Some Carleman-type inequalities in ( p , q ) $(p,q)$ -calculus

Abstract

Abstract In this paper, we construct ( p , q ) $\left (p,q\right )$ -type Jessen’s inequality using the properties of convex functions. On this basis, we generalize the classical Carleman integral-type inequality in ( p , q ) $\left (p,q\right )$ -calculus and obtain various forms of ( p , q ) $\left (p,q\right )$ -integral-type Carleman’s inequality by introducing various types of weight functions. Furthermore, we verify that under specific conditions, the ( p , q ) $\left (p,q\right )$ -integral Carleman inequality can reduce to the classical Carleman inequality in calculus.

Keywords

• ( p , q ) $(p,q)$ -Differential
• ( p , q ) $(p,q)$ -Integration
• ( p , q ) $(p,q)$ -Carleman inequality
• ( p , q ) $(p,q)$ -Jessen inequality