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