On some new estimates for a gradient of a function in product domains and related results

Some new estimates related to the gradient and derivative of an analytic or harmonic function in Bergman type spaces of analytic and harmonic functions in selfconnected domains are presented. Some new results related with $L^p$-norm estimates of the $n$-th order derivative or with the gradient of an analytic or harmonic function in Bergman type spaces of analytic and harmonic functions or such type multifunctional spaces in the case of simply connected domain in the complex plane $\mathbb{C}$ are considered. New inequalities of a similar type are also presented, in which not only simple simply-connected domains, but their Cartesian product, are involved. Proofs of inequalities of a more complex type are derived either directly from simpler inequalities of the same type, or are completely based on some interesting estimates obtained in the course of their proofs. Such inequalities attracted the attention of various authors in recent years. The theorems given in the article can have various interesting applications in the theory of function of both one and several complex variables.