Some properties and inequalities for \(h\)-geometrically convex functions. (English) Zbl 1412.26024

Zhang, Bo; Xi, Bo-Yan; Qi, Feng

J. Class. Anal. 3, No. 2, 101-108 (2013).

Summary: In the paper, the definition of properties of \(h\)-geometrically convex functions are studied, and several integral inequality for the newly defined functions are established.

MSC:

26A51	Convexity of real functions in one variable, generalizations
26D15	Inequalities for sums, series and integrals
41A55	Approximate quadratures

Keywords:

property; \(h\)-geometrically convex function; inequality

References:

[1]	R.-F. BAI, F. QI,ANDB.-Y. XI, Hermite-Hadamard type inequalities for the m - and(α,m)logarithmically convex functions, Filomat 27 (2013), no. 1, 1-7, available online at http://dx.doi.org/10.2298/FIL1301001B. · Zbl 1340.26043
[2]	S.-P. BAI ANDF. QI, Some inequalities for(s1,m1)-(s2,m2)-convex functions on the co-ordinates, Glob. J. Math. Anal. 1 (2013), no. 1, 22-28.
[3]	S.-P. BAI, S.-H. WANG,ANDF. QI, Some Hermite-Hadamard type inequalities for n -time differentiable(α,m)-convex functions, J. Inequal. Appl. 2012, 2012: 267, 11 pages, available online at http://dx.doi.org/10.1186/1029-242X-2012-267. · Zbl 1279.26043
[4]	M. E. ¨OZDEMIR, M. TUNC¸ ,ANDM. G ¨URBUZ¨, Definitions of h -logaritmic, h -geometric and h multi convex functions and some inequalities related to them, available online at http://arxiv.org/abs/1211.2750.
[5]	Y. SHUANG, H.-P. YIN,ANDF. QI, Hermite-Hadamard type integral inequalities for geometricarithmetically s -convex functions, Analysis (Munich) 33 (2013), no. 2, 197-208, available online at http://dx.doi.org/10.1524/anly.2013.1192. · Zbl 1272.26024
[6]	S. VAROˇSANEC, On h -convexity, J. Math. Anal. Appl. 326 (2007), no. 1, 303-311, available online at http://dx.doi.org/10.1016/j.jmaa.2006.02.086. · Zbl 1111.26015
[7]	B.-Y. XI, R.-F. BAI,ANDF. QI, Hermite-Hadamard type inequalities for the m - and(α,m)geometrically convex functions, Aequationes Math. 184 (2012), no. 3, 261-269, available online at http://dx.doi.org/10.1007/s00010-011-0114-x. · Zbl 1264.26033
[8]	B.-Y. XI ANDF. QI, Hermite-Hadamard type inequalities for functions whose derivatives are of convexities, Nonlinear Funct. Anal. Appl. 18 (2013), no. 2, 163-176. · Zbl 1293.26016
[9]	B.-Y. XI ANDF. QI, Some Hermite-Hadamard type inequalities for differentiable convex functions and applications, Hacet. J. Math. Stat. 42 (2013), no. 3, 243-257. · Zbl 1292.26061
[10]	B.-Y. XI ANDF. QI, Some inequalities of Hermite-Hadamard type for h -convex functions, Adv. Inequal. Appl. 2 (2013), no. 1, 1-15.
[11]	B.-Y. XI, S.-H. WANG,ANDF. QI, Properties and inequalities for the h - and(h,m)-logarithmically convex functions, Creat. Math. Inform. (2013), no. 2, in press.
[12]	B.-Y. XI, Y. WANG,ANDF. QI, Some integral inequalities of Hermite-Hadamard type for extended (s,m)-convex functions, Transylv. J. Math. Mechanics 5 (2013), no. 1, 69-84. · Zbl 1292.26062
[13]	T.-Y. ZHANG, A.-P. JI,ANDF. QI, Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions, Proc. Jangjeon Math. Soc. 16 (2013), no. 3, 399-407. · Zbl 1296.26041
[14]	T.-Y. ZHANG, A.-P. JI,ANDF. QI, On integral inequalities of Hermite-Hadamard type for s geometrically convex functions, Abstr. Appl. Anal. 2012 (2012), Article ID 560586, 14 pages, available online at http://dx.doi.org/10.1155/2012/560586. · Zbl 1253.26047
[15]	T.-Y. ZHANG, A.-P. JI,ANDF. QI, Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means, Matematiche (Catania) 68 (2013), no. 1, 229-239, available online at http://dx.doi.org/10.4418/2013.68.1.17. · Zbl 1281.26024