Some extended Simpson-type inequalities and applications

Document Type : Research Paper


1 Department of Business Administration‎, ‎Aletheia‎ ‎University‎, ‎Tamsui‎, ‎New Taipei City 25103‎, ‎Taiwan.

2 China University of Science and Technology‎, ‎Nankang‎, ‎Taipei 11522‎, ‎Taiwan‎.

3 Department of Applied Mathematics‎, ‎Aletheia‎ ‎University‎, ‎Tamsui‎, ‎New Taipei City 25103‎, ‎Taiwan.


‎In this paper‎, ‎we shall establish some extended Simpson-type inequalities‎ ‎for differentiable convex functions and differentiable concave functions‎ ‎which are connected with Hermite-Hadamard inequality‎. ‎Some error estimates‎ ‎for the midpoint‎, ‎trapezoidal and Simpson formula are also given‎.


Main Subjects

M‎. ‎Alomari and M‎. ‎Darus‎, ‎On the Hadamard's inequality for‎ ‎log-convex functions on the  coordinates‎, J‎. ‎Inequal‎. ‎Appl. 2009‎ ‎(2009)‎, ‎Article ID 283147‎, ‎17 pages‎.
S.S‎. ‎Dragomir‎, ‎Two mappings in connection to Hadamard's‎ ‎inequalities‎, J‎. ‎Math‎. ‎Anal‎. ‎Appl. 167 (1992)‎, ‎no‎. ‎1‎, ‎49--56‎.
S.S‎. ‎Dragomir‎, ‎On the Hadamard's inequality for convex functions on the‎ ‎co-ordinates in a rectangle from the plane‎, Taiwanese J‎. ‎Math. 5‎ ‎(2001)‎, ‎no‎. ‎4‎, ‎775--788‎.
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. textbf11‎ ‎(1998)‎, ‎no‎. ‎5‎, ‎91--95‎.
‎S.S‎. ‎Dragomir‎, ‎R.P‎. ‎Agarwal and P‎. ‎Cerone‎, ‎On Simpson's‎ ‎inequality and applications‎, J‎. ‎Inequal‎. ‎Appl. 5 (2000)‎, ‎no‎. ‎6‎, ‎533--579‎.
S.S‎. ‎Dragomir‎, ‎Y.J‎. ‎Cho and S.S‎. ‎Kim‎, ‎Inequalities of‎ ‎Hadamard's type for Lipschitzian mappings and their applications‎, ‎J‎. ‎Math‎. ‎Anal‎. ‎Appl. 245 (2000)‎, ‎no‎. ‎2‎, ‎489--501‎.
‎L‎. ‎Fejér‎, ‎Über die Fourierreihen‎, ‎II‎, Math‎. ‎Naturwiss‎. Anz Ungar‎. ‎Akad‎. ‎Wiss. 24 (1906) 369--390‎.
J‎. ‎Hadamard‎, ‎étude sur les propriétés des fonctions‎ ‎entiéres en particulier d'une fonction considérée par Riemann‎, J‎. ‎Math‎. ‎Pures Appl. 58 (1893) 171--215‎.
U.S‎. ‎Kirmaci‎, ‎Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula‎, Appl‎. ‎Math‎. ‎Comput. 147 (2004)‎, ‎no‎. ‎1‎, ‎137--146‎.
U.S‎. ‎Kirmaci and M.E‎. ‎Ozdemir‎, ‎On some inequalities for‎ ‎differentiable mappings and applications to special means of real numbers‎ ‎and to midpoint formula‎, Appl‎. ‎Math‎. ‎Comput. 153 (2004)‎, ‎no‎. ‎2‎, ‎361--368‎.
C.E.M‎. ‎Pearce and J‎. ‎Pečarić‎, ‎Inequalities for differentiable mappings with application to special means and quadrature formulæ‎, Appl‎. ‎Math‎. ‎Lett. 13 (2000))‎, ‎no‎. ‎2‎, ‎51--55‎.
K.L‎. ‎Tseng‎, ‎S.R‎. ‎Hwang‎, ‎and S.S‎. ‎Dragomir‎, ‎Fejér-Type‎ ‎Inequalities (I)‎, J‎. ‎Inequal‎. ‎Appl. 2010‎ ‎(2010)‎, ‎Article ID 531976‎, ‎7 pages‎.
K.L‎. ‎Tseng‎, ‎G.S‎. ‎Yang and K.C‎. ‎Hsu‎, ‎On some inequalities of‎ ‎Hadamard's type and applications‎, Taiwanese J‎. ‎Math. 13‎ ‎(2009)‎, ‎no‎. ‎6‎, ‎1929--1948‎.
‎G.S‎. ‎Yang and K.L‎. ‎Tseng‎, ‎On certain integral inequalities‎ ‎related to Hermite-Hadamard inequalities‎, J‎. ‎Math‎. ‎Anal‎. ‎Appl. 239‎ ‎(1999)‎, ‎no‎. ‎1‎, ‎180--187‎.
G.S‎. ‎Yang and K.L‎. ‎Tseng‎, ‎Inequalities of Hadamard's type‎ ‎for Lipschitzian mappings‎, J‎. ‎Math‎. ‎Anal‎. ‎Appl. 260 (2001)‎, ‎no‎. ‎1‎, ‎230--238‎.
G.S‎. ‎Yang and K.L‎. ‎Tseng‎, ‎Inequalities of Hermite-Hadamard-Fejér type for convex functions and Lipschitzian functions‎, Taiwanese J‎. ‎Math. 7 (2003)‎, ‎no‎. ‎3‎, ‎433--440‎.