Properties and integral inequalities of Hadamard Simpson type for the generalized \((s,m)\)preinvex functions
Authors
TingSong Du
 College of Science, China Three Gorges University, 443002, Yichang, P. R. China.
 Hubei Province Key Laboratory of System Science in Metallurgical Process, Wuhan University of Science and Technology, 430081, Wuhan, P. R. China.
JiaGen Liao
 College of Science, China Three Gorges University, 443002, Yichang, P. R. China.
YuJiao Li
 College of Science, China Three Gorges University, 443002, Yichang, P. R. China.
Abstract
The authors introduce the concepts of minvex set, generalized \((s,m)\)preinvex function, and explicitly
\((s,m)\)preinvex function, provide some properties for the newly introduced functions, and establish new
HadamardSimpson type integral inequalities for a function of which the power of the absolute of the first
derivative is generalized \((s,m)\)preinvex function. By taking different values of the parameters, Hadamardtype
and Simpsontype integral inequalities can be deduced. Furthermore, inequalities obtained in special
case present a refinement and improvement of previously known results.
Keywords
 Integral inequalities of HadamardSimpson type
 Hölder's inequality
 \((s،m)\)preinvex function.
MSC
References

[1]
T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 60 (2005), 14731484.

[2]
A. Barani, A. G. Ghazanfari, S. S. Dragomir, HermiteHadamard inequality for functions whose derivatives absolute values are preinvex , J. Inequal. Appl., 2012 (2012), 9 pages

[3]
F. Chen, S. Wu, Several complementary inequalities to inequalities of HermiteHadamard type for sconvex functions, J. Nonlinear Sci. Appl., 9 (2016), 705716.

[4]
S. S. Dragomir, R. P. Agarwal , Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 9195.

[5]
S. S. Dragomir, R. P. Agarwal, P. Cerone, On Simpson's inequality and applications, J. Inequal. Appl., 5 (2000), 533579.

[6]
N. Eftekhari, Some remarks on (s;m)convexity in the second sense , J. Math. Inequal., 8 (2014), 489495.

[7]
E. K. Godunova, V. I. Levin, Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, Numer. Math. Math. Phys., 166 (1985), 138142.

[8]
I. Iscan , HermiteHadamard's inequalities for preinvex function via fractional integrals and related functional inequalities, American J. Math. Anal., 1 (2013), 3338.

[9]
I. Iscan, S. Wu, HermiteHadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237244.

[10]
U. S. Kirmaci , Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004), 137146.

[11]
M. A. Latif, S. S. Dragomir, Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications, J. Inequal. Appl., 2013 (2013), 19 pages.

[12]
M. A. Latif, M. Shoaib, HermiteHadamard type integral inequalities for differentiable mpreinvex and (\(\alpha,m\)) preinvex functions , J. Egyptian Math. Soc., 23 (2015), 236241.

[13]
J. Y. Li, On Hadamardtype inequalities for spreinvex functions, J. Chongqing Norm. Univ. (Natural Science) China, 27 (2010), 58.

[14]
Y. J. Li, T. S. Du, On Simpson type inequalities for functions whose derivatives are extended (s;m)GAconvex functions, Pure. Appl. Math. China, 31 (2015), 487497.

[15]
Y. J. Li, T. S. Du, Some Simpson type integral inequalities for functions whose third derivatives are (\(\alpha,m\))GA convex functions, J. Egyptian Math. Soc., 24 (2016), 175180.

[16]
T. Y. Li, G. H. Hu, On the strictly Gpreinvex function, J. Inequal. Appl., 2014 (2014), 9 pages.

[17]
M. Mat loka, On some Hadamardtype inequalities for (h1; h2)preinvex functions on the coordinates, J. Inequal. Appl., 227 (2013), 12 pages.

[18]
M. Mat loka, Inequalities for hpreinvex functions, Appl. Math. Comput., 234 (2014), 5257.

[19]
M. A. Noor, Hadamard integral inequalities for product of two preinvex functions, Nonlinear Anal. Forum, 14 (2009), 167173.

[20]
J. Park , Simpsonlike and HermiteHadamardlike type integral inequalities for twice differentiable preinvex functions, Inter. J. Pure. Appl. Math., 79 (2012), 623640.

[21]
J. Park, HermiteHadamardlike type integral inequalities for functions whose derivatives of nth order are preinvex, Appl. Math. Sci., 7 (2013), 66376650.

[22]
R. Pini, Invexity and generalized convexity, Optimization, 22 (1991), 513525.

[23]
S. Qaisar, C. J. He, S. Hussain, A generalizations of Simpson's type inequality for dierentiable functions using (\(\alpha,m\))convex functions and applications, J. Inequal. Appl., 2013 (2013), 13 pages.

[24]
M. H. Qu, W. J. Liu, J. Park, Some new HermiteHadamardtype inequalities for geometricarithmetically s convex functions, WSEAS Trans. Math., 13 (2014), 452461.

[25]
M. Z. Sarikaya, N. Alp, H. Bozkurt, On HermiteHadamard type integral inequalities for preinvex and logpreinvex functions, Contemporary Anal. Appl. Math., 1 (2013), 237252.

[26]
M. Z. Sarikaya, E. Set, M. E. Ozdemir, On new inequalities of Simpson's type for sconvex functions, Comput. Math. Appl., 60 (2010), 21912199.

[27]
S. H. Wang, X. M. Liu, HermiteHadamard type inequalities for operator spreinvex functions , J. Nonlinear Sci. Appl., 8 (2015), 10701081.

[28]
Y. Wang, B. Y. Xi, F. Qi, HermiteHadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex, Matematiche (Catania), 69 (2014), 8996.

[29]
S. Wu, On the weighted generalization of the HermiteHadamard inequality and its applications, Rocky Mountain J. Math., 39 (2009), 17411749.

[30]
S. Wu, L. Debnath, Inequalities for convex sequences and their applications, Comput. Math. Appl., 54 (2007), 525534.

[31]
S. H. Wu, B. Sroysang, J. S. Xie, Y. M. Chu, Parametrized inequality of HermiteHadamard type for functions whose third derivative absolute values are quasiconvex, SpringerPlus, 2015 (2015), 9 pages.

[32]
Z. Q. Yang, Y. J. Li, T. S. Du, A generalization of Simpson type inequality via differentiable functions using (s;m)convex functions, Italian J. Pure. Appl. Math., 35 (2015), 327338.

[33]
X. M. Yang, X. Q. Yang, K. L. Teo, Generalized invexity and generalized invariant monotonicity, J. Optim. Theory Appl., 117 (2003), 607625.