ON THE GENERALIZED INTEGRAL INEQUALITIES FOR TWICE DIFFERENTIABLE MAPPINGS

Yıl 2019, Cilt: 37 Sayı: 1, 187 - 205, 01.03.2019

Mehmet Zeki Sarıkaya Samet Erden Hüseyin Budak

Öz

In this paper, an important integral equality is derived. Then, we establish several new inequalities for some twice differentiable mappings that are connected with the celebrated Hermite-Hadamard type and Ostrowski type integral inequalities. Some of the new inequalities are obtained by using Grüss inequality and Chebyshev inequality. The results presented here would provide extensions of those given in earlier works

Anahtar Kelimeler

Hermite-Hadamard inequality, Ostrowski inequality, Grüss inequality, Čebyšev inequality, Hölder inequality.

Kaynakça

• [1] F. Ahmad, N. S. Barnett and S. S. Dragomir, New Weighted Ostrowski and Cebysev Type Inequalities, Nonlinear Analysis: Theory, Methods & Appl., 71 (12), (2009), 1408-1412.
• [2] F. Ahmad, A. Rafiq, N. A. Mir, Weighted Ostrowski type inequality for twice differentiable mappings, Global Journal of Research in Pure and Applied Math., 2 (2) (2006), 147-154.
• [3] M. W. Alomari, A companion of Dragomir's generalization of Ostrowski's inequality and applications in numerical integration, Ukrainskij Matematytchnyj Zhurnal, 2012, Vol. 64, No. 4.
• [4] N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27(1), (2001), 109-114.
• [5] P. S. Bullen, Error estimates for some elementary quadrature rules, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 602-633 (1978), 97-103 (1979).
• [6] P. Cerone, S.S. Dragomir and J. Roumeliotis, An inequality of Ostrowski type for mappings whose second derivatives are bounded and applications, RGMIA Research Report Collection, Vol. 1, No. 1, 1998 Art.4.
• [7] S. S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett., 11(5) (1998), 91-95.
• [8] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
• [9] S. S. Dragomir, P. Cerone and J. Roumeliotis, A new generalization of Ostrowski's integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means, RGMIA Research Report Collection, Vol. 2, No. 1, 1999.
• [10] S. S. Dragomir and A. Sofo, An integral inequality for twice differentiable mappings and applications, Tamk. J. Math., 31(4), 2000.
• [11] S. S. Dragomir, P. Cerone and A. Sofo, Some remarks on the midpoint rule in numerical integration, RGMIA Research Report Collection, Vol. 1, No. 2, 1998. Art. 4
• [12] T. S. Du, J. G. Liao, Y. J. Li, Properties and integral inequalities of Hadamard--Simpson type for the generalized -preinvex functions. J. Nonlinear Sci. Appl., 9, (2016), 3112-3126.
• [13] T. S. Du, Y. J. Li, Z. Q. Yang, A generalization of Simpson's inequality via differentiable mapping using extended -convex functions. Appl. Math. Comput., 293, (2017), 358-369.
• [14] G. Grüss, Über das maximum des absoluten Betrages von Math. Z., 39, 215-226, 1935.
• [15] S. Hussain, M.A.Latif and M. Alomari, Generalized double-integral Ostrowski type inequalities on time scales, Appl. Math. Letters, 24(2011), 1461-1467.
• [16] U. S. Kirmaci and R. Dikici, On some Hermite- Hadamard Type inequalities for twice differentiable mappings and applications, Tamkang Journal of Mathematics, vol. 44, Number 1, 41-51 (2013).
• [17] N. Minculete, P. Dicu and A. Ratiu, Two reverse inequalities of Bullen's inequality, General Math. 22(1), 69-73, 2014.
• [18] D. S. Mitrinović, Analytic Inequalities, Springer-Verlang New-York, Heidelberg, Berlin, 1970.
• [19] A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10(1938), 226-227.
• [20] A. Qayyum, A weighted Ostrowski-Grüss type inequality and applications, Proceeding of the World Cong. on Engineering, Vol:2, 2009, 1-9.
• [21] A. Rafiq and F. Ahmad, Another weighted Ostrowski-Grüss type inequality for twice differentiable mappings, Kragujevac Journal of Mathematics, 31 (2008), 43-51.
• [22] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. LXXIX, 1(2010), pp. 129-134.
• [23] M. Z. Sarikaya On the Ostrowski type integral inequality for double integrals, Demonstratio Mathematica, Vol. XLV, No 3, 2012, pp. 533-540.
• [24] M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The Arabian Journal for Science and Engineering (AJSE)-Mathematics, (2011) 36:1153-1160.
• [25] M. Z.Sarikaya and S. Erden, On the weighted integral inequalities for convex function, Acta Universitatis Sapientiae Mathematica, 6, 2 (2014) 194–208.
• [26] M. Z. Sarikaya and S. Erden, On The Hermite- Hadamard-Fejer Type Integral Inequality for Convex Function, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 3, 85-89.
• [27] M. Z. Sarikaya and H. Yildirim, Some new integral inequalities for twice differentiable convex mappings, Nonlinear Analysis Forum 17, pp. 1--14, 2012.
• [28] M. Z. Sarikaya and H. Yaldiz, Some inequalities associated with the for the probability density function, Submited, 2014.
• [29] C.-L. Wang, X.-H. Wang, On an extension of Hadamard inequality for convex functions, Chin. Ann. Math. 3 (1982) 567--570.
• [30] S. Wasowicz and A. Witkonski, On some inequality of Hermite-Hadamard type, Opuscula Math. 32(2), (2012), pp:591-600.
• [31] S.-H. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, The Rocky Mountain J. of Math., vol. 39, no. 5, pp. 1741--1749, 2009.
• [32] B-Y, Xi and F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex functions and applications, Hacet. J. Math. Stat., 42(3), 243--257 (2013).
• [33] B-Y, Xi and F. Qi, Hermite-Hadamard type inequalities for functions whose derivatives are of convexities, Nonlinear Funct. Anal. Appl.. 18(2), 163--176 (2013).