Certain Inequalities Related to the Generalized Numeric Range and Numeric Radius That Are Associated with Convex Functions

In this paper, we delve into the intricate connections between the numerical ranges of specific operators and their transformations using a convex function. Furthermore, we derive inequalities related to the numerical radius. These relationships and inequalities are built upon well-established principles of convexity, which are applicable to non-negative real numbers and operator inequalities. To be more precise, our investigation yields the following outcome: consider the operators A and B, both of which are positive and have spectra within the interval m,M, denoted as σA and σB. In addition, let us introduce two monotone continuous functions, namely, g and h, defined on the interval m,M. Let f be a positive, increasing, convex function possessing a supermultiplicative property, which means that for all real numbers t and s, we have fts≤ftfs. Under these specified conditions, we establish the following inequality: for all 0≤ν≤1, this outcome highlights the intricate relationship between the numerical range of the expression gνAXh1−ν when transformed by the convex function f and the norm of X. Importantly, this inequality holds true for a broad range of values of ν. Furthermore, we provide supportive examples to validate these results.