Generalizations of the Numerical Radius, Crawford Number and Numerical Index Functions in the Weighted Case

In this article, firstly, some simple and smoothness properties of the weighted numerical radius and the weighted Crawford number functions are investigated. Then, some generalization formulas for lower and upper bounds of the weighted numerical radius function are obtained. Later on, some evaluations for lower and upper bounds of the weighted numerical index are given. The obtained results are generalized some well-known famous results about the special weighted numerical radius and the special weighted Crawford number functions in the recently literature. Also, the different and useful results are provided to the literature.