. (Wien) 128 (1991), 17-20, MR 934:26032. ZBL No. 747:26015. · Zbl 0747.26015 [5] Dragomir, S. S.: Two mappings in connection to Hadamard’s inequalities. J. Math. Anal. Appl. 167 (1992), 49-56, MR:934:26038, ZBL No. 758:26014. · Zbl 0758.26014 [6] Dragomir, S. S.: On Hadamard’s inequalities for convex functions. Mat. Balkanica 6 (1992), 215-222, MR: 934:26033. · Zbl 0834.26010 [7] Dragomir, S. S.: An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3, 3 (2002), Article 35, 1-8. · Zbl 0995.26009 [8] Dragomir, S. S.: Inequalities of Grüss type for the Stieltjes integral and applications. Kragujevac J. Math. 26 (2004), 89-122. · Zbl 1274.26035 [9] Dragomir, S. S.: Bounds for the normalized Jensen functional. Bull. Austral. Math. Soc. 74, 3 (2006), 471-476. · Zbl 1113.26021 [10] Dragomir, S. S.: Inequalities for Stieltjes integrals with convex integrators and applications. Appl. Math. Lett. 20 (2007), 123-130. · Zbl 1116.26004 [11] Dragomir, S. S., Gomm, I.: Some applications of Fejér’s inequality for convex functions (I). Austral. J. Math. Anal. Appl. 10, 1 (2013), Article 9, 1-11. · Zbl 1264.26025 [12] Dragomir, S. S., Milošević, D. S., Sándor, J.: On some refinements of Hadamard’s inequalities and applications. Univ. Belgrad, Publ. Elek. Fak. Sci. Math. 4 (1993), 21-24. · Zbl 0813.26005 [13] Dragomir, S. S., Pearce, C. E. M.: Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA Monographs, Victoria University, 2000, [online] [14] Dragomir, S. S., Pearce, C. E. M.: Some inequalities relating to upper and lower bounds for the Riemann-Stieltjes integral. J. Math. Inequal. 3, 4 (2009), 607-616. · Zbl 1182.26047 [15] Féjer, L.: Über die Fourierreihen, II. Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369-390) [16] Guessab, A., Schmeisser, G.: Sharp integral inequalities of the Hermite-Hadamard type. J. Approx. Theory 115, 2 (2002), 260-288. · Zbl 1012.26013 [17] Helmberg, G.: Introduction to Spectral Theory in Hilbert Space. John Wiley & Sons, Inc., New York, 1969. · Zbl 0177.42401 [18] Kikianty, E., Dragomir, S. S.: Hermite-Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space. Math. Inequal. Appl. 13, 1 (2010), 1-32. · Zbl 1183.26025 [19] Merkle, M.: Remarks on Ostrowski’s and Hadamard’s inequality. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 113-117. · Zbl 0946.26016 [20] Mercer, P. R.: Hadamard’s inequality and trapezoid rules for the Riemann-Stieltjes integral. J. Math. Anal. Applic. 344 (2008), 921-926. · Zbl 1147.26013 [21] Mitrinović, D. S., Lacković, I. B.: Hermite and convexity. Aequationes Math. 28 (1985), 229-232. · Zbl 0572.26004 [22] Pearce, C. E. M., Rubinov, A. M.: P-functions, quasi-convex functions, and Hadamard type inequalities. J. Math. Anal. Appl. 240, 1 (1999), 92-104. · Zbl 0939.26009 [23] Pečarić, J., Vukelić, A.: Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions. Functional equations, inequalities and applications, Kluwer Acad. Publ., Dordrecht, 2003, 105-137. · Zbl 1067.26021 [24] Pečarić, J., Proschan, F., Tong, Y. L.: Convex Functions, Partial Orderings, and Statistical Applications. Academic Press Inc., San Diego, 1992. · Zbl 0749.26004 [25] Toader, G.: Superadditivity and Hermite-Hadamard’s inequalities. Studia Univ. Babeş-Bolyai Math. 39, 2 (1994), 27-32. · Zbl 0868.26012 [26] Yand, G.-S., Hong, M.-C.: A note on Hadamard’s inequality. Tamkang J. Math. 28, 1 (1997), 33-37. [27] Yand, G.-S., Tseng, K.-L.: On certain integral inequalities related to Hermite-Hadamard inequalities. J. Math. Anal. Appl. 239, 1 (1999), 180-187. · Zbl 0939.26010
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