Some Opial type inequalities in (<em>p, q</em>)-calculus

• Main Authors: Chunhong Li, Dandan Yang, Chuanzhi Bai
• Format: Article
• Language: English
• Published: AIMS Press 2020-07-01
• Series: AIMS Mathematics
• Subjects: (<i>p; q</i>)-derivative; q</i>)-integral; q</i>)-calculus; opial inequality; opial-type integral inequality
• Online Access: https://www.aimspress.com/article/10.3934/math.2020377/fulltext.html

In this paper, we establish 5 kinds of integral Opial-type inequalities in (<em>p, q</em>)-calculus by means of Hölder’s inequality, Cauchy inequality, an elementary inequality and some analysis technique. First, we investigated the Opial inequalities in (<em>p, q</em>)-calculus involving one function and its (<em>p, q</em>) derivative. Furthermore, Opial inequalities in (<em>p, q</em>)-calculus involving two functions and two functions with their (<em>p, q</em>) derivatives are given. Our results are (<em>p, q</em>)-generalizations of some known inequalities, such as Opial-type integral inequalities and (<em>p, q</em>)-Wirtinger inequality.