导数绝对值为η-凸函数条件下的Hermite-Hadamard型不等式
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摘要
考虑由Hermite-Hadamard-Fejér不等式生成的差值和由推广的Hermite-Hadamard不等式生成的差值,通过建立涉及一阶导数和二阶导数的恒等式,分别在一阶导数的绝对值为η-凸函数和二阶导数的绝对值为η-凸函数的情况下,给出了这些差值的估计,而且通过例子说明这些结果与已有文献的结果各有强弱.
In this paper, the difference generated by the right-side of Hermite-Hadamard-Fejér inequality and the differences generated by the generalized Hermite-Hadamard inequality are considered. By establishing identities involving first derivative or second derivative, the estimates of these differences are given in the case where the absolute value of first derivative is η-convex function or the absolute value of the second derivative is η-convex function. Not only new results for estimating the difference of Hermite-Hadamard inequality are provided, but also by examples it shows that these results may be stronger or weaker than the results in existing literature.
In this paper, the difference generated by the right-side of Hermite-Hadamard-Fejér inequality and the differences generated by the generalized Hermite-Hadamard inequality are considered. By establishing identities involving first derivative or second derivative, the estimates of these differences are given in the case where the absolute value of first derivative is η-convex function or the absolute value of the second derivative is η-convex function. Not only new results for estimating the difference of Hermite-Hadamard inequality are provided, but also by examples it shows that these results may be stronger or weaker than the results in existing literature.
引文
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[6] Sarikaya M Z, Saglam A, Yildirim H. New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex[J]. Tamkang Journal of Mathematics, 2010, 5(3):353~359
[7]时统业.关于加权Hermite-Hadamard不等式[J].湖南理工学院学报(自然科学版), 2012, 25(1):8~11
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[10]匡继昌.常用不等式[M].第4版.济南:山东科学技术出版社, 2010:430~436
[11]张小明,褚玉明.解析不等式新论[M].哈尔滨:哈尔滨工业大学出版社, 2011:123~184
[12]黄金莹,赵宇.广义凸函数的Hadamard不等式[J].重庆师范大学学报(自然科学版), 2013, 30(4):1~5
[13]时统业,尹亚兰,吴涵. F凸函数的Hermite-Hadamard型不等式[J].重庆师范大学学报(自然科学版), 2016, 33(1):13~16
[14]孙文兵.分数次积分下关于s-凸函数的新Hermite-Hadamard型不等式[J].浙江大学学报(理学版), 2017, 44(5):531~537
[15]李玉娇,杜廷松.一类(α, m)-凸函数的Hadamard型不等式[J].上海大学学报(自然科学版), 2017, 23(4):583~589
[16]曾志红,时统业,钟建华,等.对称凸函数和弱对称凸函数的Hermite-Hadamard型不等式[J].西南师范大学学报(自然科学版), 2018, 43(4):24~30
[17] Gordji M E, Delavar M R, De La Sen M. Onφ-convex functions[J]. J. Math. Inequal. Journal of Mathematical Inequalities, 2016, 10(1):173~183
[18] Delavar M R, De La Sen M. Some Hermite-Hadamard-Fejér type integral inequalities for differentiableη-convex functions with applications[J]. Journal of Mathematics, 2017, 3:1~8
[19] Delavar M R, De La Sen M. On generalization of Fejér type inequalities[J]. Communications in Mathematics and Applications, 2017, 8(1):31~43
[20]时统业,曾志红,曹俊飞.η凸函数的Riemann-Liouville分数阶积分的Hermite-Hadamard型不等式[J].浙江大学学报(理学版),2018,45(5):549~554, 561.
[21]王良成.凸函数的Hadamard不等式的若干推广[J].数学的实践与认识, 2002, 32(6):1027~1030
[2] Dragomir S S, Agarwal R P. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula[J].Applied Mathematics Letters, 1998, 11(5):91~95
[3] Anastassiou G A. Handbook of analytic-computational methods in applied mathematics[M]. London/Boca Raton:Chapman and Hall/CRC, 2000:65~200
[4] Alomam M, Maslinadarus K U. Some inequalities of Hermite-Hadamard type for s-convex functions[J]. Acta Mathematica Scientia(English Series), 2011,31(4):1643~1652
[5] Sarikaya M Z. On new Hermite Hadamard Fejér type integral inequalities[J]. Studia Universitatis Babe-Bolyai Mathematica, 2012, 57(3):377~386
[6] Sarikaya M Z, Saglam A, Yildirim H. New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex[J]. Tamkang Journal of Mathematics, 2010, 5(3):353~359
[7]时统业.关于加权Hermite-Hadamard不等式[J].湖南理工学院学报(自然科学版), 2012, 25(1):8~11
[8] Gordji M E, Dragomir S S, Delavar M R. An inequality related toη-convex functions(II)[J]. International Journal of Nonlinear Analysis and Applications,2015, 6(2):27~33
[9] Dragomir S S, Pearce C E M. Selected topics on Hermite-Hadamard inequalities and applications[D]. Victoria:Victoria University, 2000
[10]匡继昌.常用不等式[M].第4版.济南:山东科学技术出版社, 2010:430~436
[11]张小明,褚玉明.解析不等式新论[M].哈尔滨:哈尔滨工业大学出版社, 2011:123~184
[12]黄金莹,赵宇.广义凸函数的Hadamard不等式[J].重庆师范大学学报(自然科学版), 2013, 30(4):1~5
[13]时统业,尹亚兰,吴涵. F凸函数的Hermite-Hadamard型不等式[J].重庆师范大学学报(自然科学版), 2016, 33(1):13~16
[14]孙文兵.分数次积分下关于s-凸函数的新Hermite-Hadamard型不等式[J].浙江大学学报(理学版), 2017, 44(5):531~537
[15]李玉娇,杜廷松.一类(α, m)-凸函数的Hadamard型不等式[J].上海大学学报(自然科学版), 2017, 23(4):583~589
[16]曾志红,时统业,钟建华,等.对称凸函数和弱对称凸函数的Hermite-Hadamard型不等式[J].西南师范大学学报(自然科学版), 2018, 43(4):24~30
[17] Gordji M E, Delavar M R, De La Sen M. Onφ-convex functions[J]. J. Math. Inequal. Journal of Mathematical Inequalities, 2016, 10(1):173~183
[18] Delavar M R, De La Sen M. Some Hermite-Hadamard-Fejér type integral inequalities for differentiableη-convex functions with applications[J]. Journal of Mathematics, 2017, 3:1~8
[19] Delavar M R, De La Sen M. On generalization of Fejér type inequalities[J]. Communications in Mathematics and Applications, 2017, 8(1):31~43
[20]时统业,曾志红,曹俊飞.η凸函数的Riemann-Liouville分数阶积分的Hermite-Hadamard型不等式[J].浙江大学学报(理学版),2018,45(5):549~554, 561.
[21]王良成.凸函数的Hadamard不等式的若干推广[J].数学的实践与认识, 2002, 32(6):1027~1030