時統(tǒng)業(yè)，曾志紅

一致分?jǐn)?shù)階Lipschitz條件下的加權(quán)Ostrowski型不等式

時統(tǒng)業(yè)1，曾志紅2

(1．海軍指揮學(xué)院，江蘇南京 211800；2．廣東第二師范學(xué)院學(xué)報編輯部，廣東廣州 510303)

對滿足一致分?jǐn)?shù)階Lipschitz條件的函數(shù)，用普通數(shù)學(xué)分析的方法建立了涉及一致分?jǐn)?shù)階積分的加權(quán)Ostrowski型不等式及其伴隨不等式．

Ostrowski型不等式；權(quán)函數(shù)；一致分?jǐn)?shù)階積分；Lipschitz條件

1 預(yù)備知識

文獻(xiàn)[4]引入一致分?jǐn)?shù)階導(dǎo)數(shù)和一致分?jǐn)?shù)階積分的定義．

關(guān)于一致分?jǐn)?shù)階積分的Ostrowski型不等式還可參閱文獻(xiàn)[6]．本文將對滿足一致分?jǐn)?shù)階Lipschitz條件的函數(shù)給出涉及一致分?jǐn)?shù)階積分的加權(quán)Ostrowski型不等式和伴隨不等式．

2 主要結(jié)果

其中，

其中

故（5）式的左邊第一個不等式得證．

同理可證（5）式右邊第一個不等式．

證明：從推論1直接可得．

由注1、定理1、推論1和推論2，可得下面的推論3，它是定理4給出的一致分?jǐn)?shù)階可導(dǎo)函數(shù)Ostrowski型不等式的加強(qiáng)和加權(quán)推廣．

其中

其中

把（9）式代入（8）式，得

其中

其中

證明：推論4的直接結(jié)果．

[1] Ostrowski A. über Die Absolute Abweichung Einer Differentiebaren Funktion Von Ihren Integralmittelwert [J]. Comment Math Helv, 1938, 10(1): 226-227.

[2] Dragomir S S. On the Ostrowski Integral Inequality for Lipschitzian Mappings and Applications [J]. Compu Math Appl, 1999, 38(11/12): 33-37.

[3] Guessab A, Schmeisser G. Sharp Integral Inequalities of the Hermite-Hadamard Type [J]. J Approx Theory, 2002, 115(2): 260-288.

[4] Khalil R, Horani M A, Yousef A, et al. A New Definition of Fractional Derivative [J]. J Comput Appl Math, 2014, 264(1): 65-70.

[5] Anderson D R. Taylor’s Formula and Integral Inequalities for Conformable Fractional Derivatives [M] // Pardalos P M, Rassias T M. Contributions in Mathematics and Engineering. Cham, Switzerland: Springer, 2016: 25-43.

[6] Usta F, Budak H, Tun? T, et al. New Bounds for the Ostrowski-type Inequalities Via Conformable Fractional Calculus [J]. Ara J Math, 2018, 7: 317-328.

Weighted Ostrowski Type Inequalities for Conformable Fractional Integral Under Lipschitz Condition

SHI Tongye1, ZENG Zhihong2

(1. PLA Naval Command College, Nanjing, China 211800; 2. Journal Editorial Department, Guangdong University of Education, Guangzhou, China 510303)

For functions satisfying conformable fractional Lipschitz condition, weighted Ostrowski type inequalities and accompanying inequalities involving conformable fractional integral are established by using ordinary mathematical analysis.

Ostrowski Type Inequality; Weighted Function; Conformable Fractional Integral; Lipschitz Condition

O178

A

1674-3563(2021)04-0001-11

10.3875/j.issn.1674-3563.2021.04.001 本文的PDF文件可以從www.wzu.edu.cn/wzdxxb.htm獲得

2020-06-14

時統(tǒng)業(yè)(1963― )，男，河北張家口人，副教授，碩士，研究方向：數(shù)學(xué)不等式

(英文審校：黃璐)

(編輯：封毅)